agit-prop, opinion manipulation and well-poisoning games Darwinist rhetorical tactics Logic and First Principles of right reason

Logic and First Principles, 7: The problem of fallacies vs credible warrant

Spread the love

When we deal with deeply polarised topics such as ID, we face the problem of well-grounded reasoning vs fallacies. A fallacy being a significantly persuasive but fundamentally misleading argument, often as an error of reasoning. (Cf. a classic collection here.) However, too often, fallacies are deliberately used by clever rhetors to mislead the unwary. Likewise we face the challenge of how much warrant is needed for an argument to be credible.

All of these are logical challenges.

Let us note IEP, as just linked:

A fallacy is a kind of error in reasoning. The list of fallacies below contains 224 names of the most common fallacies, and it provides brief explanations and examples of each of them. Fallacies should not be persuasive, but they often are. Fallacies may be created unintentionally, or they may be created intentionally in order to deceive other people. The vast majority of the commonly identified fallacies involve arguments, although some involve explanations, or definitions, or other products of reasoning. Sometimes the term “fallacy” is used even more broadly to indicate any false belief or cause of a false belief. The list below includes some fallacies of these sorts, but most are fallacies that involve kinds of errors made while arguing informally in natural language.
An informal fallacy is fallacious because of both its form and its content. The formal fallacies are fallacious only because of their logical form. For example, the Slippery Slope Fallacy has the following form: Step 1 often leads to step 2. Step 2 often leads to step 3. Step 3 often leads to … until we reach an obviously unacceptable step, so step 1 is not acceptable. That form occurs in both good arguments and fallacious arguments. The quality of an argument of this form depends crucially on the probabilities. Notice that the probabilities involve the argument’s content, not merely its form.

This focus on probabilistic aspects of informal fallacies brings out several aspects of the problem, for we often deal with empirical evidence and inductive reasoning rather than direct chained deductions. For deductive arguments, a chain is no stronger than the weak link, and if that link cannot be fixed, the whole argument fails to support the conclusion.

However, inductive arguments work on a different principle. Probability estimates, in a controversial context, will always be hotly contested. So, we must apply the rope principle: short, relatively weak individual fibres can be twisted together and then counter twisted as strands of a rope, giving a whole that is both long and strong.

Of chains, ropes and cumulative cases

For example, suppose that a given point has a 1% chance of being an error. Now, bring together ten mutually supportive points that sufficiently independently sustain the same conclusion. Odds that all ten are wrong in the same way are a lot lower. A simple calculation would be ([1 – 0.99]^10) ~10^-20. This is the basis of the classic observation that in the mouth of two or three independent witnesses, a word is established.

However, many will be inclined to set up a double-standard of warrant, an arbitrarily high one for conclusions they wish to reject vs a much softer one for those they are inclined to accept. Nowadays, this is often presented as “extraordinary claims require extraordinary evidence.”

In fact, any claim simply requires adequate evidence.

Any demand for more than this cometh of evil.

This is of course the fallacy of selective hyperskepticism, a bane of discussions on ID topics. (The strength of will to reject can reach the level of dismissing logical-mathematical demonstration, often by finding some excuse to studiously ignore and side step as if it were not on the table.)

Of course, an objection will be: you are overly credulous. That is a claim, one that requires adequate warrant. Where, in fact, if one disbelieves what one should (per adequate warrant), that is as a rule because one also believes what one should not (per, lack of adequate warrant), which serves as a controlling belief. Where, if falsity is made the standard for accepting or rejecting claims, then the truth cannot ever be accepted, as it will run counter to the false.

All of this is seriously compounded by the tendency in a relativistic age to reduce truth to opinion, thence to personalise and polarise, often by implying fairly serious ad hominems. This can then be compounded by the “he hit back first” tactic.

This also raises the issue of the so-called concern troll. That is one who claims to support side A, but will always be found undermining it without adequate warrant, often using the tactics just noted. Such a persona in fact is enabling B by undermining A. This is a notorious agit prop tactic that works because it exploits passive aggressive behaviour patterns.

The answer to all of this is to understand how arguments work and how they fail to work, recognising the possibility of error and of participants who are in error (or are in worse than error) then focussing the merits of the case.

So, as we proceed, let us bear in mind the significance of adequate warrant, and the problem of selective hyperskepticism. END

PS: As it is relevant to the discussion that emerged, let me lay out the path to intellectual decay of our civilisation, adapting Schaeffer:

Extending (and correcting) Schaeffer’s vision of the course of western thought, worldviews and culture, C1 – 21

H’mm: Geostrategic picture:

As Scuzzaman highlights the slippery slope ratchet, let me put up the Overton Window (in the context of a ratchet that is steadily cranking it leftward on the usual political spectrum) — where, fallacies are used to create a Plato’s cave shadow-show world in which decision-making becomes ever more irrational, out of contact with reality:

Likewise, here is a model of malinvestment-led, self-induced economic disaster due to foolishly tickling a dragon’s tail and pushing an economy into unsustainable territory, building on Hayek:

Let me add, a view of the alternative political dynamics and spectrum:

U/d b for clarity, nb Nil

PPS: Mobius strip cut 1/2 way vs 1/3 way across vid:

315 Replies to “Logic and First Principles, 7: The problem of fallacies vs credible warrant

  1. 1
    kairosfocus says:

    Logic and First Principles, 7: The problem of fallacies vs credible warrant

  2. 2
    Ed George says:

    KF, well presented OP, and difficult to disagree with. But the problem isn’t with selective hyperscepticism or adequate warrant. A person can claim that they have adequate warrant to declare a specific view to be highly credible. If someone disagrees they are often accused of hyperscepticism., which may or may not be true. Determining whether something has adequate warrant requires interpretation and judgement, both of which are affected by individual bias and even the perspective they approach the issue. Much like a necker cube. As such, there can be, and often is, honest disagreement without selective hyperscepticism.

    Correct me if I am wrong, but your bit about concern trolls was aimed at me. But I have made it clear that I like using the “devil’s advocate” approach to discussions. I find it to be a very powerful technique to focus and solidify various view-points. If a play the devil’s advocate on a specific point (abortion for example), and the counterargument fails, it is possible that the original view needs adjusting.

  3. 3
    ET says:

    I have met a Devil’s Advocate. I can say with 100% certainty that Ed George is not one.

  4. 4
    Ed George says:

    ET

    I have met a Devil’s Advocate. I can say with 100% certainty that Ed George is not one.

    I agree with ET. I have always found him to be an astute individual with remarkable intellectual abilities and superhuman insights. He is also the most fair, honest and civil person I have ever interacted with.

  5. 5
    ET says:

    See? The absolute extremes. The desperation to blame others. The patronizing, egotistical and condescending tone.

    Definitely an evo trying to undermine Uncommon Descent by pretending to be an ID proponent. But it’s OK to keep it around, as a pet. 😎

  6. 6
    Brother Brian says:

    Ed George@4, I saw what you did there. ????

  7. 7
    kairosfocus says:

    EG, . In the last thread you were objecting to logical-mathematical demonstrations, one of the strongest classes of warrant there is. That is actually beyond selective hyperskepticism. The practical test for that fallacy is in the OP: double-standards in warrant. A useful case in point is that many skeptics about moral truth expect those they argue with to acknowledge duties to truth and right reason, meanwhile they are trying to dismiss objective moral truth. You also seem to have an exaggerated view of your significance; my remarks are based on eleven years of experience at UD, where we have had to hammer out understanding of many rhetorical devices and combinations commonly used by objectors, up to and including attempts to hack, cyberstalk and stalk relatives at several degrees of remove on the ground. Concern trollery is one of the tactics we have met and it seems to be a growing phenomenon across our civilisation, so it is well worth noting. I should add, long ago, I cut my intellectual eyeteeth dealing with marxist agit prop operators and strategists, against the backdrop of a low-grade civil war in a country that had become a theatre of operations in the Cold War . . . hence some of the connexions I made when PaV raised some questions recently. (I now have a context for the 1976 Cubana aircraft bombing I had not made before.) More broadly, fallacies are a bane of serious discussion and are well worth a place of note in a series on foundations of reasoning — a series that is there because of the poor quality of reasoning out there on ID matters and on far broader matters of significant controversy. And in general, the rule is as Bob Marley put it, who de cap fit, let ‘im wear it. KF

  8. 8
    kairosfocus says:

    PS: It is worth noting Simon Greenleaf from vol 1 of his famous treatise on evidence:

    Evidence, in legal acceptation, includes all the means by which any alleged matter of fact, the truth of which is submitted to investigation, is established or disproved . . . None but mathematical truth is susceptible of that high degree of evidence, called demonstration, which excludes all possibility of error [–> Greenleaf wrote almost 100 years before Godel], and which, therefore, may reasonably be required in support of every mathematical deduction. [–> that is, his focus is on the logic of good support for in principle uncertain conclusions, i.e. in the modern sense, inductive logic and reasoning in real world, momentous contexts with potentially serious consequences.]

    Matters of fact are proved by moral evidence alone; by which is meant, not only that kind of evidence which is employed on subjects connected with moral conduct, but all the evidence which is not obtained either from intuition, or from demonstration. In the ordinary affairs of life, we do not require demonstrative evidence, because it is not consistent with the nature of the subject, and to insist upon it would be unreasonable and absurd. [–> the issue of warrant to moral certainty, beyond reasonable doubt; and the contrasted absurdity of selective hyperskepticism.]

    The most that can be affirmed of such things, is, that there is no reasonable doubt concerning them. [–> moral certainty standard, and this is for the proverbial man in the Clapham bus stop, not some clever determined advocate or skeptic motivated not to see or assent to what is warranted.]

    The true question, therefore, in trials of fact, is not whether it is possible that the testimony may be false, but, whether there is sufficient probability of its truth; that is, whether the facts are shown by competent and satisfactory evidence. Things established by competent and satisfactory evidence are said to be proved. [–> pistis enters; we might as well learn the underlying classical Greek word that addresses the three levers of persuasion, pathos- ethos- logos and its extension to address worldview level warranted faith-commitment and confident trust on good grounding, through the impact of the Judaeo-Christian tradition in C1 as was energised by the 500 key witnesses.]

    By competent evidence, is meant that which the very-nature of the thing to be proved requires, as the fit and appropriate proof in the particular case, such as the production of a writing, where its contents are the subject of inquiry. By satisfactory evidence, which is sometimes called sufficient evidence, is intended that amount of proof, which ordinarily satisfies an unprejudiced mind [–> in British usage, the man in the Clapham bus stop], beyond reasonable doubt.

    The circumstances which will amount to this degree of proof can never be previously defined; the only legal [–> and responsible] test of which they are susceptible, is their sufficiency to satisfy the mind and conscience of a common man; and so to convince him, that he would venture to act upon that conviction, in matters of the highest concern and importance to his own interest. [= definition of moral certainty as a balanced unprejudiced judgement beyond reasonable, responsible doubt. Obviously, i/l/o wider concerns, while scientific facts as actually observed may meet this standard, scientific explanatory frameworks such as hypotheses, models, laws and theories cannot as they are necessarily provisional and in many cases have had to be materially modified, substantially re-interpreted to the point of implied modification, or outright replaced; so a modicum of prudent caution is warranted in such contexts — explanatory frameworks are empirically reliable so far on various tests, not utterly certain. ] [A Treatise on Evidence, Vol I, 11th edn. (Boston: Little, Brown, 1888) ch 1., sections 1 and 2. Shorter paragraphs added. (NB: Greenleaf was a founder of the modern Harvard Law School and is regarded as a founding father of the modern Anglophone school of thought on evidence, in large part on the strength of this classic work.)]

  9. 9
    kairosfocus says:

    F/N: looks like the new WP editor forces every word in a title to begin with a capital letter, even when it was not typed in that way. And as for inserting diagrams . . . KF

    PS: Got a diagram to load.

  10. 10
    kairosfocus says:

    F/N: See 2014 article on selective hyperskepticism here: https://uncommondescent.com/atheism/darwinian-debating-devices-12-selective-hyperskepticism-closed-mindedness-and-the-saganian-slogan-extraordinary-claims-require-extraordinary-evidence/ A very clear case study from elsewhere is the elevatorgate sexual harassment issue that popped up at an atheist convention several years ago. Of course, another fallacy is tainting and conviction by accusation — which goes all the way back to Joseph’s case at the hands of Potiphar’s wife. KF

  11. 11
    hazel says:

    Hmmm. I don’t recall ED, nor me, objecting to a “logical-mathematical demonstration, one of the strongest classes of warrant there is.” For the record.

  12. 12
    Ed George says:

    Hazel

    Hmmm. I don’t recall ED, nor me, objecting to a “logical-mathematical demonstration, one of the strongest classes of warrant there is.” For the record.

    I agree. Nobody is suggesting that quantity is not inherent in the universe. One sun, nine planets (or eight), dozens of moons, etc. But beyond that is where I have a problem. We use calculus to model orbits and acceleration. A tool invented by man. Just because we can use a man-made tool to model what we see does not mean that this man-made tool is inherent in the universe. And to propose that mathematics was used by the designer to create the universe is beyond preposterous.

  13. 13
    hazel says:

    Hi Ed. I agree with most of what you say: “Just because we can use a man-made tool to model what we see does not mean that this man-made tool is inherent in the universe” is a good line.

    However, I disagree when you write, “ And to propose that mathematics was used by the designer to create the universe is beyond preposterous.”

    I don’t think it is preposterous to believe that an intelligent designer of some cosmic kind conceived first of math and then used math when creating the universe. That’s a philosophical position with a long history. However, it is not the only philosophical position with a long history, and it certainly isn’t validated by any “logical-mathematical demonstration”, to use kf’s phrase.

    So when I wrote, “I don’t recall ED, nor me, objecting to a “logical-mathematical demonstration,” what I meant to imply was a distinction between disagreeing with kf about a philosophical position, which I believe we do, and disagreeing with him about the logical validity built-in to mathematical systems, which I believe we don’t.

    That’s the distinction I want to make.

  14. 14
    ET says:

    Ed George:

    We use calculus to model orbits and acceleration. A tool invented by man.

    That is your opinion, Ed. Calculus is a tool discovered by man. Srinivasa Ramanujan is a great argument for discovery

    And to propose that mathematics was used by the designer to create the universe is beyond preposterous.

    Such a childish “argument”, Ed. Do you have anything coherent to back that up?

  15. 15
    Ed George says:

    Hazel@12, I can’t really disagree with you. You know far more about mathematics than I ever will. But I am interested in your view on something as simple as the mathematics involved in orbital mechanics. Do you think that they are inherent in the universe, or do you think we invented the math required to describe it?

  16. 16
    ET says:

    hazel:

    I don’t think it is preposterous to believe that an intelligent designer of some cosmic kind conceived first of math and then used math when creating the universe. That’s a philosophical position with a long history.

    Except it isn’t a philosophical position. See? I can just dismiss what you say because you just say it. Your say-so isn’t evidence.

    I don’t see anything that prevents us from determining that mathematics permeates the universe. And that when we do encounter intelligent ET’s their mathematics will be ours, with advancements (perhaps).

  17. 17
    math guy says:

    In response to E.G. @ 11 who writes
    “And to propose that mathematics was used by the designer to create the universe is beyond preposterous.”

    I give you quotes by Galileo and Kepler, respectively
    “The book of nature is written in the language of mathematics.”
    and
    “Geometry, which before the origin of things was coeternal with the divine mind and is God himself (for what could there be in God which would not be God himself?), supplied God with patterns for the creation of the world, and passed over to Man along with the image of God; and was not in fact taken in through the eyes. ”

    Now I will conclude with a quote attributed to Lincoln, Twain, and several others
    “Better to remain silent and be thought a fool than to speak out and remove all doubt.”

  18. 18
    ET says:

    “If you accept the idea that both space itself, and all the stuff in space, have no properties at all except mathematical properties,” then the idea that everything is mathematical “starts to sound a little bit less insane,” Tegmark said in a talk given Jan. 15 here at The Bell House. The talk was based on his book “Our Mathematical Universe: My Quest for the Ultimate Nature of Reality” (Knopf, 2014)

  19. 19
    Ed George says:

    Well, I just got a phone call letting me know that I am a first time grandfather. So, all I can say is that all of the petty bickering I have seen over the short time I have been here means absolutely nothing.

  20. 20
    ET says:

    Ed:

    So, all I can say is that all of the petty bickering I have seen over the short time I have been here means absolutely nothing.

    The petty bickering has you as the common denominator and instigator, Ed.

  21. 21
    hazel says:

    That is wonderful, Ed. Your life is changed. You will enjoy, and be inspired about the wonders of life, by your grandchild in ways that will be different, and bigger, than when you had you own children. Congratulations!

    But I’m still going to respond somewhat extensively to your question at 15. 🙂

  22. 22
    Ed George says:

    ET

    And that when we do encounter intelligent ET’s…

    Do you think that there will ever be an intelligent ET?

  23. 23
    Ed George says:

    Hazel@21, thank you for the good wishes. I appreciate it. I would love to say that I am ready for this, but ….

  24. 24
    ET says:

    Ed George:

    Do you think that there will ever be an intelligent ET?

    When compared to you, Ed, there are, have been and will continue to be, many. 😛

  25. 25
    ET says:

    Ed:

    Do you think that they are inherent in the universe, or do you think we invented the math required to describe it?

    It will be interesting to see if whatever hazel responds with is supported by something other than what hazel says.

  26. 26
    hazel says:

    Ed, you ask, “Do you think that they (for instance, laws describing orbital mechanics) are inherent in the universe, or do you think we invented the math required to describe it?”

    This sentence gets to the heart of the issue, and there are several components.

    Let me break this into two parts.

    First, I think that there is general agreement that the particular words and symbols in which we express our mathematics are human creations. However, once certain ideas are formulated, logical deductive trains lead us to discoveries of mathematical facts, including some/many that are not at all obvious. So within a mathematical system, there is a feedback loop, so to speak, of invention and discovery, with insightful inventions leading to large sets of discovered facts, and the further invention of new symbols and concepts leading to further discoveries.

    (I used to spend a class period at the start of second semester of my pre-calculus class giving a lecture on the development of the number system from counting numbers up to complex numbers: It’s a fascinating subject.)

    The second part of your question is the philosophical one about math being able to describe the physical world – as you say, using math to model the world – as opposed to just developing logical mathematical systems.

    Things in the physical world behave in orderly ways which can be accurately modeled by mathematics. This is an empirically true fact. In my opinion, this doesn’t mean that somehow the components of our mathematics themselves are out there in the real world.

    Let me illustrate with a somewhat lengthy example.

    I used to start a chapter on differentials in calculus by asking, Why doesn’t the moon fall down?” I then showed that if we consider the moon as a point on a circle with the earth, we can represent two forces on the moon by vectors: one pointing down to represent the force of gravity and one tangent to the circle representing the moon’s movement. By looking at the hypotenuse created by the the vectors, we can see that the moon will move at an angle.

    Now since the moment the moon moves, the directions of the two vectors change, any one vector diagram is only an approximation of what the moon will do at any moment. However, if we shrink the vectors down to infinitesimal size, then the resulting triangle represents the instantaneous direction of motion.

    And then, if you learn how to do the calculus, I would point out, mathematicians can show that movement describes not a circle, but an ellipse. This process, elaborated extensively, is how the mathematical models of celestial orbits were created, and one of the main vehicles for Newton’s development of calculus.

    Pardon the long story, but here is now the big question. Are there really little infinitesimal vectors in the world? No, there are not. Gravity is acting, the moon is moving with inertia: those are true facts, but the vectors are a model, not something “inherent” in the world.

    Let me emphasize: it is absolutely true, and a mystery which arouses philosophical questions, that the world is such that mathematical models such as those using vectors work. Various people have different beliefs about the philosophy about why this is true. In my opinion there is no sure-fire way to determine which philosophical views are correct. We all, to use kf’s metaphor (which I like), wind various strands of thought into a rope of belief, but the ropes are not the same as logical deductive chains. Furthermore, people have different attachments to their beliefs (here the rope analogy breaks down) – but that’s a different topic.

  27. 27
    ET says:

    hazel:

    First, I think that there is general agreement that the particular words and symbols in which we express our mathematics are human creations.

    I strongly disagree.

    I will emphasize that this is not a philosophical question. It is a question that can be scientifically explored.

  28. 28
    kairosfocus says:

    EG, the evidence of your attempt to dismiss mathematical-logical demonstration as imposition of opinion is there. That, you cannot wave away rhetorically. KF

    PS: On the assumption of truth regarding declared personal circumstances, I wish you well on that front.

  29. 29
    kairosfocus says:

    H, you would be well advised to start from the principle of distinct identity and its consequences which give us a large core of structure and quantity. FYI, too, vector quantities and fields are indeed embedded in the world in many relevant ways. Where, field implies infinitesimal components, so yes, calculus is embedded in the world as rates, flows, gradients and thus also vectors. As just one example a magnetic field is a vector field. There is much more. KF

  30. 30
    hazel says:

    OK, ET: where did the term imaginary number and the symbol i to represent it come from? I claim a human being (or set of human beings) created it. What do you think?

  31. 31
    hazel says:

    kf writes, “PS: On the assumption of truth regarding declared personal circumstances.”

    Why in the world would you start that sentence “on the presumption of truth …”? Did you seriously think there is a chance that Ed would have made up a story about becoming a new grandfather?

  32. 32
    ET says:

    I claim that all information, mathematics included, comes from The Source. That is from where Srinivasa Ramanujan discovered his many formulae.

    What is your evidence that humans invented mathematics?

  33. 33
    hazel says:

    Could the person who first used i used a different word and symbol, like maybe “pretend” number and p?

  34. 34
    Ed George says:

    KF

    PS: On the assumption of truth regarding declared personal circumstances, I wish you well on that front.

    On the assumption of truth???? You are a really sad little man. I think this is my last attempt at a discussion with you.

  35. 35
    Ed George says:

    Hazel@26, I think we are saying the same thing, although you are saying it far more eloquently than I have.

    I think the difference of opinion I have with KF, WJM and maybe ET, is simply how we look at reality. They see an organized system and think that the system was designed to obey specific mathematical constructs. I see the system as something that we, because of the abilities we have been given, can describe in terms we have defined mathematically. It is really the chicken and egg problem.

    But, again, thank you for your best wishes. I have just been forwarded a picture of my new grandson. I think he is the most beautiful baby ever born. But, I might be a little biased. 🙂

  36. 36
    Brother Brian says:

    KF “PS: On the assumption of truth regarding declared personal circumstances, I wish you well on that front.“
    I have to agree with Hazel. Your comment is insensetive and beyond the pale. An apology might be in order. Just a suggestion.

  37. 37
    kairosfocus says:

    Folks, this is not a verandah coffee break. I noted the possibility of what intel agencies term legends in order to underscore the realities of the sort of polarisation we have to routinely deal with, especially when anonymous and often adverse commenters are involved. Indeed, several years ago we faced an elaborate hoax by someone who was given a guest post and had put up a false front (changing his sex, using a handle that properly belongs to a college professor of Mathematics, and much more), in a case that was quite close to the classic definition of concern trolling. I add, that case was in part broken when the persona failed to recognise a mathematical property that was pivotal to a heuristic model developed to quantify a threshold metric for functionally specific complex organisation and/or associated information. He also used part of the penumbra of attack sites to boast of his hoax. That context must not be forgotten. The issue is not paranoia, but whether we are sufficiently aware of what sort of agit prop operations may be going on and the dangers involved. KF

    PS: The case also highlights the question of degree of warrant required for acknowledging credible truth. That is, it brings to bear selective hyperskepticism and Clifford-Sagan evidentialism.

    PPS: Definition, FYI:

    Legend

    A spy’s claimed background or biography, usually supported by documents and memorized details

  38. 38
    kairosfocus says:

    H, we know the history of invention of i in the context of cubic polynomials. We also know the history of recognising rotation operators and vector numbers. You were part of an exchange in recent weeks here on precisely that topic. Going further, you are again failing to acknowledge the force of the distinction between the world-embedded substance of structure and quantity and the culturally influenced study of it. I have already pointed out how physical length obtains in a spatially extended world (as opposed to how in any world we may see a mathematical space), which is antecedent to our culturally influenced creation of a standard unit for length such as the metre or the yard (or the fathom, ell or cubit, etc). Likewise, I have pointed out the difference between numbers in themselves and culturally influenced numerals and representations. Pi can be expressed in many ways, the decimal place value notation being only one. The fact that in a planar space corresponding to Euclid’s framework circles and diameters are such that circumference length to diameter will stand in a specific numerical ratio is a case of world-embedded structure and quantity which we discover rather than invent. Then, we find that pi is very relevant in a world of approximately round objects, e.g. ponder how round gearing attains to evenly spaced, uniform, properly meshing teeth. (Nice slideshow: http://ocw.uc3m.es/ingenieria-.....es-1/gears Also notice for example the significance of pi in various tabulated formulae: https://www.engineersedge.com/gear_formula.htm ) That applies to the power train of cars and other vehicles, it applies to the watches and clocks we use to tell time, it applies even to fishing reels. The balance I am pointing to is critical, and the consistent refusal to address both aspects in a balanced way as was just evident again, is a tell relevant to the focus of this thread. And as we see also the tendency to seize upon any pretext to trumpet taking offence (and to thereby suggest that the offended party is therefore in the right) I will not name names on fallacies involved. But, they clearly are there. KF

  39. 39
    kairosfocus says:

    ET,

    You are quite right to highlight Ramanujan as an example of discoveries of world-embedded intelligible rational principles of structure and quantity. I here clip Wikipedia as a testimony against known ideological agenda:

    Srinivasa Ramanujan
    From Wikipedia, the free encyclopedia
    Jump to navigation
    Jump to search
    “Ramanujan” redirects here. For other uses, see Ramanujan (disambiguation).
    In this Indian name, the name Srinivasa is a patronymic, not a family name, and the person should be referred to by the given name, Ramanujan.
    Srinivasa Ramanujan

    FRS
    Srinivasa Ramanujan – OPC – 1.jpg
    Born 22 December 1887
    Erode, Madras Presidency, British India (present-day Tamil Nadu, India)
    Died 26 April 1920 (aged 32)
    Kumbakonam, Madras Presidency, British India (present-day Tamil Nadu, India)
    Residence

    Kumbakonam, Madras Presidency, British India (present-day Tamil Nadu, India)
    Madras, Madras Presidency, British India (present-day Chennai, Tamil Nadu, India)
    London, England, United Kingdom of Great Britain and Ireland (present-day United Kingdom)

    Nationality Indian
    Education

    Government Arts College (no degree)
    Pachaiyappa’s College (no degree)
    Trinity College, Cambridge (BSc, 1916)

    Known for

    Landau–Ramanujan constant
    Mock theta functions
    Ramanujan conjecture
    Ramanujan prime
    Ramanujan–Soldner constant
    Ramanujan theta function
    Ramanujan’s sum
    Rogers–Ramanujan identities
    Ramanujan’s master theorem
    Ramanujan–Sato series

    Awards Fellow of the Royal Society
    Scientific career
    Fields Mathematics
    Institutions Trinity College, Cambridge
    Thesis Highly Composite Numbers (1916)
    Academic advisors

    G. H. Hardy
    J. E. Littlewood

    Influences G. S. Carr
    Influenced G. H. Hardy
    Signature
    Srinivasa Ramanujan signature

    Srinivasa Ramanujan FRS (/??ri?ni?v??s? r???m??n?d??n/;[1] About this soundlisten (help·info); 22 December 1887 – 26 April 1920)[2] was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems considered to be unsolvable. Ramanujan initially developed his own mathematical research in isolation: “He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered”.[3] Seeking mathematicians who could better understand his work, in 1913 he began a postal partnership with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognizing the extraordinary work sent to him as samples, Hardy arranged travel for Ramanujan to Cambridge. In his notes, Ramanujan had produced groundbreaking new theorems, including some that Hardy stated had “defeated [him and his colleagues] completely”, in addition to rediscovering recently proven but highly advanced results.

    During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).[4] Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research.[5] Nearly all his claims have now been proven correct.[6] The Ramanujan Journal, a peer-reviewed scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan,[7] and his notebooks—containing summaries of his published and unpublished results—have been analyzed and studied for decades since his death as a source of new mathematical ideas. As late as 2011 and again in 2012, researchers continued to discover that mere comments in his writings about “simple properties” and “similar outputs” for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death . . .

    It often takes genius to discover many things and here we have a case of isolation leading to independent witness. Where, many of the results in question are in theory of numbers. Numbers being demonstrably embedded in the structure of any world, as they emerge from the import of distinct identity.

    This is indeed an empirical case in point.

    KF

  40. 40
    hazel says:

    kf writes, you are again failing to acknowledge the force of the distinction between the world-embedded substance of structure and quantity and the culturally influenced study of it.”

    No kf, I specifically addressed this distinction in post 26.

  41. 41
    hazel says:

    kf writes, “Folks, this is not a verandah coffee break.”

    Kf, this is a verandah coffee break. It’s just a little internet forum populated by a couple dozen regular or semi irregular participants.

  42. 42
    kairosfocus says:

    H, you have again chosen to ignore highly relevant context; here, of actual hoaxing and of a linked long-term penumbra of abusive objector sites . I must note, that in such a context, we must take due note of patterns of behaviour and where they may well point. Not what we desire, what we must face in a world where, decades ago now many major objectors to design theory chose not to primarily engage on substance but on agit prop operations, administrative lockout, expulsion, tainting and lawfare. KF

  43. 43
    hazel says:

    No, kf, I am not “ignoring” anything. You are paranoid, and attaching vastly more significance to the discussions here then they deserve. There is no excuse for your unkind remark to Ed, whoever he may be. ‘Nuf said.

  44. 44
    ET says:

    hazel, I will ask again-

    What is the evidence that demonstrates humans invented mathematics (as opposed to discovering it)?

  45. 45
    ET says:

    hazel:

    You are paranoid, and attaching vastly more significance to the discussions here then they deserve

    Wow. Why are you even here, hazel? Clearly you must think the discussion is important as you and Ed have both put your time into it- no effort but time

  46. 46
    kairosfocus says:

    H, what you indeed acknowledge in part on the one hand you have then repeatedly refused to recognise on critical points where it becomes relevant and in fact decisive. That there is demonstrable substantial structure and quantity embedded in any possible world is already decisive, and especially so when that substance ranges from the naturals to the reals and transfinites, with direct relevance to space. So, for concrete example, that i was proposed as a controversial solution to roots of polynomials in general, strictly does not overthrow the point that we must distinguish the world-embedded, demonstrable substance of structure and quantity from our culturally influenced study of it. And yet, above you clearly posed i as though it were a counter example to the basic point. In addition, in earlier threads you tried to set aside the onward discovery that i introduced vectors and rotations so has a very natural meaning that in fact historically broke the back of the controversy. The vector approach extends to quaternions (and the linked ijk basis vector system) and to octonions which may yet turn out to be a key tool for understanding high energy physics and the infamous zoo of particles. So, no, the issue is settled and there should be no remaining controversy on understanding maths as the (study of the) logic of structure and quantity. Yes, what it means for abstract entities to have credible reality is an onward debate, but it does not help to see that when logic of being and the linked concepts, possible vs impossible beings and contingent vs necessary beings were raised, you were found raising dismissive objections. That numbers such as 2 are necessary beings, framework for any possible world is material to the significance of mathematics in the actual world. Likewise, as mathematical systems build abstract logic-model worlds, such are possible worlds and necessary beings discovered in such explorations will be applicable to any world. So, which is it: is Math as defined, why or why not: _____ ? Are the counting numbers and associated sets necessary beings, why/why not? ______ Is the vector-rotation approach to the complex numbers anchored in objective reality or a figment of cultural imagination like Shapespeare’s Hamlet? _______ Does or does not mathematical praxis set out to erect abstract logic-model worlds (such as classically, Euclidean Geometry and today axiomatised systems and models that may or may not have real world significance)? ______ In such, do we or do we not discover necessary beings? _______ Does this then raise implications for understanding Mathematics in our world? ____ KF

  47. 47
    kairosfocus says:

    H, I find it out of order for you to infer that it is — in the context as explained — “unkind” to wish someone well, noting the proviso of truth. We did not set the context of hoaxing etc, the objector side did. That has consequences and those consequences must be recognised. That wider pattern of abusive and destructive behaviour by a known circle of objectors and activists (which, note, has reached as far as falsity and abuse embedded into educational materials, court decisions and policy, corrupting science also . . . ) is why UD is not a verandah coffee break. KF

  48. 48
    hazel says:

    ET, let’s take this one step at a time, with an example.

    A group of people (Tartaglia, Cardano, Descartes) in the 1500’s adnd 1600’s decided to use the word imaginary and the letter i to talk about and study the idea of the square root of negative 1.

    Do you agree that that happened?

  49. 49
    hazel says:

    kf writes, “H, I find it out of order for you to infer that it is — in the context as explained — “unkind” to wish someone well, noting the proviso of truth.”

    There is nothing “out of order” to point out that the implicit judgment that Ed might have been lying was unkind. That you somehow justify that is, as you are fond of saying, telling.

  50. 50
    hazel says:

    kf writes, “Yes, what it means for abstract entities to have credible reality is an onward debate,”

    Yep, that seems to be what we are debating.

  51. 51
    ET says:

    hazel:

    ET, let’s take this one step at a time, with an example.

    Already tried that

    A group of people (Tartaglia, Cardano, Descartes) in the 1500’s adnd 1600’s decided to use the word imaginary and the letter i to talk about and study the idea of the square root of negative 1.

    Good for them. All the information they used to discuss it already existed. Everything they discussed already existed.

  52. 52
    kairosfocus says:

    H, the substantial matter has just been illustrated again. By again refusing to acknowledge the significance of the vector-rotation perspective, a material factor is omitted. Do you not see how that then fits a pattern with very unfortunate associations that on track record you would take as offensive if I were to even hint at them? KF

    PS: I simply note that this issue of consequences of selective focus feeding argument patterns that omit material factors extends to side-issues above also. EG, unfortunately, has also fit a pattern through many weeks of commentary.

  53. 53
    hazel says:

    I do believe that this looks like these could be interminable conversations! 😉 Perhaps I’ll do us a favor and quit, and then in a few weeks kf can post “chirp, chirp, chirp” because I wasn’t willing to go around and around and around …

  54. 54
    ET says:

    hazel:

    I do believe that this looks like these could be interminable conversations!

    Right now it may appear that way but that is only because you refuse to support anything that you have said.

  55. 55
    Ed George says:

    Hazel@26, thank your for the response. I had to read it several times to get my head around it. But it makes perfect sense.

  56. 56
    hazel says:

    Thanks, Ed. The example was a little hard to follow without a picture, and I don’t think one can insert pictures in comments. But the main point was to illustrate how the math accurately modeled the world without the elements of the math actually being in the world. The gravitational force is modeled by a vector, but gravitational force isn’t itself a vector. I think that is the distinction you and I have been trying to make, and are in agreement about.

  57. 57
    kairosfocus says:

    H, any force is a vector, having magnitude and direction, [and it will act at a point of application — a push or pull], i.e. quantity and spatial structure. The gravitational force field is a vector field, and its strength and direction at any given point are readily measured, in N/kg. As a first 6th form lab, I used to make my students measure force on a 1 kg mass, then give them earth radius, allowing calculation of Earth’s mass, approx 6 * 10^24 kg. We of course symbolically represent by flow field lines, or locally by arrows we draw, but those are only symbols. If you wish to say that 9.8 N/kg is not somehow manifest in a way that we can put a camera on space and it magically lights up with the value somehow floating there, that is a different matter, but the quantitative structure, the gravity field is certainly physically present, just ask NASA about why they build rockets. KF

  58. 58
    hazel says:

    kf, I know all that. I know that the world behaves the way it does. Gravitational fields exist. That is not the issue. The issue is that the mathematical symbols we use to model the field are not the same as the field. The fields exist, but the idea that they can be represented by vectors, both visually and numerically, is a mathematical model. The map is not the territory. The abstraction is not the reality. That is the philosophical point.

  59. 59
    ET says:

    hazel, you ignore all of the evidence to the contrary and prattle on without even addressing it.

    Sad, really.

  60. 60
    ET says:

    hazel:

    But the main point was to illustrate how the math accurately modeled the world without the elements of the math actually being in the world

    And you FAILed. Nicely done.

    Next

  61. 61
    kairosfocus says:

    H, we are yet again at the issue of the substance of structure and quantity manifested in the world vs the study of same using cultural traditions, symbols, syntax, semantics, language. The world-embedded substance is real, that needs to be faced; starting with how distinct identity necessitates numbers thence the ladder, N –> Z –> Q –> R –> C etc. For simple example the fingers of my right hand stand in 1:1 correspondence with the toes of my left foot, and again with the chain of symbols used in counting, 1 – 2 -3-4-5. They have the same cardinality, reflecting THE five-set in the von Neumann order type sequence {0,1,2,3,4} –> 5. Fiveness is real and will appear in any possible world that holds distinct identity. How we may symbolise or manipulate it using symbols etc does not invent that reality, it responds to it. I must also point out that the gravity field is precisely a classic example of a vector field, one that takes a direction and size at any particular point in it. (BTW, the old trick of using iron filings to illustrate lines of force in a magnetic field is only illustrative of tendencies, the field itself is smoothly present between the lines.) Vectors are commonplace in the world, the mathematical apparatus allows us to study them quantitatively and systematically. KF

    PS: I trust that by now the astute onlooker will appreciate the depth of polarisation in our civilisation, and will therefore see that if actual mathematical-logical demonstration is patently not enough to secure mutuality across the rift, the rift is likely to be fatal. Of course, once grand worldview agendas and commitments are locked into this sort of polarisation, the power of responsible, rational freedom governed by duty to truth, right reason etc has been undercut. That undercutting did not come from despised design thinkers, theists, creationists or the like. It has come from the evolutionary materialistic establishment and those who travel with it. No wonder we see people insisting that there are 112 “genders”or that the unborn child is little more than a parasitical blob of tissue that if unwanted can be excised at will, etc and hoping to impose such by force of agit prop agitation and lawfare, as just one or two manifestations of how far wrong our civilisation has gone. And no, I am not merely projecting from a few online or in person encounters, I am pointing to widespread problems in our civilisation that frankly point right over the cliff.

  62. 62
    hazel says:

    The model of the counting numbers works for things that appear to us as having distinct identity: rocks, fingers, etc. The model doesn’t work for other things:how do you count clouds, for instance?

    However, the hyperbolic paranoia expressed in kf’s second paragraph, which is totally out of proportion to the subject at hand, is why we can’t have a constructive conversation: to kf it’s an inevitable slippery slope from acknowledging that there other legitimate views other than the Platonic view to mass murder and civilization going over a cliff.

    A bit bizarre, in my opinion.

  63. 63
    hazel says:

    More to the point, kf says “Fiveness is real .”

    The question is in what way is “fiveness” real.

    The position I am describing, as an alternative to the Platonic one, is that “fiveness” is an abstract concept that exists in our mind, and is instantiated in the physical world with a word and symbols, But “fiveness” as an abstraction does not exist outside of our mind. The world has things in in it, and we can describe some groups of those things with our concept of five, but “fiveness” resides as a mental concept in our minds: “fiveness” does not reside in the physical world itself. Fiveness is about the world, not in the world,

    This is another way of looking at this alternative view.

  64. 64
    ET says:

    exerpt from “Is God a Mathematician?”

    As the British physicist James Jeans (1877-1946) once put it: The universe appears to have been designed by a pure mathematician.” Mathematics appears to be almost too effective in describing and explaining not only the cosmos at large, but even some of the most chaotic of human enterprises.

    Again, there is only one good reason that is so.

    Looks like a duck. Quacks like a duck. Waddles and paddles like a duck. Has the DNA of a duck. Likes to land an take off from lakes and ponds like a duck.

    But is “duckness” real? Is that the argument now? Really?

  65. 65
    ET says:

    Now to revisit what Ed George posted above:

    And to propose that mathematics was used by the designer to create the universe is beyond preposterous.

    “The universe appears to have been designed by a pure mathematician.”– physicist James Jean

    😀 😀 😀 😀 😀

  66. 66
    Ed George says:

    Hazel

    A bit bizarre, in my opinion.

    A lot bizarre, in mine.

  67. 67
    ET says:

    hazel:

    However, the hyperbolic paranoia expressed in kf’s second paragraph, which is totally out of proportion to the subject at hand, is why we can’t have a constructive conversation:

    Again we can definitely agree to disagree. What he says is out of frustration from dealing with people who refuse to deal with the evidence presented.

  68. 68
    Ed George says:

    ET

    Again we can definitely agree to disagree. What he says is out of frustration from dealing with people who refuse to deal with the evidence presented.

    No, he is frustrated by people who he has not been able to convince of the truth of his arguments. Under similar circumstances most of us would conclude that we have failed to present our arguments as well as we would like. KF, however, concludes that those he has not been able to convince are agit prop marxists from the penumbra of hate sites….

    And, sadly, he is being enabled by those who refuse to call him on his paranoid delusions. I take no pleasure in pointing this out, but I honestly don’t see how any serious discussion can be had given the level of paranoia that is evident here.

  69. 69
    ET says:

    Ed George- There is no convincing the willfully ignorant. And yes, you have totally failed to present any arguments at all. You have managed to ignore and avoid them, though.

    Look, Ed, you are the problem. Any delusions are all yours. The paranoia is also yours. You cannot carry on a discussion. You think your handwaving is somehow coherent.

    I am still waiting for the evidence that mathematics is a human invention. And waiting…

  70. 70
    ET says:

    And Ed, you “argue” like a child:

    And to propose that mathematics was used by the designer to create the universe is beyond preposterous.

  71. 71
    kairosfocus says:

    Folks,

    If the import of the onward discussion were not so inadvertently sadly revealing, I would call myself amused.

    First, the basic facts still stand on their own merits. For example, vectors are real, are physically manifested, are even commonplace, e.g. we live in a gravity field. Likewise, pilots have to calculate effects of wind speed and direction in navigating, take-off or landing. Even the evident rotation of a hurricane is an effect of multiple vector effects that give rise to Coriolis forces. Similarly, once a distinctly identifiable world exists, distinct identity exists, thus duality, unity and nullity thence the counting numbers, thereafter integers, rationals, reals, the complex numbers (which brings in vectors), etc. Where, it cannot properly be an objection to the necessity of discrete counting numbers, that there are also continua, there may be superpositions, etc. I will go so far as risking further taking umbrage by naming the first fallacy involved, the red herring.

    Next, resistance to mathematical-logical demonstration compounded by the claim that one is not persuaded simply shows the destructive effects of radical relativism and subjectivism in our civilisation, as well as the underlying agendas that have pushed such. Evolutionary materialistic scientism by description, naturalism (in its various forms) by name, along with fellow travellers.

    I am not frustrated or surprised to see such manifested, that is expected and is a strong sign of the fatal disintegration of rational, responsible freedom and soundness that is currently destroying the intellectual heart of our civilisation. (BTW, the repeated appeal to, oh he is frustrated may point to a rhetorical stratagem: refusal of assent or responsiveness to well warranted results, in order to undermine moving to a conclusion; but I suspect the typical situation is more likely the error of redefining truth as opinion.)

    At this stage, I expect this sort of error. We are going to see a lot more of it as the collapse of our civilisation proceeds on current track towards the cliff.

    In effect, through the dominance of essentially irrational ideologies — their name is legion — and through linked agit prop and lawfare as well as institutional subversion — we are collectively losing contact with empirical and abstract-logical reality.

    It is not too hard to see that, all over.

    The consequences are all too predictable: our civilisation is on a collision-path with reality and predictably we will lose that collision. (My usual metaphor is a lemming-like march towards the cliffs, which under stress will crumble underfoot. The fall itself will not do the damage, it is the collision with hard, painful reality far below that will break our civilisation’s back, on current track. And yes, I know real lemmings don’t act like that. I cannot say the same for people, with the history of the past 100+ years as exhibit no. 1.)

    A likely manifestation of this march of folly will be adoption of willfully blind economic and/or geostrategic policies that are collectively ruinous, reflecting a fatal decline of the quality of political, legal, military, academic, media, financial and economic leadership.

    Where, already, demographic collapse looms.

    I note, ill-informed populism, environmentalism (especially watermelon variety), socialism and nietzschean superman- above- law political messianism are all viable candidates to lead the charge to economic and/or geostrategic collapse.

    Economic suicide is of course a key flash-point as it is an inherently abstract discipline and sound policy requires thinking that is very likely to be unpopular. Fallacies with economic and/or geostrategic impact are very likely to seize control of policy agendas in civilisations where debased, fundamentally irrational thinking has taken over. And, one does not have to look very far to find such fallacies on the march.

    So, there are many signs of just such a collapse, but de Nile is a river in Egypt, just as much as wishful thinking and denial of evident but unwelcome reality through mechanisms tied to cognitive dissonance open the door to a raft of fallacies.

    Now, is agit prop a term that I imagine that there are commies under every bed?

    No, the Bolsheviks were the pioneers of modern agitation and propaganda, but now the habits and strategies have become a common property of those who exploit those they can mislead, everywhere. Political messianism of lawless character can pop up anywhere along the political spectrum as conventionally conceived.

    I do not like the spectrum, and think a more useful one is from autocratic tyranny to oligarchic dominance to the lawful state to constitutional democracies to libertarianism to anarchic chaos and/or the state of nature. Tyranny is a hard to escape vortex and anarchic chaos is a repeller pole. One may slide down into tyranny or break down into threatened chaos and snap down into said vortex in desperation for a deliverer figure. B ut it is very hard to escape a strong vortex.

    Signs of such dynamics are everywhere.

    That said, I do note the massively destructive influence of cultural marxism (thus of so-called critical theories) and linked Alinsky nihilism.

    As for the diagnosis from a comfortable distance of “paranoia,” that itself speaks to the trifecta fallacy: red herrings, led away to strawman caricatures soaked in ad hominems and set alight to cloud, confuse, poison and polarise the atmosphere.

    (Where, BTW, the penumbra of attack sites is very real; I just won’t bother to name them, other than to note that TSZ is the more or less semi-respectable front operation, not the core of the problem. And when malevolent activists identify the business of a distant relation who has nothing to do with the ID debate in a context of implicit threat, that speaks to on the ground stalking with highly specific local knowledge. )

    But in the end, it is the prophets of old who truly nailed it 2700 years ago:

    Isa 5:18
    Woe to those who draw iniquity with cords of falsehood,
    who draw sin as with cart ropes,
    19
    who say: “Let him be quick,
    let him speed his work
    that we may see it;
    let the counsel of the Holy One of Israel draw near,
    and let it come, that we may know it!”
    20
    Woe to those who call evil good
    and good evil,
    who put darkness for light
    and light for darkness,
    who put bitter for sweet
    and sweet for bitter!
    21
    Woe to those who are wise in their own eyes,
    and shrewd in their own sight! [ESV]

    So, let us look on and understand where we are collectively headed if we continue to lose contact with reality.

    Just maybe, it is not too late to turn back from self-induced ruin.

    But this, I firmly believe: the lessons of sound history were bought with blood and tears; those who neglect, reject or dismiss them doom themselves to pay the same coin over and over and over again.

    If you doubt me, ask the ghosts of the Athenians who saw the Peloponnesian war and its aftermath, demonstrating the suicidal tendencies of democracies. Where BTW, it is precisely that sobering precedent that led the American framers to be so cautious in their political design.

    Those are the matches we are playing with.

    KF

  72. 72
    john_a_designer says:

    Here is something which I wrote on an earlier thread which I think is worth repeating here.

    Penrose describes his metaphysical world view as a tripartite one consisting of the physical world, the mental world and separate and distinct mathematical world. He goes on to explain that… ’there is the relationship between these three worlds which I regard, all three of them, as somewhat mysterious or very mysterious. I sometimes refer to this as “three worlds and three mysteries.” Mystery number one is how is it that the physical world does in fact accord with mathematics, and not just any mathematics but very sophisticated, subtle mathematics to such a fantastic degree of precision. That’s mystery number one.’

    However, since Penrose is a non-theist (according to Wikipedia, which quotes a BBC interview) I don’t see that he has any other choice but to postulate the existence of a separate transcendent Platonic realm. But this is probably too high of a cost for other naturalists to pay (of course, it’s unthinkable for a died-in-the-wool materialist.) [That’s no doubt why we have been seeing such a resistance to the idea that mathematics is discovered by several of our interlocutors.] But if we reject the idea of a transcendent mathematical realm where does our mathematical knowledge and know-how come from? From our minds– which is an epiphenomena of our brains… which is the product of a long mindless evolutionary process. If you begin with those assumptions that’s where the logic leads you, therefore, mathematics must be a human invention. The problem is that you first need to prove that your metaphysical presuppositions are true or that they are more probably true than not.

    Let me add to this an argument William Lane Craig made in his 2013 debate at Purdue University with Alex Rosenberg who is a Professor of Philosophy at Duke University. Craig took the theistic perspective, Rosenberg the atheistic naturalistic perspective.

    Philosophers and scientists have puzzled over what physicist Eugene Wigner called the uncanny effectiveness of mathematics. How is it that a mathematical theorist like Peter Higgs can sit down at his desk and by pouring over mathematical equations predict the existence of a fundamental particle which experimentalists thirty years later after investing millions of dollars and thousands of man-hours are finally able to detect? Mathematics is the language of nature. [8] But, how is this to be explained? If mathematical objects are abstract entities causally isolated from the universe then the applicability of mathematics is, in the words of philosopher of mathematics Penelope Maddy, “a happy coincidence.” On the other hand, if mathematical objects are just useful fictions, how is it that nature is written in the language of these fictions? In his book, Dr. Rosenberg emphasizes that naturalism doesn’t tolerate cosmic coincidences. But the naturalist has no explanation of the uncanny applicability of mathematics to the physical world. By contract, the theist has a ready explanation. When God created the physical universe, he designed it on the mathematical structure he had in mind. We can summarize this argument as follows:

    *1. If God did not exist, the applicability of mathematics would be a happy coincidence.

    *2. The applicability of mathematics is not a happy coincidence.

    *3. Therefore, God exists.

    https://www.reasonablefaith.org/media/debates/is-faith-in-god-reasonable/

    The point is if you believe that mathematics is discovered, as its applicability to the physical world suggests, then metaphysically you’re left with only a few options: (1) an eternally existing Platonic realm, (2) and eternally existing (or self-existing) transcendent mind (God) or (3) a disingenuous shrug of the shoulders, “Oh, we just don’t know… Maybe it’s just a brute fact about the universe”– in other words, a non-explanation explanation. (At least those are the only three I can think of.)

    On the other hand, the atheistic naturalist is stuck with no way to explain the “uncanny applicability of mathematics to the physical world.” So obviously, since mathematics is an abstract function of the mind and minds are just epi-phenomena of our brains, which in turn are the result of millions of years of a mindless/ purposeless naturalistic evolutionary process, mathematics MUST be something we invented. That does follow logically if you assume all your presuppositions are true. But even if you make that kind of leap of faith you still haven’t explained why there is such an “uncanny applicability of mathematics to the physical world.”

  73. 73
    kairosfocus says:

    F/N: I added to the OP, two charts showing the intellectual decay of our civilisation and how malinvestment can trigger collapse. Now, plus a chart on the spectrum of politics and a couple in the main article on truth. KF

  74. 74
    kairosfocus says:

    JAD, sobering point. KF

  75. 75
    math guy says:

    KF’s point is that logical reasoning is the basis for discovering truth. Those incapable of such reasoning will base their decisions on emotion, propoganda, and unverified opinion, easily swayed by Screwtape and his associates in the mass media.

    Two of our interlocutors stubbornly refuse to accept that without adherence to logical reasoning, the masses (like lemmings) will follow the most persuasive modern incarnation of Pericles over the proverbial cliff.

  76. 76
    kairosfocus says:

    MG, yup — though I think Alcibiades and co were more directly responsible for the actual collapse with the folly of the Sicilian expedition and events leading up to Aegospotami being central to my thoughts — we prefer to lose on our own than to take the counsel and aid of a twice exiled traitor. Pericles and co committed the antecedent folly of converting a defensive league into an empire. Hubris. And (coming up to date) if logical-mathematical demonstration is not adequate to move opinions, only a hard collision with unyielding reality will. That’s patently, horribly, our civilisation’s current track. My projection is, economic or geostrategic fallacies are most likely to be the way the path of folly will lead over the cliff. Neither of these disciplines is reducible to popular memes and simplistic talk-points, but they have extremely severe possible consequences. In the case of the USA in particular, it is in cold, lawfare and agit prop civil war already. Europe is seeing mobs in the streets and has had mass burnings of cars, terrorists marching on the streets and shooting bystanders at will or mowing them down with vehicles, etc. Lurking beneath all are the consequences of enabling the ongoing holocaust of living posterity in the womb, 800+ millions in 40+ years, mounting at another million or so per week. The warping of the fabric of political understanding, policy-making, law and law enforcement to enable that is directly fatal. KF

  77. 77
    ScuzzaMan says:

    The OP is deficient in reference to the Slippery Slope argument.
    Not even the people making the argument 50 years ago would have given credence to legislatures forcing schools to let boys dress as girls and therefore – due to their stated opinion in opposition to their plain biology – be allowed entry to the girls bathroom, i.e. they had no possible clue as to the probabilities involved and nor in principle could they have.
    Here in Germany several years ago the State decided that Germany should join the rest of the EU in explicitly banning sex with animals (although one should note, not as inherently deranged behaviour but as an affront to the rights of the animal). Announcement of this intention was followed by a public protest by over 200 literal “animal lovers” and their animals, claiming the contrary right to their previously legal practice.
    Nobody ever made the argument based on the probability of Step 10 occurring in the context of the culture and mores at the time of Step 1. But the Overton Window is merely another description of the Slippery Slope: it describes not only the fact that certain subjects, facts, and opinions are officially forbidden or mandatory, but also that these categories are not only malleable but are actively manipulated over time in order to degrade public opposition to the managerial classes’ excesses.
    We have overwhelming warrant from history to presume, a priori of any particular current demonstration, that no political action is ever solely what it appears and that long-term agendas inimical to our collective and several interests are perpetually being advanced. The “Boiling the Frog” illustration is also merely another version of the Slippery Slope argument. They all three (Slippery, Overton, and Frog) exist as common wisdom because the phenomenon they describe IS so common as to be inescapable.
    Are misapplications of the argument possible?
    Well, duh.
    That doesn’t make the argument a fallacy, or else all forms of argument are fallacious.

  78. 78
    kairosfocus says:

    SM, you raise a very interesting case. I have added the Overton Window. The key to the slippery slope is the strength of the underlying [leftward] ratchet. I actually once had an experience. There is a parking lot that has been undermined by iguanas making burrows, but with an older vehicle I had, the drive control/traction management software made it safe to park there — and others would not do so so you could count on space. Then, when I tried to park a replacement car there, once it went beyond a certain point I was hopelessly stuck. I had to be winched out by a rescue vehicle on solid ground and have never parked there again. The point is, in Overton Window terms, whether the BATNA borders are shifting and why. If there is a ratchet that will pull the borders ever leftward, then one must fight here and now as lost ground will not be recovered and the slope gets worse and worse until catastrophic collapse happens: take the walk-away from the negotiation table and fight, for you are not dealing with good faith negotiators. The 1930’s clearly show that taking courage to fight early (say 1935 or 36) would have averted catastrophe. But of course, the dominant thought made that unthinkable. BTW, in 1934 it was Mussolini who challenged Hitler and checked him. Likewise, 50 years ago, it was unimaginable that we would now have 112 so-called “genders” and activists demanding to enforce such in law. Now, we are there. And we have holocaust of 800+ millions, growing at another million or so per week. KF

  79. 79
  80. 80
    hazel says:

    Math guy writes,

    Two of our interlocutors stubbornly refuse to accept that without adherence to logical reasoning, the masses (like lemmings) will follow the most persuasive modern incarnation of Pericles over the proverbial cliff.

    Assuming I am one of the two people you are referring to, I don’t see how, in the slightest, I am refusing to accept adherence to logical reasoning.

    I disagree with kf about a philosophical point that began between Plato and Aristotle, and that has had adherents on both sides of the issue for centuries. But I believe strongly in the power of logical reasoning, and have made many statements about that during the threads in which we have been discussing these topics.

    So I’d be interested in why you think I refuse to accept logical reasoning?

  81. 81
    Ed George says:

    Hazel

    So I’d be interested in why you think I refuse to accept logical reasoning?

    Nobody is refusing to accept logical reasoning. All we are doing is disagreeing on the conclusions drawn from this reasoning. And, possibly, disagreeing on some assumptions used in presenting the logic. That sort of thing happens all the time.

    For example, if you randomly select a Democrat and a republican, chances are that they each use logically sound reasoning. Where they differ is in the assumptions that they each take for granted when presenting their logic.

    Given the impact that government has over all of our lives, I understand why people get so emotional about the left/right arguments. But I can’t, for the life of me, see why some get so emotionally wrapped up with whether or not mathematics is a human invention.

  82. 82
    hazel says:

    I agree, Ed. As several people have pointed out here, except in pure math, logical reasoning uses premises of various kinds to, using kf’s analogy, weave a rope of conclusion that intertwines logic with other kinds of belief.

    Speaking for myself, I also don’t get the strong concerns about how our discussion is related to all these other dire topics about the state of the world. I think somehow I represent, or am saddled with, a stereotype that is important to many people here but hasn’t actually been a part of my concerns. The two topics I have been interested in are the nature of math and its relationship to the physical world, and the nature of consciousness and the mind. I have a different perspective than most people here, but how that is related to civilization going over a cliff is beyond me.

    (Also, to be clear, Ed and I are not in complete agreement about what the issue is. He phrases it as “whether or not mathematics is a human invention”, but that isn’t how I would describe what the issue is to me. Post 63 above is a fairly succinct summary about what I consider the philosophical point, and I’ve written a lot about how just a small part of math is invented, and that the bulk of what we learn both within math and in respect to the physical world is discovered.)

  83. 83
    ET says:

    hazel:

    Assuming I am one of the two people you are referring to, I don’t see how, in the slightest, I am refusing to accept adherence to logical reasoning.

    Of course you don’t.

    I disagree with kf about a philosophical point …

    Except it isn’t a philosophical point.

    So I’d be interested in why you think I refuse to accept logical reasoning?

    That is what the evidence demonstrates. But then again you don’t seem to understand evidence.

  84. 84
    ET says:

    Ed George:

    Nobody is refusing to accept logical reasoning.

    The evidence says you and hazel are doing just that.

  85. 85
    hazel says:

    At 63, I said that kf and I disagree about a fundamental philosophical point, which I’ve pointed out goes clear back to Plato and Aristotle.

    kf says “Fiveness is real.”

    The question is in what way is “fiveness” real.

    The position I am describing, as an alternative to the Platonic one, is that “fiveness” is an abstract concept that exists in our mind … But “fiveness” as an abstraction does not exist outside of our mind. The world has things in in it, and we can describe some groups of those things with our concept of five, but “fiveness” resides as a mental concept in our minds: “fiveness” does not reside in the physical world itself. Fiveness is about the world, not in the world.

    Since then I’ve done some reading and discovered some details about this perennial issue. The following is based in part on the Problem of Universals

    We are discussing the problem of universals, which I have been calling abstractions, and which the article refers to as properties: “In metaphysics, the problem of universals refers to the question of whether properties exist, and if so, what they are. …While philosophers agree that human beings talk and think about properties, they disagree on whether these universals exist in reality or merely in thought and speech.

    The bolded part is the issue under discussion.

    There are three major schools of thought.

    1. Platonic realism: properties exist “in an ideal form independently of any mind or description.”

    2. Aristotelian realism: properties exist only when real objects exist which exhibit the quality.

    The difference between the two is that Platonic realism posits that “universals are real entities existing independent of particulars”, and Aristotelian realism that “universals are real entities, but their existence is dependent on the particulars that exemplify them.”

    3. The third school is anti-realist, one form of which is nominalism (which is the term I will use). Nominalism posits that “universals are a product of abstract human thought”: universals exist in the mind, but not in the external world. Interestingly enough, William of Ockham, of Occam’s razor fame, was a medieval proponent of nominalism.

    To be very succinct, kf holds to Platonic realism, and I have been discussing the position of nominalism. We are at opposite ends of a philosophical spectrum.

    Given that both of these positions have a long history and numerous credible proponents, I think the useful thing to do is to try to articulate the positions we hold and offer the various lines of thought that we think support our beliefs (that is, describe the rope we have woven) while understanding that other positions have been held valid by many over the centuries.

    Given that these are metaphysical beliefs, as the article states, I don’t think we should expect that there is any route to showing that any one position is “really” correct. Our beliefs about these kinds of things are ropes of belief, not purely deductive chains of proof.

  86. 86
    john_a_designer says:

    Also on an earlier thread I argued:

    that to make a logically valid argument you need to begin with premises and propositions that are either (1) self-evidently true, (2) provably true or (3) at least probably true, otherwise your conclusion does not follow. (That’s deductive logic 101.) Unfortunately, TRUTH is not served to us on a silver platter so we seldom have the advantage of beginning with #1 or #2. Of course, the problem with #3 is: do arguments based on probabilities ever give us certainty? The answer is no. Nevertheless, that is what we are left with– there is no such thing, in most cases, with absolute proof or certainty. However, that doesn’t justify that one can throw up one’s hands and say “Since, I believe in X therefore X is true” or “I don’t believe in Y therefore Y is not true.” Fideism and nihilism are really just two sides of the same coin. Arguments need to be about the Truth not about beliefs. The pursuit of truth requires both intellectual and ethical honesty and some degree of humility. But how can one have either intellectual or ethical honesty if one doesn’t believe in truth to begin with?

    On the other hand, deductive arguments work very well in mathematics. For example, starting with just a few self-evident definitions and postulates Euclidean geometry we are able to prove (as were the ancient Greeks) that that there are– indeed, there only can be– five regular polyhedral or Platonic solids in three dimensional space.

    Descartes no doubt was attracted to the power of that kind of logic when he tried to used cogito ergo sum as an ontological and epistemic presupposition for his philosophy. However, we don’t find the same logically conclusive stepping stones in metaphysics that we do in the axioms and postulates of mathematics. I don’t think any metaphysical system can really avoid that.

    However, no where did I suggest that using deductive and inductive logic (as well as abductive logic) are useless in determining whether or not a philosophical belief is true or false. Quite to the contrary logic is the only tool we really have in such discussions and debates.

    What is totally useless, on the other hand, are ungrounded personal beliefs and opinions. Doubling down on the same ungrounded personal beliefs and opinions is not advancing an argument, rather it’s being argumentative. That’s all we are getting from some of our interlocutors. Basically their argument is as follows:

    Mathematics is something that is primarily (A) discovered or (B) invented.

    I believe it’s B.

    Therefore B is true.

    For example, the proposition, “to propose that mathematics was used by the designer to create the universe is beyond preposterous,” is not something that is self-evidently true. So it is an ungrounded assertion– just an opinion or belief. Stand-alone opinions and beliefs are not arguments. Doubling down on opinions and beliefs is a waste of everyone’s time.

  87. 87
    hazel says:

    Hi JAD. I am assuming I am one of the interlocutors to whom you refer when you write,

    That’s all we are getting from some of our interlocutors. Basically their argument is as follows:

    Mathematics is something that is primarily (A) discovered or (B) invented.

    I believe it’s B.

    Therefore B is true

    That certainly has not been my position, not my argument. I think you have oversimplified the discussion. However, I can only speak for myself, so you might have been referring exclusively to someone else.

    You write,

    However, no where did I suggest that using deductive and inductive logic (as well as abductive logic) are useless in determining whether or not a philosophical belief is true or false. Quite to the contrary logic is the only tool we really have in such discussions and debates.

    But as you have said, and so has kf in the OP, except in chains of pure deductive reasoning, as happens in math, we always have some premises which are not necessarily true, and we do our best to weave strands together to make solid beliefs.

    I don’t recall your ever saying that “using deductive and inductive logic (as well as abductive logic) are useless in determining whether or not a philosophical belief is true or false.”

    I think the whole point of kf’s rope analogy is that we use all three types of reasoning (as well as other things such empirical observations) to come to our philosophical conclusions. However, for many issues, such as the one concerning universals that I discussed in 85, given the role of “probably true” premises, I don’t think there is any route to determining which position on universals is “really” true.

    And P.S., it was not I that made the “preposterous” remark: in fact I pointed out that I disagreed with that remark.

  88. 88
    kairosfocus says:

    H, As a quick note, showing the reality of natural counting numbers in any possible world is antecedent to how is that done. And, that embeddedness as part of the framework for a world to be then points to the principle that there is objective albeit obviously abstract substance for us to discover and explore; including structure and quantity, the “stuff” of Mathematics. Where too, that logical implicit structure has ontological import for the nature, properties and relationships that are possible in any world, including for physical entities in our own; I have already pointed to gear trains and their properties as round objects close enough to ideal circles for the properties of circularity to apply, e.g. to tooth numbers; getting teeth to mesh aright is an onward fascinating and technologically important exercise. BTW, propositions in general are abstracta too, as are many other things. We can then ponder things like, such things are mental, in our experience. However, such are secondary to the import of there being a world, any world, with distinct identity. KF

  89. 89
    hazel says:

    Platonic realism, one of three possible philosophies mentioned in post 85. Another is nominalism, which I am suggesting. I am making the point that your philosophy, although it seems absolutely certain to you, is seen differently by other respected philosophers.

    Of course there are circular things: your fishing reel has many of them. But the abstract circle, or the property of circleness, exists only in our minds. That is the position of nominalism (which a word I didn’t know until a couple of days ago, but it fits what I have been trying to say.)

    The world is full of individual things, and it is our mental ability to isolate general characteristics that produces abstractions about various aspects of those things.

  90. 90
    kairosfocus says:

    H,

    showing that in any distinct possible world, there will necessarily be natural counting numbers as a corollary to distinct identity is ANTECEDENT to any particular philosophy of what such abstracta are.

    This, I believe you have acknowledged to be so, which leads to my puzzlement over the prolonged exchanges on what that hard to deny fact implies.

    Where also, once we have N in context of ordinal succession, we have transfinites etc, with the hyperreals and surreals beckoning. That is, we have the quantities and structures of number systems, continua etc, thus also abstract spaces of arbitrary dimensionality. Such quantities and structures are embedded in the simple fact of distinct identity.

    They also constrain the logic of being, starting with simple properties such as additivity: || + ||| –> |||||.

    How that works out in terms of metaphysics is an onward question, one not of primary importance to recognising what seems to be the central disputed point: there is a world-embedded, intelligible logical substance of structure and quantity that is present in any possible world, which is distinct from and objectively constrains our culturally influenced study of the logic of said structure and quantity.

    That is, in such core parts — I do not claim “for the most part” — Mathematics discovers and elaborates on what objectively exists antecedent to human (or Kzinti or Tree-Cat etc) creativity and cultural tradition. This is where we find such mathematical considerations to be so powerful and pervasive in our study of the physical and even social sciences.

    Such facts “on the ground” then constrain our worldviews choice per the comparative difficulties across factual adequacy, coherence and balanced explanatory power.

    There are major and pervasive facts of Mathematics — structure and quantity — antecedent to arguing about worldview options.

    The fact that most working Mathematicians (and those who work in allied fields such as Computer Science, Physics, Chemistry, Engineering, Statistics etc) are in effect platonists may simply reflect which worldview option is most plausible to those familiar with the force of the facts.

    That is an onward issue, all along I have simply stood up for recognising that there are mathematical — logic of structure and quantity — facts antecedent to axiomatic schemes, models and other manifestations of how we use that logic to create abstract logic-model worlds and often apply results to the observed world.

    I find it a tell, that it seems very hard for some to accept in practice when concrete cases are on the table that there are two aspects to Mathematics: the substance of structure and quantity and our disciplined, culturally influenced study of structure and quantity.

    KF

    PS: the roundness of gears in my fishing reels and requisites of functioning gear trains make extremely abstract entities such as the transcendental number pi, deeply embedded in and constraining of actuality. True, if I am careless and get abrasive volcanic sand embedded things will literally grind to a halt, precisely because the requisite geometry has been compromised. Yes, the abstracta of circularity are abstract, so mental: we reflect on them. That does not mean they are arbitrary mental creations, they manifest themselves in gear trains of reels, watches and vehicles alike. That is, there are commonalities between concrete cases that we may recognise as in effect laws or rational principles of reality — that should not be controversial in a sci-tech age but it obviously does not sit well with popular ideologies of our time. For example, there will be a very specific number of teeth on each member of the train, which have to match geometrically for the train to work. The strength, elastic properties etc of materials are relevant, and are highly quantitative and structural. Even, drag — slipping clutch — materials; substituting certain popular advanced materials leads to a real risk of stripping the gears on reels not designed for such. Likewise, one is well advised to set the slippage point for the drag at no more than 25% of the breaking strain of the line. Which will vary with how much line is left on the reel as that materially affects a radius, thus a moment-arm . . . back off the drag when a lot of line is out but there is still a reserve, and use side strain to shift direction of pull.

  91. 91
    john_a_designer says:

    Again in an earlier thread I pointed out that there are a couple of atheists who agree that mathematics was discovered not invented. For example, one was mathematician Roger Penrose, who said, “It is very important in understanding the physical world that our way of describing the physical world, certainly at its most precise, has to do with mathematics. There is no getting away from it. That mathematics has to have been there since the beginning of time. It has eternal existence. Timelessness really.

    https://uncommondescent.com/intelligent-design/responding-to-ed-george-about-mathematics/#comment-669708

    I also cited MIT physicist Max Tegmark,

    Take a look at the first ten minutes of this episode of Nova, which is a regular science program on PBS. The episode gives several examples of where the Fibonacci sequence as well as the value of pi appear, sometime quite unexpectedly, in nature. It’s followed by an interview with MIT physicist Max Tegmark who believes that everything in nature can be reduced to mathematics. He appears to suggest that we could all be living in some sort of virtual reality.

    https://uncommondescent.com/intelligent-design/responding-to-ed-george-about-mathematics/#comment-669738

    Add the late American astronomer Carl Sagan to the list:

    “The astonishing fact is that similar mathematics applies so well to planets and to clocks. It needn’t have been this way. We didn’t impose it on the Universe. That’s the way the Universe is. If this is reductionism, so be it.”

    Carl Sagan, The Demon-Haunted World: Science as a Candle in the Dark

    And in his science novel Contact Sagan’s main protagonist Ellie Arroway says,

    “Mathematics is the only true universal language.”

    Is mathematics truly a universal language? Could it potentially us give a way to communicate with an extraterrestrial civilization? How could it if it is our invention?

    For example, in his novel Sagan explored in some depth how a message might be constructed to allow communication with an ET civilization, using prime numbers as a starting point, followed by various universal principles and facts of mathematics and science. Is that a viable idea if mathematics is not universal? How could it be universal if it’s not in some sense grounded in something transcendent?

  92. 92
    math guy says:

    Let me supplement JAD@91

    When SIR Roger Penrose has something to say about physics, math, or even philosophy, we would do well to consider his ideas instead of spontaneously rejecting them. I highly recommend his book “Shadows of the Mind” in which he refutes the hopeless task of equating minds with Turing machines or equivalent computational models. (News, are you listening?)

    Although his suggested replacement model (based on QM) is not supernatural, his demolishing of the A/Mat notion of mind as computer is thorough and convincing.

  93. 93
    hazel says:

    Hi math guy. Did you see my post 80? Any comment?

  94. 94
    ET says:

    And still no one has presented any evidence that math was invented.

  95. 95
    hazel says:

    GIve it up, ET, You have declared that all information is discovered because it all comes from The Source, whatever that is. You have an evidence-free belief, so no one could offer anything that could contradict it. I certainly am not going to spend any time responding to you about that any more.

  96. 96
    kairosfocus says:

    H, in all fairness, if you have a right to your opinion so does ET to his. Also, can you kindly show us how minds emerged from blind chance and/or mechanical necessity, or how mind is an epiphenomenon of computing substrates, or how the mathematical substance demonstrably embedded in this or any world is little more than labels we attach to things, forming essentially arbitrary, culturally determined clusters that could just as easily have gone another way? Or, specifically, how circularity and pi could have been otherwise? Etc? KF

  97. 97
    kairosfocus says:

    JAD, most working mathematicians are in effect platonists about the substance of mathematics. KF

  98. 98
    hazel says:

    Kf writes,

    H, in all fairness, if you have a right to your opinion so does ET to his.

    Sure he does. And I returned my opinion, and told him why I wasn’t going to bother to respond, so he could give up his version of “chirp, chirp, chirp”.

    Also, can you kindly show us how minds emerged from blind chance and/or mechanical necessity

    That is not my position. I’ve never made any claim like that.

    or how mind is an epiphenomenon of computing substrates

    That is not my position. I’ve never made any claim like that.

    , or how the mathematical substance demonstrably embedded in this or any world is little more than labels we attach to things, forming essentially arbitrary, culturally determined clusters that could just as easily have gone another way?

    That is not my position. You mischaracterize what I am trying to say badly.

    I really enjoyed my conversation with Gpuccio about consciousness and the mind a few threads ago because we paid attention to each other, working to both explain ourselves and understand the other. When we disagreed, we acknowledged our disagreement and working to refine our understanding of our disagreements.

    However, most of the conversations I have here are unfortunately not like that. For kf to post so much that doesn’t apply to me, as if it did, makes it seem fairly useless for me to spend any time going over any of this again: there is no value in trying to explain what I think if there is such an inability and/or unwillingness to properly understand the position of another with whom one disagrees.

  99. 99
    ET says:

    hazel- Obviously you have series issues and should seek help. I have provided the evidence and you ignored it.

    And yes, if there was evidence to the contrary I would take a look. However you have not provided anything but your raw spewage.

    OK, people, hazel has its mind made up and no amount of reasoning and evidence will ever change it. Add that to the fact it will never support its position and what we have here is a kluge of willful ignorance.

  100. 100
    ET says:

    Also, this is not a philosophical question. That hazel thinks it is just further exposes its agenda

  101. 101

    hazel said,

    The position I am describing, as an alternative to the Platonic one, is that “fiveness” is an abstract concept that exists in our mind … But “fiveness” as an abstraction does not exist outside of our mind. The world has things in in it, and we can describe some groups of those things with our concept of five, but “fiveness” resides as a mental concept in our minds: “fiveness” does not reside in the physical world itself. Fiveness is about the world, not in the world.

    If your alternative to Platonic Realism is that fiveness exists in our mind but not in the physical world, what does “exist in our mind” mean, if it is not part of the physical world? It seems to me you need to first define what “mind” means, and then what “exists in mind”, means, if those things do not reside in the physical world.

  102. 102
    hazel says:

    “It” and “its”. LOL

    Interesting: ET has doubts about my gender. It is telling, as kf would say, that if I used a man’s name he would not do that. Very off topic, but sort of fun. 🙂

  103. 103
    hazel says:

    Yes, wjm, those are related issues. I know you have different thoughts, but throughout all these conversations I have been working from a dualist perspective whereby mind and matter are separate, but connected by some kind of interface, possibly via quantum effects. This was all in a previous thread with Gpuccio, but I forget which one.

  104. 104
    Ed George says:

    Hazel

    Interesting: ET has doubts about my gender. It is telling, as kf would say, that if I used a man’s name he would not do that. Very off topic, but sort of fun.

    Actually he does that to anyone he disagrees with. Odd behaviour, but consistent.

  105. 105
    hazel says:

    “Actually it does that to anyone it disagrees with” There, fixed it for you. 🙂

  106. 106

    Hazel,
    My point is that the phrase “exists in mind,” especially in contrast to “not found in the physical world”, is functionally meaningless unless one explains what those terms mean. In contrast, I (and others) have provided an explanation of what is meant by the phrase “exists in mind.” You say you are offering an alternative to what others here mean by that phrase, but do not explain how it is different.

    Until you explain the distinction, you haven’t actually offered an alternative at all.

  107. 107
    kairosfocus says:

    H,

    Wikipedia, that humble but handy reference, gives us a point of reference:

    Nominalism
    From Wikipedia, the free encyclopedia
    Jump to navigation
    Jump to search

    In metaphysics, nominalism is a philosophical view which denies the existence of universals and abstract objects, but affirms the existence of general or abstract terms and predicates.[1] There are at least two main versions of nominalism. One version denies the existence of universals – things that can be instantiated or exemplified by many particular things (e.g., strength, humanity). The other version specifically denies the existence of abstract objects – objects that do not exist in space and time.[2]

    Most nominalists have held that only physical particulars in space and time are real, and that universals exist only post res, that is, subsequent to particular things.[3] However, some versions of nominalism hold that some particulars are abstract entities (e.g., numbers), while others are concrete entities – entities that do exist in space and time (e.g., pillars, snakes, bananas).

    Nominalism is primarily a position on the problem of universals, which dates back at least to Plato, and is opposed to realist philosophies, such as Platonic realism, which assert that universals do exist over and above particulars. However, the name “nominalism” emerged from debates in medieval philosophy with Roscellinus.

    The term ‘nominalism’ stems from the Latin nomen, “name”. For example, John Stuart Mill once wrote, that “there is nothing general except names”.

    Under physicalism, there are no abstracta, just stored signals in some computational substrate that likes to call itself — oops — an individual mind. It is not too hard to see that such physicalism falls apart under the force of self-referential incoherence and GIGO. Evolutionary materialism self-falsifies and takes its fellow travellers down with it. Where, of course, on such failed premises — oops, there are no premises as there are no propositions (as abstract a thing as we can get) — there is no logic of being and no logic of structure and quantity as there is no logic, just signals in some GIGO-driven substrate that somehow assembled itself from cumulative noise and whatever mechanical cascades of events obtained. There is no logic, of course radically self-refutes. We cannot have a REASON to believe there is no right reason as there are no principles, there is no right and there is no reason.

    Utter confused chaos.

    And before we get there there is no answer — oops, that is a logical commodity — on such premises (oops) to how we get to functionally specific, coherent, complex organisation and associated information for us to have computational substrates.

    Perhaps, we have a modest version, where there is mind. But as WJM highlighted, what then is mind? Surely, not just another name — oops, an abstract commodity again — for a GIGO-limited computational substrate that will happily process nonsense nonsensically until the system crashes.

    Far better, to start with undeniable fact 1: the self-aware, conscious mind that perceives, reflects, reflects on itself, infers, reasons, understands, accesses and uses abstracta and in our cases is embodied. Where, I on reflection chose to type out these reflections, character by character. Among those abstract entities and processes is reasoned judgement under principles of reason, logic. The logic that tells me that a world with distinct identity, W, will have distinction, W = {A|~A}. so also, duality, unity, nullity. Thus, by way of the von Neumann construction the naturals. Further, integers, rationals, reals, complexes and other vectors and similar structures. Thus also abstract space where a circle of rad r centred on origin has r^2 = x^2 + y^2, and we can make any circle in the space by applying a vector displacement and a scaling. Where, we may proceed similarly to other figures. We may observe a circumference C and a diameter D = 2r, leading to pi = C/D, a transcendental real number.

    These are indeed abstracta but by logic will necessarily be present in any possible world, and will affect possibilities and actualities in any such world. For example, in ours, ponder the ways relevant structures and quantities affect a gear train in a fishing reel, a watch or a car.

    So, what are these abstracta?

    Not, traces on paper or ion gradients — oops, another abstract commodity — in nerve cells. Such may be associated, but very similar ion gradients readily mean something else. Oops, meaning is another abstraction. Just like man, woman, mind, truth, love.

    The candidate to beat is: something contemplated by a mind.

    Which, in the context of roots of a world, points to eternal mind.

    Which, in turn seems to be the real problem.

    Ideological commitment to block any shadow of Him with whom we wish to have no dealings.

    Never mind the resulting utter incoherence.

    But, I have not been primarily interested in debates over theism, just with the nature of Mathematics.

    KF

  108. 108
    hazel says:

    To wjm at 106:

    I have enjoyed learning about some philosophical views, including yours.

    However, as was a key point in my discussion with Gpuccio, I start with and accept the experiential reality of my own consciousness and mind, and the experiential reality of there being an external world, including my own body, that is different from and separate from my mind. I don’t think that distinction is “functionally meaningless” if I can’t define what those terms mean. Given how given they appear to my experience, I’m willing to consider their existence as undefined aspects of my views. (This is somewhat analogous to how geometry starts with some undefined terms like point, line, and plane.)

    I feel pretty comfortable with philosophies that support the distinction between mind and matter, and not so much to philosophies that don’t.

  109. 109
    hazel says:

    to kf: all your remarks about physicalism and evolutionary materialism, etc. have nothing to do with me, and I’m baffled why you can’t get that.

    I read the Wikpedia article earlier. One line in there says; “Conceptualists hold a position intermediate between nominalism and realism, saying that universals exist only within the mind and have no external or substantial reality.”

    That is probably closer to what I have been trying to describe.

  110. 110
    ET says:

    hazel:

    ET has doubts about my gender.

    Not really. But if saying that makes you feel good, then have at it.

    It is telling, as kf would say, that if I used a man’s name he would not do that.

    And yet I do so all of the time. It is what I do when people just ignore the evidence and arguments and prattle on regardless. People- I don’t care about their gender seeing that they choose to be willfully ignorant. And I don’t want to insult any gender by calling a willfully ignorant person by that type of pronoun.

  111. 111
    ET says:

    hazel:

    I feel pretty comfortable with philosophies that support the distinction between mind and matter, and not so much to philosophies that don’t.

    And I feel pretty comfortable with the reality and evidences that support the distinction between mind and matter, and not so much to philosophies that don’t. 😎

  112. 112
    ET says:

    Ed George:

    Actually he does that to anyone he disagrees with.

    That is not true, Ed. But I understand why you would say it.

    It is what I do when people just ignore the evidence and arguments and prattle on regardless. People- I don’t care about their gender seeing that they choose to be willfully ignorant. And I don’t want to insult any gender by calling a willfully ignorant person by that type of pronoun.

  113. 113
    john_a_designer says:

    Math Guy @ 92,

    Thanks for the comment. Here is an interesting quote from a paper Penrose wrote:

    Gödel appears to have believed strongly that the human mind cannot be explained in terms of any kind of computational physics, but he remained cautious in formulating this belief as a rigorous consequence of his incompleteness theorems. In this chapter, I discuss a modi?cation of standard Gödel-type logical arguments, these appearing to strengthen Gödel’s conclusions, and attempt to provide a persuasive case in support of his standpoint that the actions of the mind must transcend computation. It appears that Gödel did not consider the possibility that the laws of physics might themselves involve noncomputational procedures; accordingly, he found himself driven to the conclusion that mentality must lie beyond the actions of the physical brain. My own arguments, on the other hand, are from the scienti?c standpoint that the mind is a product of the brain’s physical activity. Accordingly, there must be something in the physical actions of the world that itself transcends computation. We do not appear to ?nd such noncomputational action in the known laws of physics, however, so we must seek it in currently undiscovered laws going beyond presently accepted physical theory. I argue that the only plausibly relevant gap in current understanding lies in a fundamental incompleteness in quantum theory, which reveals itself only with signi?cant mass displacements between quantum states (“Schrödinger’s cats”). I contend that the need for new physics enters when gravitational effects just begin to play a role. In a scheme developed jointly with Stuart Hameroff, this has direct relevance within neuronal microtubules, and I describe this (still speculative) scheme in the following.

    https://philpapers.org/rec/PENGTM

    If I am understanding him correctly Penrose does not believe that a computer could ever have the capability to derive any of Gödel’s incompleteness theorems. In other words, “proving” such theorems requires the insight of an intelligent conscious mind. (As I understand Gödel’s incompleteness theorems they’re proving that there are true statements in a given system of mathematics that are undecidable.) Whether or not they are “intelligent” computers clearly are not conscious or minds. Of course this view is not uncontroversial.

    However, this brings up another interesting question: could computers using advanced AI algorithms ever get to the point where they could independently solve unproven conjectures such as the twin prime conjecture (whether or not the set of twin primes is infinite,) the Goldbach conjecture or the Riemann hypothesis? In other words, we feed a problem into a supercomputer programmed with an advanced AI algorithm named Math Savant and after some processing it solves one of the unsolved conjectures in mathematics.

    This question, along with Penrose’s thesis, I think would make an interesting topic for an OP or a Mind Matters article.

    I am not a mathematician or a computer geek so presently I have no opinion one way or the other what a computer could do. Anybody else (like a mathematician or computer geek) have any thoughts?

  114. 114
    kairosfocus says:

    H,

    it seems clear that a main form of nominalism is physicalism (and its fellow travellers); where the associated naturalism is known to be a major ideological motive in our time. So, it is entirely in order to show that this form of nominalism falls apart into absurdity.

    Then, once one accepts the reality of mind beyond epiphenomena and delusions — if that is possible! — of a computational substrate, then the problems of rejecting the reality of archetypes instantiated in cases are seen to also follow. One cannot even posit propositions, what is meant and asserted by a sentence of certain types. Logic follows, poof. Rational principles, poof too.

    If one says, no, we accept minds (and presumably their contents) — and not merely computational substrates — then one has accepted abstracta. There is no good reason to reject archetypes, which will include quantities and structures. The real issue is, what are these things that are not matter, motion and clumps of same.

    Let me again clip Wikipedia as a handy source:

    In medieval philosophy, the French philosopher and theologian Roscellinus (c.?1050 – c.?1125) was an early, prominent proponent of nominalism. Nominalist ideas can be found in the work of Peter Abelard and reached their flowering in William of Ockham, who was the most influential and thorough nominalist. Abelard’s and Ockham’s version of nominalism is sometimes called conceptualism, which presents itself as a middle way between nominalism and realism, asserting that there is something in common among like individuals, but that it is a concept in the mind, rather than a real entity existing independently of the mind. Ockham argued that only individuals existed and that universals were only mental ways of referring to sets of individuals. “I maintain”, he wrote, “that a universal is not something real that exists in a subject… but that it has a being only as a thought-object in the mind [objectivum in anima]”. As a general rule, Ockham argued against assuming any entities that were not necessary for explanations. Accordingly, he wrote, there is no reason to believe that there is an entity called “humanity” that resides inside, say, Socrates, and nothing further is explained by making this claim. This is in accord with the analytical method that has since come to be called Ockham’s razor, the principle that the explanation of any phenomenon should make as few assumptions as possible. Critics argue that conceptualist approaches only answer the psychological question of universals. If the same concept is correctly and non-arbitrarily applied to two individuals, there must be some resemblance or shared property between the two individuals that justifies their falling under the same concept and that is just the metaphysical problem that universals were brought in to address, the starting-point of the whole problem (MacLeod & Rubenstein, 2006, §3d). If resemblances between individuals are asserted, conceptualism becomes moderate realism; if they are denied, it collapses into nominalism.[10]

    So, is there or is there not an archetype in common that allows us to recognise the fiveness of say my right fingers and left toes, or those of my wife or mother? That archetype is also found embedded in the logic — notice how impossible it is to avoid speaking or reasoning in categories and using universals — of there being any distinct world. That is, there is an abstract substance of structure and quantity we may call fiveness. It will be an in common property of any set of discrete things that will match the successive counting glyphs: 1, 2, 3, 4, 5. Those things may be concrete — my fingers — or abstract, the fingers of my late father.

    For that matter, I may conceive a plan without giving it physical instantiation, and of several competing plans, the logical performance of say C will lead me to give it approximate physical effect, say as the main gear train of a 6500 C3 fishing reel. Which, will embed pi.

    The distinctions we may make indicate distinct identifiable entities, some physical, some abstract. And no, I need not suggest a weird, independent world of forms separate from the mind of God and from the logic of being for worlds that are possible of instantiation. God, here, being reason himself.

    KF

  115. 115

    Hazel @ 108 said:

    I start with and accept the experiential reality of my own consciousness and mind, and the experiential reality of there being an external world, including my own body, that is different from and separate from my mind. I don’t think that distinction is “functionally meaningless” if I can’t define what those terms mean.

    What I meant was that the phrase “exists in mind” was functionally meaningless in providing a contrasting alternative to Platonic Realism, which is what the post I was referring to was about. I didn’t say it was functionally meaningless in terms of practically sorting out behavioral options and distinguishing broad classes of experience.

    Until you provide a way that something “exists in mind” other than platonic realism, even though you don’t require such an explanation to live and function successfully, you haven’t actually offered an alternative to Platonic Realism, which does offer an explanation of what “exists in mind” means.

    Exists in mind – but NOT Platonic Realism … okay, so then what? “I don’t know” is a perfectly fine answer, but neither “nominalism” or “conceptualism” mean anything unless one defines what they mean by “exists in mind”.

  116. 116
    hazel says:

    wjm, I mean “exists in my mind”. I, right now, am contemplating Euler’s Identity, e^(i*pi) = -1. I understand what it means and how it is related to the rest of the number system. It exists as a concept in my mind. I assume it exists in other people’s minds who have studied the math also. I have no idea as to whether there is any larger mind, or Platonic realm, where it exists independently of those of us in whose individual minds it exists.

  117. 117
    hazel says:

    kf, I have no idea how the ideas I’m an expressing have to do with physicalism. See my previous remark to wjm. Abstractions exist in the minds of human beings. As I said in 98, my position has nothing to do with physicalism and all those other things: why can’t you get that into your head?

  118. 118
    ET says:

    why can’t you get that into your head?

    It’s an interface issue. Most likely there is either too much SWR at the transmitter or receiver.

  119. 119
    hazel says:

    kf quotes someone from the Wikipedia article on nominalism:

    If the same concept is correctly and non-arbitrarily applied to two individuals, there must be some resemblance or shared property between the two individuals that justifies their falling under the same concept and that is just the metaphysical problem that universals were brought in to address, the starting-point of the whole problem (MacLeod & Rubenstein, 2006, §3d).

    The kf adds,

    So, is there or is there not an archetype in common that allows us to recognise the fiveness of say my right fingers and left toes, or those of my wife or mother?

    This is a good question, and I’d like to share some thoughts.

    Each of us have a unique set of concepts in our mind, and they vary widely in how similar they are to other people’s.

    Some concepts, to some extent, are based on experiences with the physical world, and since we are all approximately alike in our sensory experience, share some commonality for that reason. However, the vast majority of our concepts are in large part brought to us through language, either verbal or written, and are thus held in common through learning with symbols.

    Thus, I don’t believe the fact that we share approximately common concepts with others is an argument that those concepts reside outside of us an archetype or some other type of independently existing abstraction.

    A few further thoughts on what I said above. Everyone’s concepts are unique no matter how much common substance they have. Paying attention to my own consciousness, I know that every concept I have has a cloud of peripheral associates that add to its overall meaning. Everyone’s overall cloud, then, is different based on their individual background and experience.

    An example: take the number seven. Almost all people over about the age of four understand it to be a certain number that can be used to count things. Some people also know it is a prime number. I now think about how 7^2 – 1 is a multiple of 24, and know why, but I didn’t know that a few weeks ago. That has expanded the cloud of knowledge around the concept. I also know you can’t construct a regular seven-sided polygon with a ruler and compass, but that you can using the seven 7th roots of 1 in the complex plane. These are things most people don’t know.

    Thus, the overall concept of seven, even though it shares a common core with others, exists in me differently, at least in some ways, then it does in anyone else.

    Of course, in math concepts can be formally defined. How most people learn that aspect of the concept is by being taught. The main reason we have a large body of shared concepts, both mathematically and otherwise, is that we are taught them, both formally and informally, through language. We share a lot of common sensory experiences, but the structure for applying abstractions to those experiences comes from sharing ideas through language and then building our own unique world of understanding in our minds.

    So i disagree with MacLeod and Rubenstein, quoted above, when they say, “If the same concept is correctly and non-arbitrarily applied to two individuals, there must be some resemblance or shared property between the two individuals that justifies their falling under the same concept,”

    We share the same concepts because we pass those ideas from mind to mind, and because both our minds and our sensory experiences share a great deal of a common human nature, but that doesn’t mean that the abstract property that we both, approximately, share, has a reality outside of our minds.

  120. 120
    math guy says:

    Hazel @80 claims to accept logical reasoning (whose rules form a Boolean algebra used in computer architecture, an abstract non-physical entity).

    I claim it is false to suggest that all humans need to be taught rules of logic in order to use them, or even think about them. In fact, young children recognize logical implication early on, as witnessed by me (a parent).

    Let us examine the alternate position that abstract objects only exist in human minds. But JAD has already presented us with a gedanken experiment where SETI finally detects a nearby signal. We respond by sending pulses corresponding to an initial segment of the sequence of primes. Of course, ETI has no “human mind” with which to share our idea of primes and cannot make sense of our reply, even though ETI has somehow made a signal generator and sent intelligible signals across space to us.

  121. 121
    math guy says:

    There is lots more I’d like to say, but I have a day job which inhibits posting reams of material (unlike BA77 and Hazel, although BA77 tends to copy/paste from his vast and impressive collection).

    Nominalism, as per Hartry Field, cannot model QM, since the latter requires an infinite dimensional Hilbert space. On the other hand QM is the most successful scientific theory in history in that it perfectly models small-scale physics. QM is as “true” as any physical model could be hoped for (and its failed reconciliation with GR is most likely because our theory of gravity is lacking; cf the many discussions about dark matter).

    So using logical syllogism, nominalism cannot explain the physics that we observe and hence cannot be a correct model of reality.

  122. 122
    hazel says:

    mathguy, I agree that human beings have some common cognitive abilities, and that children know how to use logic, just as they learn to speak. Rationality includes both the ability to learn, create, and use abstractions and the ability to manipulate them logically. Rationality is one of the main properties of human beings. I guess I’ve just been assuming that is part of the background of our discussion. Of course, people have to be taught formal logic, but the informal ability to reason from premise to conclusion is part of human nature.

    As to SETI, I’ve just talked about human minds, but if there are other minds like ours out in the world, it seems reasonable to me that they would develop some similar concepts, and thus we could establish some commonality by sharing something like a sequence of primes.

    I don’t see how QM affects anything. Our understanding of QM, and the tools we have developed to explore it mathematically, are abstractions in our minds, just as the rest of math is.

  123. 123
    kairosfocus says:

    H,

    again, I already explained precisely why I started with physicalism and its fellow travellers, then extended to claimed non-physicalist forms of nominalism, including of course conceptualism. I noted how attempts to reduce mind to computational substrates fail to account for the substrates. They also fail to account for abstracta, which include logic, inference, propositions, numbers etc. It turns out that our rational discussion is inextricably entangled with reference to any number of abstracta so that we in effect cannot reason without such abstracta being implicitly accepted as accurately referring to realities. Realities that are not mere concrete entities or collections — oops, abstractum — and which are not merely arbitrary ideas or concepts — oops — going no further than somehow mutually agreed — oops — games — oops — that we play together. Games we call Mathematics that then inexplicably — poof, magic — apply effectively to concrete realities.

    (And, “reality” — the world of entities in themselves as opposed to our error-prone perceptions — is itself yet another abstraction in this context. Here comes that Kantian ugly gulch and fail again. As F H Bradley pointed out c 1897, s/he who imagines that one may mot know about reality in itself, has already proposed a knowledge claim regarding reality. Where, knowledge and propositions expressing same, warrant that supports the claim, etc are all again abstracta. )

    If I were in a more light-hearted mood, I would chuckle; but you have objected to my noting such, so I refrain.

    Instead, I note that WJM is right to highlight, what do you mean by mind, noting too your tendency to confine to human ones. Let me clip what looks as close to describing mind as you seem willing to posit:

    Some concepts, to some extent, are based on experiences with the physical world, and since we are all approximately alike in our sensory experience, share some commonality for that reason. However, the vast majority of our concepts are in large part brought to us through language, either verbal or written, and are thus held in common through learning with symbols.

    Thus, I don’t believe the fact that we share approximately common concepts with others is an argument that those concepts reside outside of us an archetype or some other type of independently existing abstraction.

    A few further thoughts on what I said above. Everyone’s concepts are unique no matter how much common substance they have

    Do you see how many abstracta are involved? How many collections that do not reckon with the inherently abstract concept of such collection into a cluster — a universal? How many linked propositions, another abstract commodity? How many assumptions rooted in inferred import of family resemblance? How much seems to parallel the physicalist appeal to computational substrates and seems to echo the Kantian ugly gulch? Etc?

    In short, you illustrate the inextricable entanglement I have pointed out already.

    If we are indisputably thinking, inferring, reasoning, warranting, and if such inextricably are entangled with abstracta, then it seems reasonable to accept the inevitable.

    As, not a demonstration, but as a start point: abstracta, whatever they will turn out to be ontologically, are an inescapable fact of rational thought.

    Where, likewise, ability to communicate and come to mutuality of sufficient degree on key abstracta is inescapable in being a community of minded — whatever that is — thinking practitioners. So, human thriving is also entangled, thus moral government too. Where, known duties to truth, right reason, fairness, prudence etc are prominent in discussion, argument, attempts to persuade, and are — again — inextricably entangled. Even knowledge is an abstractum.

    The logical import — abstract, again — is that the world of thought and its applicability to reality as experienced through embodiment and consciousness, intentionality etc (notice, the abstracta and collectives) are inescapably tied to the truthfulness of some propositions as well as to accepting that a great many abstract entities, such as sets, numbers, broader quantities and structures are real in some sense that transcends any particular human mind or community. That is, they are objective, which has in it openness to extension, adjustment or correction but entails having enough reliability to be routinely and confidently used. Including, hypothetically towards instantiating a design. For a circle, once we get to the reals and use i* to rotate, we have a conceptual, planar, flat space. In that space, we can readily specify circles as fulfilling the relationship: r^2 = x^2 + y^2, with translations and reflections and scaling allowing for arbitrarily many circles. Where in the r –> inf. , we have a straight line, where at any point along an arbitrary curve we can define a radius and centre of curvature (as well as extending such reflections to cumulative arc length).

    The relations of circularity and pi etc as extended will obtain for Kzinti, angels and God.

    These relations would extend to more or less round objects, including gear trains for watches, fishing reels, bicycles and wheeled or tracked vehicle drive trains for one and all. (Does God favour Patek Philippe or Casio? Automatic or Quartz? Pope Francis goes for Casio quartz, US$ 12; unique among world leaders. Right now, I am a bit concerned that I have a time finding a battery for a Wenger quartz movement wristwatch., my favoured timepiece. Our best house clocks are all Casios BTW. And yes, there is at least one clock in every room of the house.)

    We see that abstracta credibly hold reality, especially relevant, rationally pondered structures and quantities.

    This is not mere individual or collective opinion, we have explored warrant, including logic of being. It being effectively undeniable that at least one world exists, we can assess possible worlds and find that on distinct identity, the naturals must exist as framework to any possible world. That generality of result transcends humanity. And yes, I know you tried to lock out such considerations some weeks ago. All that showed is a gap in your considerations.

    In that light, it is then reasonable to hold that there is a core of rationally accessible structure and quantity in this or any possible world. Such, being inextricably part of the framework for any distinct world to be. It is reasonable to ponder an abstract, shadow framework logic-world embedded in the fabric for any world, and term that core mathematical reality. And yes, that echoes Plato. And Augustine, who considered this a view into the contemplation of God, considered as root of reality, creator of this and any other actualised domains of reality. That is phil, it is enough for Math, that a core abstract intelligible framework necessarily obtains in this or any other possible world. We can be confident about the power of Mathematics. Wigner is answered.

    Which, is a big part of my context of thought.

    Further, we may extend from this core of necessary mathematical facts, creating various abstract logic-model worlds exhibiting structures and quantities. That is, we may ponder mathematical contingencies of more or less restricted scope, including designs for technological entities, theories of science, economic or financial models, statistical models, etc. Then, we may test and sufficiently confirm reliability and zone of applicability to use them confidently but responsibly.

    Where, too, going back to the OP, we can see how our thoughts can go amiss, and particularly, how important it is to see that just to think and operate as rational, responsible creatures, there are many core points that are inevitably involved and which must be taken as true beyond ability to demonstrate. For, demonstration itself relies on such.

    KF

  124. 124
    kairosfocus says:

    MG, I use insomnia power. This week has been particularly busy with a cluster of meetings connected to moving MNI from hard-won breakthroughs to a redevelopment breakout. The unfolding brexit chaos is also material; I wonder how Mrs May soldiers on; but then I suspect no one else would do materially better — and I see the just walk away and use WTO. More, later today — solar energy initiatives — so I will try to catch a nap or two. Y/day was sea port. Day before, annual financial aid mission. KF

  125. 125
    kairosfocus says:

    H, QM is a capital example of mathematics deeply embedded in the experienced physical world as key substructure. KF

  126. 126
    kairosfocus says:

    H,

    I note from another thread and suggest to you — i/l/o the actual OP topic (talk about side tracks!), the painful subject of fallacies vs credible warrant thus substitution of opinion for objective truth — that too often you have set up in effect a strawman target. Sorry if you find that objectionable or painful, but sometimes something has to be forced through resistance and even pain.

    For example, my primary point on this side-tracked discussion (red herrings for breakfast, anyone?) has always been that to get a distinct world, W — any such world — we must have the generic distinction A vs ~A. Consequently, we contrast and partition: W = {A|~A}. Instantly, we have two distinct entities so duality. A is a unit, and the dichotomy implies nothing inside the partition. Likewise W has nothing outside A and ~A. So, 0, 1, 2. Mix in succession and per von Neumann, the naturals, with Z, Q, R, C to follow, etc. Thus a flat space in which circles etc. All of this is inherent in the distinction of identity for any distinct world. So, these entities are necessary beings embedded in the framework for a world to exist.

    There is therefore a logic of structure and quantity embedded in any world and that is a substance of mathematics.

    Mathematics is not reducible to the culturally influenced study of effectively arbitrary conceptual entities we want to amuse ourselves by playing games with. Nor is it a mere clash of opinions, we have a demonstration on the table of necessary being mathematical entities which form a considerable body of facts on the ground antecedent to any axiomatisation that sets up an abstract, logic-model world. And I note that for many weeks, you have never presented a refutation of the core point. You have tried to convert it into a discussion of opinions and have posed on a different opinion. Indeed, you have effectively tried the rhetorical dismissal that I labour under the delusion that I have perceived an established, certain truth.

    At this point, my response is, that distinct identity is the core of logic, and its corollaries will be equally necessary, framework entities in any world. So, having shown that N, Z, Q, R and C (from the vector perspective you scarcley will acknowledge as valid) are such corollaries and/or constructions on such, I may freely draw conclusions on substantial matters.

    First, there is mathematical substance antecedent to our error-prone subjectivity, cultural influences and traditions at work in any possible world. This being a collection of propositions that for relevant purposes sufficiently describes a possible state of affairs for this or another world. Henceforward, core mathematical facts: CMF’s. Such, also, patently being abstract but often serving as archetypes reflected in concrete specific particular entities, e.g. a 6500 C3 gear train which manifests several phenomena linked to circles. Imperfect reflection is valid reflection.

    Thus, I freely posit on such CMF’s, that Mathematics is of dual character. Substantial and objective, as well as a study constrained by CMF’s. Thus, a well supported good enough fer gvv’mint work definition: Mathematics is (the study of) the logic of structure and quantity.

    Going further, one may speculate as to how such abstract entities may have some existence. It is clear that nominalism is incoherent due to how abstracta are inextricably entangled in every act of serious conceptual thought. This particularly holds for evolutionary materialistic scientism — which happens to be ideologically dominant and so must be addressed first or else one is open to the you set up a strawman objection. It then extends to fellow traveller ideologies commonly seen among today’s educated classes, by the same token of entanglement. It extends to conceptualism, and to effectively any other species one cares to erect.

    Nominalism is dead.

    That means that we must address an abstract domain that collects relevant abstracta. Call it what you will, a neo-Platonic domain or whatever. Labels are not the issue — nominalism being dead. Substance is.

    Going forward, we can take due note on the merits, holding that we have answered Wigner’s challenge:

    the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, “What is mathematics?”, then, “What is physics?”, then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.

    Note, how he focusses on Mathematics as study:

    mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms. Furthermore, whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics. Thus, the rules for operations with pairs of numbers are obviously designed to give the same results as the operations with fractions which we first learned without reference to “pairs of numbers.” The rules for the operations with sequences, that is, with irrational numbers, still belong to the category of rules which were determined so as to reproduce rules for the operations with quantities which were already known to us. Most more advanced mathematical concepts, such as complex numbers, algebras, linear operators, Borel setsãand this list could be continued almost indefinitelyãwere so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty. In fact, the definition of these concepts, with a realization that interesting and ingenious considerations could be applied to them, is the first demonstration of the ingeniousness of the mathematician who defines them. The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used. The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess. However, this is not our present subject. The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity. [3 M. Polanyi, in his Personal Knowledge (Chicago: University of Chicago Press, 1958), says: “All these difficulties are but consequences of our refusal to see that mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting” (p 188).]

    The complex numbers provide a particularly striking example for the foregoing. Certainly, nothing in our experience suggests the introduction of these quantities. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius. [4 The reader may be interested, in this connection, in Hilbert’s rather testy remarks about intuitionism which “seeks to break up and to disfigure mathematics,” Abh. Math. Sem., Univ. Hamburg, 157 (1922), or Gesammelte Werke (Berlin: Springer, 1935), p. 188.]

    Of course, the vector-rotation view actually does embed complex numbers in relevant physical contexts. Where the span from N to C is necessary so on exploring in an abstract possible world, we will readily extend necessary entities to any world and useful possible ones to any relevant one. Such as, gear trains.

    And so forth.

    KF

  127. 127

    Hazel @116;

    In other words, you don’t have an alternate explanation, you are simply rejecting the Platonic Realism explanation of what “exists in mind’ means. IOW, under the category of what “exists in mind” means, there exists under discussion two sub-categories of explanation of (at least some) mental phenomena; (1) physicalism/materialism (such thoughts are generated by material forces) and (2) Platonic Realism – such thoughts represent things that objectively exist in a shared mental landscape, which we all can discover independently by exploring a bit in our mind.

    If I remember correctly from other threads, you have rejected both. Furthermore, you don’t object to Platonic Realism based on an evidential or logical basis (in fact, I think you agreed to at least some of the evidence and logic), but rather because the enormity of what would reside in such a realm if true is something you cannot accept (if I remember correctly), which I’m sure you understand is not a logical objection.

    As I have said before, having a coherent theory of mind is not necessary to practically function and succeed in the world (in such terms as success is usually defined), but if you’re going to enter into philosophical discussions about the nature of mind and thought and the like, it might be interesting to explore what it is that you do believe (via your own personal introspection) about what is going on when you think and imagine, about the nature the existence of universal mental forms and values and how it is they occur. Come up with your own positive ideas to present.

  128. 128
    kairosfocus says:

    WJM, one may argue p => q, ~q so ~p. This means we are back at competing premises, in factual adequacy, coherence and explanatory balance of power. Nominalism is in a problem of inextricably entangled abstracta, across its various forms. It wants to dismiss abstracta and cannot escape them. It is dead. The real issue going forward is, how do we account for a world in which abstracta are inextricably in the roots of reality. This of course makes mind antecedent to matter. KF

  129. 129
    hazel says:

    wjm writes that there are two categories,

    (1) physicalism/materialism (such thoughts are generated by material forces) and (2) Platonic Realism

    I don’t think those are the only two. My mind, a non-material part of me with cognitive, rational abilities, generates abstractions, embodied in symbolic language but existing within me as broader, non-verbal, holistic thoughts, based on my experience of the world, including what other teach me.

    You then say

    it might be interesting to explore what it is that you do believe (via your own personal introspection) about what is going on when you think and imagine, about the nature the existence of universal mental forms and values and how it is they occur.

    But I don’t see my thoughts as being about “universal mental forms and values”. I see them as being about the world I experience, a combination of my own rational ability and the heritage of symbolic understandings, including math, that have been developed and handed to me. I don’t see any of that as existing, as a mental concept, outside of me (other than in shared symbolic language, which is the means by which I create a commonality of understanding with other human beings.)

    And you write,

    If I remember correctly from other threads, you have rejected both. Furthermore, you don’t object to Platonic Realism based on an evidential or logical basis (in fact, I think you agreed to at least some of the evidence and logic), but rather because the enormity of what would reside in such a realm if true is something you cannot accept (if I remember correctly), which I’m sure you understand is not a logical objection.

    I appreciate it that you do remember correctly. The issue is not whether I can develop a logical objection to Platonic Realism. The issue is whether I can think of logical and evidentiary reasons that seem sufficient to believe in such, and I can’t.

    My theory of mind, with I discussed quite a bit with Gpuccio, is that it is something (I don’t know what, and I don’t think anyone does) that we experience “internally” through the medium of consciousness. Somehow (I don’t know how and neither does anyone else) it has a two-way interface with the external world, though my body. It is capable of, among other things, logical rational thought and the embodiment of abstractions in symbolic form which it can then manipulate to reach new conclusions.

    I think that is sufficient for me. It isn’t sufficient for you. I think we will have to leave it at that.

  130. 130
    hazel says:

    I am not interested in continuing this conversation with kf, as it continues to cover the same ground, and he doesn’t seem to really engage what I am saying.

    I don’t dismiss abstractions. I think they reside in my mind. Kf thinks, “The real issue going forward is, how do we account for a world in which abstracta are inextricably in the roots of reality. This of course makes mind antecedent to matter.”

    This is not the “real issue” to me, nor a conclusion I agree with, but I understand this is the heart of kf’s philosophy. I think it’s time to leave it at that also.

    But I’m going to succumb to a temptation:

    In the Wigner article, Wigner writes, “mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts.

    As kp would say, oops! Wasn’t arguing against the idea that mathematics is invented one of the stimuli behind this whole series of threads. 🙂

  131. 131

    KF,

    I think Hazel is putting forth a good faith contribution to the discussion inasmuch as she can. While many of us understand the first principles arguments and the necessity of their validity in all matters both esoteric and practical, it takes a high degree of commitment to discovery and self-analysis to navigate that terrain. For some it also takes a willingness to break down what is a very comfortable (and operationally successful) ambiguity when it comes to these things. They don’t see (IMO) how the mental discipline and commitment can ever pay off.

    For me, these are essential question that go to the root of who and what I am, where I am, what I’m doing, what existence means, how it works and what it is ultimately about. For most people, these things don’t even rise to the level of conscious consideration, much less considering them worth the kind of commitment required to dive in and learn to swim. I appreciate that you at least attempt to point out the danger of such superficial and vague mindsets (when it comes to these issues), but it’s sort of a chicken and egg problem – it’s difficult to understand how not understanding these things is dangerous if you don’t understand them and their importance.

    I know you try to explain that as well, but without that commitment to understanding them in the first place, the danger part falls on deaf ears, unfortunately.

  132. 132

    Hazel said:

    I don’t think those are the only two.

    I didn’t say there are only two; I said that there have been two presented and that rejecting those is not the same as offering a third alternative.

    My mind, a non-material part of me with cognitive, rational abilities, generates abstractions, embodied in symbolic language but existing within me as broader, non-verbal, holistic thoughts, based on my experience of the world, including what other teach me.

    What you are doing here for the most part is describing your experience, which is not an explanation of that experience. The single explanatory term you use is “non-material”, which explains that you are not talking about a material phenomenon (or, at least, a phenomena generated by material causation). Both the physicalist and Platonic Realism explanations for that experience have been given; you’ve rejected both.

    Part of what Platonic Realism attempts to explain is the universality of certain aspect of mind – certain values, forms, principles, etc. That was really the meat of those first threads – how does anyone explain that any mind can discover the universality of 2+3+5? Or A=A? Or circles, pi, the equations of mass and velocity?

    Your description of your personal mental experience is not an explanatory model for the universal consensuality (and, indeed, necessity & absolute nature) of certain mental, non-material abstractions, as is Platonic Realism. Until you have an alternative model that explains these things, we are left with physicalism vs Platonic Realism.

    My theory of mind, with I discussed quite a bit with Gpuccio, is that it is something (I don’t know what, and I don’t think anyone does) that we experience “internally” through the medium of consciousness. Somehow (I don’t know how and neither does anyone else) it has a two-way interface with the external world, though my body. It is capable of, among other things, logical rational thought and the embodiment of abstractions in symbolic form which it can then manipulate to reach new conclusions.

    I think that is sufficient for me. It isn’t sufficient for you. I think we will have to leave it at that.

    I’m certainly not here to try and force it to not be sufficient for you. I’m primarily here just to develop and challenge my own thoughts and views. I do hope you will take into account, though, in future discussions with others, when they appear to lose their patience, become frustrated and start utilizing appeals to motive, character, etc., that it can be quite challenging to engage with someone who rejects explanatory models for no logical or evidential reason, offers no alternative and then ends further discussion by saying, essentially, that they are comfortable not pursuing the matter further and accepting their own internal ambiguity on these issues.

    I’m not saying there’s anything wrong with such a personally held position – IMO, there isn’t. IMO, such accepted internal ambiguity offers a certain experiential value. My point is rather that from the perspective of many that take these things very seriously, that internal acceptance of conceptual ambiguity while also rejecting other ideas that certainly do not violate any principle within that ambiguity looks like something else entirely. Something to keep in mind going forward.

  133. 133
    kairosfocus says:

    WJM, H is by her admission, a mathematician, so she understands the logic at work. That she clearly has no logical counter is significant. We can take the basic point as made: there is in any world that is possible, a substantial core of mathematical facts that will necessarily be present, starting with N and progressing therefrom. Given that historically, imaginary numbers were proposed to solve polynomials, it would seem plausible that they are an arbitrary invention — which gave rise to much resistance. However, it was later realised that we are dealing with an algebraic expression of vectors, with rotations. Consequently, they actually have a natural sense and should not be regarded as in effect an artifact of human ingenuity. All of this suffices to ground that certain key abstract entities exhibiting structure and quantity are mathematical facts that constrain what is possible of being and constrain certain relational possibilities. In a simple case, self evidently || + ||| –> ||||| . Beyond such, it is clear enough that nominalism fails, as abstract entities are inextricably entangled in reasoned thought. KF

  134. 134
    kairosfocus says:

    H, it seems that on the contrary, the core point has been sufficiently warranted that there is no effective counter. Going beyond, I took up the nominalism, addressing first its role in evolutionary materialism, which cannot be ignored. Onwards, I addressed the challenge that abstracta are so inextricably entangled in our thinking that nominalism falls apart. Further, once we see mind beyond claimed emanations of computational substrates, one deals with the abstract domain. All of this is before one gets to actual claims about mathematical platonism as such. I would suggest that the only viable views will be those that allow key facts of structure and quantity as shown, to materially affect reality by being part of the fabric of what give a world its distinct identity. KF

  135. 135
    StephenB says:

    Hazel

    Each of us have a unique set of concepts in our mind, and they vary widely in how similar they are to other people’s.

    Everyone has different experiences and sense impressions about this or that triangle, but everyone’s concept of a right triangle is the same if they know what it is.

  136. 136
    hazel says:

    True, Stephen, everyone’s formal understanding of a right triangle is the same, assuming they know the very basic definition. However, as I mentioned in 119, every concept we have, I think, has what I called a cloud of associations around it – peripheral knowledge that enriches the core concept – and those are, at least in theory, unique to the person. My guess is that I have more such associations about right triangles than you do just because I taught about them in numerous contexts for years, so when I think about right triangles there are many aspects to the thought beyond the formal definition. That is an example of what I meant when I wrote, “Each of us have a unique set of concepts in our mind, and they vary widely in how similar they are to other people’s.”

  137. 137
    Ed George says:

    Hazel, please correct me if I am wrong. Which is quite possible. But our current vector based math is based on three vectors at 90 degree to the others. X,Y,Z. But is there anything preventing the X,Y, Z coordinates being anything different from 90 degrees from each other? Certainly, it would make the mathematics more complicated. But is this mathematically impossible?

  138. 138
    hazel says:

    kf, I really don’t know why you think nominalism implies physicalism, but if that is the case, then nominalism doesn’t describe my position. I’m not interested in labeling myself with some philosophical description. I’m interested in understanding my own thoughts about things. But I believe that consciousness and the mind are different than the body, so all your talk about physicalism and “”evolutionary materialism” just doesn’t have anything to do with me.

    I keep thinking I won’t bother to say this again, because you seem to ignore me, but here I am saying it again.

  139. 139
    hazel says:

    Ed, try googling “axes not perpendicular to each other” and see what you find out. This is not a topic I have ever studied.

  140. 140
    hazel says:

    Comment deleted: On second thought, I’ll just move on.

  141. 141
    math guy says:

    Hazel, I don’t intend to be rude when I liken the discussion involving KF, WJM, and yourself to interaction with one of the many online Turing test bots. Such a bot is incapable of understanding a reasoned argument (although they are programmed to feign understanding) but responds with trivialities and changes of subject.

    Unlike a bot, I believe you ARE capable of some understanding but are wed to a philosophical position/world view which you must maintain at all costs. Your motivation for maintaining such a world view is not openly acknowledged.

    WJM on the other hand has acknowledged his honest and deep search for understanding and actual TRUTH (yet another platonic abstraction), with the implied admission to not deny the results of such a time/effort consuming search, regardless of whether such results are subjectively agreeable or not.

  142. 142
    StephenB says:

    Hazel

    True, Stephen, everyone’s formal understanding of a right triangle is the same, assuming they know the very basic definition.

    That is correct. It follows, therefore, that people do not have varying understandings of what a right triangle is. They either grasp the concept or they don’t.

    However, as I mentioned in 119, every concept we have, I think, has what I called a cloud of associations around it – peripheral knowledge that enriches the core concept – and those are, at least in theory, unique to the person.

    The individual’s “association” with the concept of a right triangle, as opposed to the concept itself, is unique because it is based on personal experience (and imagination), which is different for everyone. That goes without saying. However, the concept itself is the same for everyone; it is universal and unchanging.

    My guess is that I have more such associations about right triangles than you do just because I taught about them in numerous contexts for years, so when I think about right triangles there are many aspects to the thought beyond the formal definition.

    Other than those elements included in the definition, there are no “aspects” to the concept of a right triangle, which is a *universal* formulation that applies objectively to all right triangles and subjectively to all knowers of right triangles. There are, however, different experiences, examples, or sense impressions produced by this or that *particular* right triangle. Knowledge of concepts is about universals; sense impressions associated with concepts are about particulars.

    That is an example of what I meant when I wrote, “Each of us have a unique set of concepts in our mind, and they vary widely in how similar they are to other people’s.”

    By concept, I mean an abstract idea that can be defined and known for what it is. If you mean something else, then we are not talking about the same thing.

  143. 143
    kairosfocus says:

    EG, the classic xyz co-ordinates and ijk vectors are only one (or two) of various possible co-ordinate systems, e.g. we may use an origin, a defined polar axis and elevation plus azimuth angles — telescopes, satellite dishes, gunlaying etc. We can have right vs left hand axes, cylindrical coordinates are used etc. What is clear is that there are three degrees of freedom for location in “ordinary” space, and they need to be sufficiently mutually independent. We can use more complicated ways to capture the degrees of freedom (a different “basis”) but the underlying characteristics will remain. In effect, so long as three vectors (or relevant quantities) are mutually independent and can be combined to “cover” every point in the space, they can serve. This does include vectors that are not mutually perpendicular; you are here looking at vector spaces, a subject in so-called linear algebra, an advanced topic. Where, vectors can be generalised beyond the simple arrow with magnitude and direction familiar from school physics. Beyond, we can extend to n-dimensional hyper-spaces, though those are beyond visualising. To do so, we usually resort to algebraic representations of vectors similar to multi-axis coordinates: r = (x1, x2, . . . xn), something you may have seen in simple cases in matrix algebra in school. In statistical thermodynamics, this can extend readily to about 10^22 degrees of freedom, due to location, momentum and rotation of molecules etc. This opens up configuration spaces, state spaces and phase space, which are again useful advanced topics. BTW, this influences why I give priority to the vector-rotation view of complex numbers (which also makes them far more evidently reasonable and even natural.) . In Quantum theory, limitless — try here https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation and here https://en.wikipedia.org/wiki/Matrix_mechanics as a start. I hope this opens up enough possibilities for your exploration. KF

  144. 144
    kairosfocus says:

    H,

    pardon but above I quite explicitly spoke — repeatedly — to different varieties of nominalism, including conceptualism.

    I have explained that I indeed repeatedly first addressed the nominalistic aspects of evolutionary materialism because of ideological importance and also as it points to the wider problem. At UD and elsewhere, we must always be aware of wider context and often need to speak more broadly than the specific question or argument posed. For example in a non-controversial context see the just above to EG on vector spaces and linear algebra etc.

    On nominalism in general (see one reason I use emphases?), I point out yet again that the core problem is entanglement of the abstract in all effective thought.

    We can start with Truth; which is an abstract relationship, accurate description of reality beyond mere appearances.

    Propositions, what is asserted, affirmed or denied in truth or falsity claims, are patently abstract.

    Logical inference on entailment, implication or support [used in induction] is abstract.

    Numbers are abstract, as are ever so many other quantities and linked structures.

    Structure is an abstractum too.

    The very conception of a name denoting a collection of individual, particular cases in a set, class or more loose grouping is an abstract.

    Neither you nor I have ever touched or handled or seen: man, woman, truth, love, number, a point, length, volume, area, velocity, speed, direction, angle, size, big, small, hot, cold, lukewarm, red, blue, grue, bleen, pi, e, energy, time, momentum, mass, morality, ethics, democracy, law, government, governance, policy, politics, tree, tree-diagram, logic, proof, probability and much more.

    Nominalism thus falls apart on being requested to state its case without being forced to use entities it denies the reality of. It is forced to use or imply what it denies, once it asserts something to be so or not so, i.e. truth or falsity.

    This undeniability is a characteristic of first principles of reason, that we cannot do without them, so we literally are forced to use them in the attempt to deny them or to “prove” them.

    For example let me add: given that arguments inevitably appeal to known duties to truth, right reason, prudence and fairness etc, it is futile to try to dismiss the inextricable entanglement of IS and OUGHT, so the only viable worldviews are those that take bridging the IS-OUGHT gap seriously. This then extends to the super-class, the problem of coherently unifying the ONE and the MANY, without sacrificing the whole or the part, thus also ORDER (and intelligibility including discoverability as opposed to observability) vs CHAOS. Such issues go to the heart of many vexed problems.

    The principle of distinct identity and its corollaries LNC and LEM are in that class, with the natural numbers and extensions to complex numbers, abstract spaces and the transfinites and hyperreals to surreals as credibly necessary extensions. (I do NOT claim, self-evidence for such extensions; a SET is true, is seen as necessarily true by those with the experience-base to understand, and are so on pain of immediate, patent absurdity on the attempt to deny.)

    In this series, I have already explored how thinking in terms of archetypes in common to different entities and onward distinct features allows us to see how distinct identity operates on entities that can be grouped. And yes, computational objects and inheritance are very much in view. From this, for example, I saw how in inductive, observational exploration, if we come across enough to form a picture of the in-common, we can justify a generalisation from a core of cases to other ones that are sufficiently similar. In short, induction is an exercise in identification and extending properties or characteristics on close enough family resemblance. This is also tied to the legitimate use of analogy, which turns out to be a facet of inductive reasoning, argument by support. Likewise, abductive reasoning is of inductive form. Induction is the vast majority of our reasoning.

    I trust this will help us break through the resistance created by today’s intellectual climate so that we can break out into a new abstract space of logic and first principles of reasoning. And, pardon a military metaphor. Such is not necessarily toxic but may be unwelcome. Just as, masculinity.

    KF

  145. 145
    kairosfocus says:

    SB & H: denotation vs connotation is relevant. Objective truth and the reality we thereby access, will lie in the main in what is denotation. A right angle triangle is by easiest formal definition, any triangle that stands on the diameter of some circle and has third vertex along the arc of circumference of the circle. The diameter will be its hypotenuse and the ends of the diameter will have complementary angles phi and psi summing to a right angle. The angle on the circumference will perforce be a right angle. This approach gets out of the seeming arbitrariness of choosing one class of triangles as special, which — in an age haunted by radical subjectivism and relativism taught to use the hermeneutic of suspicion towards any authority that is not fashionable — is important. In instruction, we can set up a semicircular body, maybe plywood, use a string fixed at A on the diameter and run it out to B on the arc then down to C on the other end of the diameter, with the ball of further string beyond then invite students to measure angles and line lengths. The angle sum triangle relations, complementarity, right angle opposite the hyp, Pythagorean relations and more will be empirically evident. Given the above and exchanges over the years, I am beginning to think this should be an early exercise in physics, exploring properties of our apparent euclidean space. Power cut, pardon the late edit. KF

    PS: I begin to think that those inclined to believe that mathematical properties are cultural creations may profit by creating such an exercise. Use a diameter of five units and set up a case where AB is three units. H’mm, if AC is five inches, using one-foot rulers will be convenient. See how BC = 3 units naturally emerges. Notice, 3 + 4 + 5 = 12. Try out 60 and 30 degree set square triangles. Run a cord along the arc of the half-circle and measure, taking ratio to diameter. Ask what happens when B and C coincide or nearly coincide. And more. I think this will do more to blow up some intellectual strongholds than almost anything else I know. Next exercise: set up a hexagon inscribed inside a circle with vertices at 60 degree points and explore the world of relationships — I dare you.

  146. 146
    kairosfocus says:

    F/N: It is instructive to see Wikipedia’s struggle with the concept, proposition:

    A proposition is a tentative and conjectural relationship between constructs that is stated in a declarative form. An example of a proposition is: “An increase in student intelligence causes an increase in their academic achievement.” This declarative statement does not have to be true, but must be empirically testable using data, so that we can judge whether it is true or false. Propositions are generally derived based on logic (deduction) or empirical observations (induction). Because propositions are associations between abstract constructs, they cannot be tested directly. Instead, they are tested indirectly by examining the relationship between corresponding measures (variables) of those constructs. The empirical formulation of propositions, stated as relationships between variables, is called hypotheses [1]. The term proposition has a broad use in contemporary analytic philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other “propositional attitudes” (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of declarative sentences. Propositions are the sharable objects of attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens which are not sharable, and concrete events or facts, which cannot be false.[2]

    Collins English Dict: >>2. countable noun [oft NOUN that]
    A proposition is a statement or an idea which people can consider or discuss to decide whether it is true.
    [formal]
    The proposition that democracies do not fight each other is based on a tiny historical sample.>>

    Merriam-Webster: >> 2a : an expression in language or signs of something that can be believed, doubted, or denied or is either true or false
    b : the objective meaning of a proposition>>

    OED: >>proposition
    noun

    1A statement or assertion that expresses a judgement or opinion.
    ‘the proposition that high taxation is undesirable’

    1.1Logic A statement that expresses a concept that can be true or false.

    1.2Mathematics A formal statement of a theorem or problem, typically including the demonstration.>>
    KF

  147. 147
    ET says:

    And still no evidence that humans invented mathematics. No wonder hazel thinks this is a philosophical issue (which it isn’t but hazel cannot find anything that supports its claims)

  148. 148
    hazel says:

    No one here believes “that mathematical properties are cultural creations.” Another large mischaracterization.

  149. 149
    hazel says:

    Stephen writes, “By concept, I mean an abstract idea that can be defined and known for what it is. If you mean something else, then we are not talking about the same thing.”

    How about the concept of “tree”? Does your sentence above apply?

  150. 150
    hazel says:

    Math guy writes, “Hazel, I don’t intend to be rude .”
    Hmmm. I think you are coming across as rude, though, FWIW.

  151. 151
    ET says:

    hazel:

    No one here believes “that mathematical properties are cultural creations.”

    So what are they, then? In your opinion

  152. 152
    hazel says:

    Properties of logical symbolic systems.

  153. 153
    ET says:

    hazel:

    Properties of logical symbolic systems.

    According to hazel, mathematical properties are properties of logical symbolic systems.

    What does that mean? Were those properties invented by us, discovered by us or what?

  154. 154
    hazel says:

    I’ve answered those questions several times, ET, but you don’t pay attention nor remember. If you look back at previous threads you can find out what I’ve had to say.

  155. 155
    ET says:

    hazel, From what I have read you don’t have anything to say- anything of substance, anyway.

    Thank you for not disappointing.

  156. 156
    hazel says:

    Then why did you ask again?

  157. 157
    kairosfocus says:

    H, do you wish to revisit the issue of complex numbers as an example of why the invented vs discovered issue is key, on a relevant core. The vector-rotation approach is a key bit of relevant evidence, brought to your attention for weeks now. The history on solving polynomials is not the whole story, and it shows how what we think we are inventing artificially may actually reflect something that has a very natural manifestation. I note that I have highlighted that our study, symbol systems, base of numerals and much more are culturally influenced so are invented, but these respond to a base of antecedent facts embedded in reality, in key, core part. That includes how we get to the Naturals and a chain up to C, thus space and spatial properties. Where all of these are abstract but real and reality-constraining entities. I suggest that the trend of your argument has not been misconstrued or misrepresented. The core disagreement seems to be that you have persistently tried to delimit Mathematics to the study, and have — while occasionally and too often vaguely admitting that there are world embedded structures and quantities — sought to define Mathematics as the discipline. I and others have pointed out the substance, which is in key part demonstrated and unanswered implying that there is no credible refutation. Once such abstracta have reality as world-embedded entities, that is enough to bring out that they are real and can be understood as standing independent of our cultural discovery, symbolism and recognition. Further, the nominalism you attempted to champion fails, as was again summarised but so far side stepped again. I can say that as for days the reason why the failure is so was explained but you stepped aside to talk about how you are not a materialist. Why the materialistic view has had to be addressed was explained and the wider problems with nominalism were pointed out. The onward conclusion unless you give cause to infer otherwise, is that you do not have an answer as to how nominalism can be expressed without resort to the abstracta it denies. KF

  158. 158
    ET says:

    hazel- I was just checking if your cowardice was still in place. It is.

  159. 159
    hazel says:

    kf writes, “H, do you wish to revisit the issue of complex numbers?”

    No thank you, kf.

  160. 160
    hazel says:

    ET, you’re a petty guy.

  161. 161
    kairosfocus says:

    H,

    I can take it as effectively conceded then that the vector-rotation- complex exponentials and power series approach sufficiently demonstrates that C is a set that fits in with major natural phenomena. That is, there are world-embedded structures and quantities that allow us to use numbers of form z = x + i*y, x and y real, to address vectors, rotations, extended exponential phenomena, frequency and transient response etc. Almost as a side effect i*i*x = -x so i^2 –> sqrt (-1).

    Indeed, I would argue that when we look at an expression like y = 3x + 1 and set out to plot a graph, noting slope, y intercept and x intercept our convention of assigning axes to x and y is an implicit vector approach. Indeed the coordinates of a point (x,y) are a position vector.

    So, we are already looking at vectors in a plane and can apply the ijk extension of the i* operator — yes, that shadow of the Quaternions — to see that vectors were there all along. Going further, integers are vectors one dimension vectors and addition and subtraction of integers are vector operations of stepping right or left from a start position. That is indeed what we do when we use the number line or the counting on or back approach to simple addition or subtraction.

    All of this brings us back to how there are considerably many world-embedded intelligible, discoverable rational principles of structure and quantity that provide a rich, naturally occurring motherlode of the substance of Mathematics.

    A motherlode that is full of world-embedded abstracta that have massive ontological effects.

    Things that we have traced back to the numerical implications of the principle of distinct identity.

    KF

  162. 162
    hazel says:

    kf writes, “I can take it as effectively conceded.”

    No, you cannot.

  163. 163
    hazel says:

    To be clear, kf, about the part I don’t concede, it’s the “full of world-embedded abstracta that have massive ontological effects” part.

  164. 164

    Hazel:
    “Just when I thought I was out, they pull me back in!”

  165. 165
    hazel says:

    That’s a good point, wjm, although not actually a real quote from me: I’m having trouble letting go, but I’m winding down. 🙂

  166. 166
    kairosfocus says:

    H, with all due respect, at this stage mere disagreement is not good enough. It is demonstrated that on the distinct identity for any particular world to be, the naturals necessarily follow, thence Z, Q, R and C. These are abstract, they are necessary entities and are part of the framework of any possible world. There is no world where || + ||| –> ||||| does not obtain. That is part of the logic of being for a world to possibly be. Logic of being being a phrase that expands ontology. There are necessary, rationally intelligible structures and quantities that must obtain in any world, with direct import for all sorts of things. For example, in any world, we may use R and i*R to construct a flat conceptual, abstract space. In that space we can have r^2 = x^2 + y^2, defining a circle centred on the origin. In such a circle we may freely posit a triangle standing on the diameter where the oX is cut at +r [B] and -r [A], with the third vertex somewhere along the upper arc, say, C. The angle at C MUST be a right angle and those at A and B MUST sum to a right angle. Pythagoras’ theorem will obtain on the three sides of the triangle, there will be all sorts of trig relationships and more. Those are logic of being phenomena which are necessary and obtain in any possible world. If we were to define some unit so that AB were five units long and AC 3, BC must be 4 and the other angles are specified. If the world had plywood in it, we could build such a triangle and within the precision of manufacture such would obtain. Indeed, we could copy the Egyptians and mark twelve equal length — handy abstraction, length — segments of rope. This would specify a 345 triangle and could be used to square up a building or a fence etc. Much the like goes on and on. The ontological import of structural, quantitative abstracta is plain. And of course, it is still an unmet challenge to assert nominalism without implicitly using abstracta. Starting with, I believe nominalism is true. Believing is abstract, non-concrete, nominalism is an abstract label for an abstract concept, and truth is an abstract relationship between the intent of a claim and actual reality; what is. KF

  167. 167
    Ed George says:

    Hazel

    That’s a good point, wjm, although not actually a real quote from me: I’m having trouble letting go, but I’m winding down. ????

    I don’t know how you lasted this long. I do not have that much patience or tolerance.

  168. 168
    hazel says:

    kf writes, “with all due respect …”
    It’s not at all clear how much respect you think is due., FWIW, so I’m not sure what that opening phrase means.

    But it baffles me why you keep retyping the same things over and over. You don’t need to tell me about the Egyptians and the ropes with 12 knots ever again, or about the use of vectors: I used to teach both those things as examples of how math could be applied to the real world.

  169. 169

    Hazel,

    It was a famous quote from Al Pacino’s character in Godfather, Part 3.

    The point KF is making is that this is not about whether or not you are convinced or agree, but rather that you’ve provided no logical or evidential rebuttal to the argument for platonic realism. Leaving the conversation (debate) without any such rebuttal is, in effect, conceding the point that you have have no such counter-point or rebuttal to the logic/evidence provided. You’re not conceding that you agree or that you have changed your views, but rather that you cannot effectively counter the argument on the table for Platonic Realism. You essentially agreed back in another thread when you said “I cannot accept that ….” about the ramifications of Platonic Realism but offered no logical counter.

    Just because you cannot accept Platonic Realism doesn’t change the fact you have conceded the debate by (1) not countering the argument on the table with a logical rebuttal, and (2) walking away from it. It’s like walking off from a chess game after the opponent has made a move. That’s conceding the game whether or not you accept that their move actually wins the game.

  170. 170
    Brother Brian says:

    WJM

    Leaving the conversation (debate) without any such rebuttal is, in effect, conceding the point that you have have no such counter-point or rebuttal to the logic/evidence provided.

    I have not read all the comments on this thread but from the few I have read, Hazel appears to have provided ample argument and rationale to support his philosophical views. Leaving a discussion that has run its course and is going nowhere is not conceding anything. It is just a case of effective time management.

  171. 171

    Brother Brian said

    I have not read all the comments on this thread but from the few I have read…

    Unless you have read the entire run of the debate which has gone on through several threads and argued by the same participants in each, you really have no basis for making any worthwhile judgement whatsoever.

  172. 172
    Brother Brian says:

    WHM

    Unless you have read the entire run of the debate which has gone on through several threads and argued by the same participants in each, you really have no basis for making any worthwhile judgement whatsoever.

    You are certainly entitled to your opinion. But I have read enough to spot juvenile piling on simply because someone has a different philosophical viewpoint. And I have read enough of Hazel’s comments to know that she presents logical arguments in spite of mischaracterizations of her view by people who disagree with her.

  173. 173
    Ed George says:

    BB

    But I have read enough to spot juvenile piling on…

    KF has certainly mischaracterized Hazel’s arguments, but I am sure that it is an innocent misunderstanding. To call it “juvenile” is uncalled for and an apology should be offered.

  174. 174
    hazel says:

    Thanks, Brother Brian. I think if you read even more, going back to the first post in this series of posts, you would continue to feel like supporting me. Obviously other people’s mileage differs.

    wjm, this has been a discussion, not a debate. I don’t see discussions as zero-sum games. This was not like a chess game that I choose to play until the end.

    I have benefitted quite a bit from the discussions. Whether kf or anyone else feels they have benefitted from the discussion is up to them. At a certain point, a bit back, I decided (for about the third time) that it wasn’t worth my time to continue to go around and around over the same ground, especially given some of the negative comments made about and to me, and the seeming inability or unwillingness for some to even characterize what I’ve said accurately.

    That is not “conceding the debate.” That is, as Brian said, making a reasonable judgment, from a cost/benefit point of view, about whether I should continue or not.

  175. 175
    hazel says:

    Ed, I agree that “juvenile” probably doesn’t apply. However, I think enough other more or less uncalled negative statements have been made about people (see 141 and 158, for instance) that calling out Brian probably isn’t necessary. My 2 cents.

  176. 176
    ET says:

    Strange that we are still waiting for any evidence that mathematics was invented and not discovered. On the other hand there has been evidence presented for mathematics being discovered- because the universe was designed using mathematics

  177. 177
    hazel says:

    This isn’t evidence, but it is the word of someone commonly quoted, as kf did in 126: Eugene Wigner, the author of “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”.

    Wigner wrote, and kf quoted,

    mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts.

  178. 178

    Hazel,

    I understand your position and how you feel about it, which is perfectly fine. However, whether or not you feel like you have conceded, and whether or not you feel like it’s a discussion rather than a debate, isn’t the point of what I said. If you don’t want your part of the “discussion” to be taken as a challenge to debate based on logic and evidence, and how you proceed in that “discussion” to be potentially taken by those who engage you as a de facto concession, you’re in the wrong place. That’s not an admonishment or a complaint, that’s a heads up.

    BTW, Brian called himself out by admitting:

    I have not read all the comments on this thread but from the few I have read …

    Note further, Brian says:

    I have read enough of Hazel’s comments to know that she presents logical arguments…

    … interesting, since Hazel says just afterward:

    this has been a discussion, not a debate. I don’t see discussions as zero-sum games.

    Note, arguments are, essentially, zero-sum games. You either rebut a logical criticism, or you lose the argument. You either support your position logically and evidentially, or you have in effect conceded the point. You either provide a logical criticism that cannot be effectively rebutted, or you’ve lost the game. In such scenarios, “I cannot accept that … ” and “I am unconvinced.” are concessions of defeat in the argument.

    IOW, Hazel (consistently, I might add, throughout the multi-thread discussion) has repeatedly stated she is not engaging in an argument or a debate, but rather is exploring her own experience of mind, thought, physicality, etc. Therefore, she is not making logical rebuttals against the views of others, or even making a positive logical/evidential case for her own position (which would necessarily include defense of criticisms).

    And, this has actually been the case throughout the discussion – Hazel repeatedly informing others that she is not obligated to provide a logical/evidential argument or rebuttal when challenged to do so, or when others complain about it or characterize her negatively for NOT providing such arguments rebuttals, (Correct me if I am mischaracterizing your position/statements/sentiments, Hazel) because Hazel is engaging in a discussion while others, such as KF, are engaging in an argument/debate.

    This is why you really should have either stayed silent or read more than “a few” comments, Brian. You’re uninformed. Hazel hasn’t been making “logical arguments” for or against anything. She’s been expressing her perspective, her views, and her thinking about her mind/physicality experience. Expressing your views for discussion is not “making logical arguments.”

  179. 179
    hazel says:

    I don’t think you’ve done a very good job of summarizing the situation, wjm. I have provided “logical arguments and evidence” in my thoughts, both for the position I have been taking and against some of the ideas of other. Having a discussion rather than a debate doesn’t mean that evidence and logic aren’t being used.

    However, the subjects we have been discussing, as I have repeatedly pointed out, are perennial issues about which different philosophies, and different premises, exist: they are not issues about which logic and evidence can conclusively resolve. Nobody is going to settle these issues in a discussion forum among a few people on the internet.

    P.S. Also, wjm, I have responded to criticisms. However the bulk of the criticisms I have read have mischaracterized my position, or just said I didn’t know what I was talking about. Relevant criticisms have been in the minority, I think.

  180. 180
    ET says:

    hazel- with everything we have presented to you what you have presented in 177 is nothing short of a joke.

    Your quote-mine doesn’t seem to be supported by anything

  181. 181
    ET says:

    The only evidence that could possibly show that mathematics does not permeate the universe (meaning we merely discovered it) is that in support of materialism- nature producing itself just because it did and here we are.

    Any evidence for the intelligent design of the universe has mathematics permeating said universe because mathematics was used to design and implement that design of the universe. You can disagree with that but then you would have to show that such a design feat could be had in the absence of mathematics. Good luck with that.

    Then there is that math genius from India hazel wants to ignore- the evidence mathematics is discovered…

  182. 182

    Hazel said:

    I have provided “logical arguments and evidence” in my thoughts, both for the position I have been taking and against some of the ideas of other. Having a discussion rather than a debate doesn’t mean that evidence and logic aren’t being used.

    Careful, Hazel. You can’t have it both ways. Yes, one can use logic and evidence in a discussion, but if you’re now going to say that you’ve provided logical arguments for and against propositions and criticisms, then you are subject to the rules and expectations of what it means to engage in a logical argument. You don’t get to argue up to a point, then say “I can’t accept that …” or “I’m finished with this conversation” and expect it to not be taken as a concession that you have no logical rebuttal against the point being made.

    Perhaps you want it to be a discussion when it suits you, and a logical argument when it suits you? What determines when we’re supposed to react as if it’s just a discussion, or when you are pursuing a logical argument? Can you see how that might be frustrating for your interlocutors when you seem to be engaging in a logical argument, and then turn around and say this is just a discussion where you are not compelled to either rebut a criticism with valid argument or concede the point?

    However, the subjects we have been discussing, as I have repeatedly pointed out, are perennial issues about which different philosophies, and different premises, exist: they are not issues about which logic and evidence can conclusively resolve. Nobody is going to settle these issues in a discussion forum among a few people on the internet.

    That’s a very interesting, very broad set of philosophical assertions from someone who, if I remember correctly, has said they are not a philosopher, is not that interested in philosophy, and who (if I remember correctly) doesn’t know that much about it. Would you care to back any of those assertions up? Would you care to explain what brought you to these conclusions?

    Or, are you just saying things you really have no idea about whether or not they are valid or supportable in order to provide justification for your fence-sitting between “logical argument” and “discussion” (which, to me, appears to be a matter of convenience when you are faced with logical points you have no logical rebuttal against, or when you are out of your depth.)

  183. 183
    hazel says:

    182 illustrates my P.S, at 179, and supports my last paragraph at 174.

  184. 184
    Brother Brian says:

    Hazel

    182 illustrates my P.S, at 179, and supports my last paragraph at 174.

    Yes, they certainly do.

  185. 185
    kairosfocus says:

    H,

    Pardon, but I am by no means committed to accept all Wigner said in his famous paper.

    I cited (and linked) so we can have context of the famous argument by a Nobel prize winner.

    In that context, I took it for granted that arguments already present across several threads and above would suffice to see that Wigner spoke to the study of Mathematics, and was astonished to see how that study seemed so aptly fitted to address physically instantiated empirical reality.

    Wigner and others of that sort, are my primary context.

    The answer is, that Mathematics is of dual character. There is a study, one that is shaped by a world-embedded substance, intelligible rational principles of structure and quantity present in the observed world.

    Further to this — and you were found in disagreement and/or not wishing to take context from findings on the subjects, logic of being and possible worlds [which I and many others find to be highly relevant and powerful] — we can see that just from the distinct identity necessary for a particular world to be possible, distinction must be, so we may partition W = {A|~A}. From this, immediately we see duality, unity and nullity as has been repeatedly drawn out and which you seem to find as an irrelevant repetition of things you have “answered” by trying to impose a datum line of exclusion or disagreement without provided counter-warrant.

    However, I for cause view such a response is unresponsive to the logical force and cogency of the matter on the table. And, I am not simply being idiosyncratic, puerile, juvenile, personally hostile or the like. Pardon if such words will stir negative reactions, but given onward remarks I find it regrettably necessary to indicate precisely why the sorts of objections or answers repeatedly made fail to cogently counter the core issue.

    I continue, summarising. The von Neumann construction of succession on order types applies and from 0, 1, 2 we can proceed limitlessly. I find JvN gives substance to Peano’s succession. Thus, we have the naturals as abstract quantities that must be present, embedded in the logical fabric of what makes a distinct world distinct.

    With N [in the form that includes 0] on the table, additive inverses give us Z as has been outlined already. Ratios give us the rationals, power series sums — the underlying framework of place value notations — give us the reals, thus continuum and therefore space. The vector and rotation operator i* gives us i*R, orthogonal to R so that we have complex numbers as vectors with rotational properties. The ijk extension gives us a 3-d space and in fact we here echo Quaternions and Octonions (I never thought that obscure subject would have to be brought up!) and onward n-dimensional vectors and structures such as matrices and tensors. Thus, with N on the table as embedded in the distinct identity of any given world, we see a panoply of Mathematical facts that are in material part necessary, abstract entities embedded in any possible world. Where, as this is tied to the logic of being, we find it plausible that we should see such reflected in physically instantiated entities. Which are the objects studied in the physical sciences.

    This is the substance I have spoken of and which point you have often demurred and set aside, refusing to reckon with.

    I further find that were I to go back to the physics classroom, I would introduce the triangle in a semicircle lab exercise to stamp vividly into the minds of students the point that there are intelligible rational principles manifest in space itself. Laws of reality that show the embedded substance of structure and quantity in action.

    I would also take a far more forceful view of parallelogram of forces lab exercises and the inferring of orthogonal basis vectors that span space.

    Coming back to Wigner, I trust you can see why I conclude that he has overlooked the other side of Mathematics, to the extent that he has emphasised the culturally influenced study.

    Going further, I have noted of axiomatic systems, that they came along in the context of established core mathematical facts, which therefore subtly constrained how they are set up. This even holds for Geometry, e.g. ponder the 12-segment rope 345 triangle trick used by the Egyptians etc to establish a right angle for practical purposes. I have repeatedly discussed gear trains.

    Coming forward, ET is of course correct to point to Ramanujan and his explorations. This is an empirical case in point of independent discovery by in effect an initially isolated lone genius. If only he had enjoyed a long life!

    I think I should again point to some Dictionary definitions as showing that it is not idiosyncratic to hold that Mathematics has dual character — substance and culturally influenced study — as the (study of the) logic of structure and quantity:

    AmHD: The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.

    Collins: a group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape, and space and their interrelationships by using a specialized notation

    RHK Webster: the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.

    Merriam Webster: the science of numbers and their operations (see operation sense 5), interrelations, combinations, generalizations, and abstractions and of space (see space entry 1 sense 7) configurations and their structure, measurement, transformations, and generalizations Algebra, arithmetic, calculus, geometry, and trigonometry are branches of mathematics.

    Cambridge: the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them

    Wikipedia: Mathematics (from Greek ?????? máth?ma, “knowledge, study, learning”) includes the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5]

    Mathematicians seek and use patterns[6][7] to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

    Rigorous arguments first appeared in Greek mathematics, most notably in Euclid’s Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[8]

    Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics, or mathematics for its own sake, without having any application in mind. Practical applications for what began as pure mathematics are often discovered.[9][10]

    Hom, Livescience: Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.

    Since the beginning of recorded history, mathematic discovery has been at the forefront of every civilized society, and in use in even the most primitive of cultures. The needs of math arose based on the wants of society. The more complex a society, the more complex the mathematical needs.

    The above suffices to draw out the dual character I have so often spoken to.

    Going forward, I find a short essay on foundations of Mathematics stimulating:

    Mathematics is the study of systems of elementary objects, whose only considered nature is to be exact, unambiguous (two objects are equal or different, related or not; an operation gives an exact result…). Such systems are conceived independently of our usual world, even if many of them can resemble (thus be used to describe) diverse aspects of it. Mathematics as a whole can be seen as «the science of all possible worlds» of this kind (of exact objects).

    Mathematics is split into diverse branches, implicit or explicit frameworks of any mathematical work, which may be formalized as (axiomatic) theories. Each theory is the study of a supposedly fixed system that is its world of objects, called its model. But each model of a theory may be just one of its possible interpretations, among other equally legitimate models. For example, roughly speaking, all sheets of paper are systems of material points, models of the same theory of Euclidean plane geometry, but independent of each other. . . . .

    The content of a theory, describing its model(s), is made of components which are pieces of description (concepts and information, described in 1.3). A theory starts with a choice of foundation made of a logical framework and an initial version of its content (hopefully rather small, or at least simply describable). The components of this initial version are qualified as primitive.

    The study of the theory progresses by choosing some of its possible developments : new components resulting from its current content, and that can be added to it to form its next content. These different contents, having the same meaning (describing the essentially same models), play the role of different presentations of the same theory. Any other possible development (not yet chosen) can still be added later, as the part of the foundation that could generate it remains. Thus, the totality of possible developments of a theory, independent of the order chosen to process them, already forms a kind of «reality» that these developments explore.

    To express the properties of its models, the content of a theory includes a list of statements, which are formulas meant as true when interpreted in any model. Primitive statements are called axioms. Further statements called theorems are added by development to the content, under the condition that they are proven (deduced) from previous ones : this ensures them to be true in all models, provided that previous ones were. Theorems can then be used in further developments in the same way as axioms. Other kinds of developments (definitions and constructions) which add other components beyond statements, will be described in 4.8 and 4.9.

    A theory is consistent if its theorems will never contradict each other. Inconsistent theories cannot have any model, as the same statement cannot be true and false on the same system. The Completeness theorem (4.6) will show that the range of all possible theorems precisely reflects the more interesting reality of the diversity of models, which indeed exist for any consistent theory.
    There are possible hierarchies between theories, where some can play a foundational role for others. For instance, the foundations of several theories may have a common part forming a simpler theory, whose developments are applicable to all.

    A fundamental work is to develop, from a simple initial basis, a convenient body of knowledge to serve as a more complete “foundation”, endowed with efficient tools opening more direct ways to further interesting developments . . .

    We see here the abstract, logic model possible world approach I have pointed to several times. I simply note that structure and quantity are central to the subject matter and that there is a considerable body of core mathematical facts that shape onward discussion, including construction of axiomatic systems.

    KF

    PS: If one demurrs from cogently countering a substantial body of rational warrant, one implicitly concedes the point, even if one objects. In the end, warrant must turn on the force of the merits on fact and logic, of course including self-evident first truths. I note here that nominalism is on the table as being forced to implicitly appeal to precisely the sort of abstracta it would deny. As I pointed out, that we cannot avoid phenomenon is a characteristic of first principles. That is why for example we are forced to accept LOI, LNC, LEM, and why they are unprovable first points from which proofs hang. That is, so soon as we argue and articulate the claims we have, we are already using them.

    PPS: H, it will probably be distasteful to you but in the sort of highly polarised context we face, I must note that BB has popped up out of nowhere in particular, late in a thread as a pseudonymous support character who has not made the sort of substantial contributions say MG has, but is in effect voting for the objector party while making dismissive rhetorically presented claims about those s/he ostensibly supports — “brother” carries a connotation which given lines of argument by NCSE, ACLU, Wikipedia and many other sources already carries a huge rhetorical load. Where, it would be easy to show that many of his/her remarks are materially false, perhaps as s/he has not sufficiently followed the discussion and may not grasp the relevance of say the natural numbers then Z, Q, R, C etc being rooted in the requisites of distinct identity of any possible world. Namely, it is demonstrated from first principles that such are part of the fabric of any possible world. What remains is to discuss how, what sort of existence such entities have. A clue being, that we find pervasive intelligible rational principles embedded in the fabric of any possible world. Real abstracta that shape how a world and its contents can be, interact, manifest themselves etc. Where, abstracta are characteristically objects of contemplation of mind. I do not instantly conclude that s/he is a sock-puppet or concern troll, but I do draw attention to the sort of pattern that also applies to EG etc. Where, in the case of a CT, there will be what intel agencies call a legend. Yes, you took umbrage earlier at my calling attention to this fact of life at UD or in the like domain, but it is a reality we have had to squarely face ever since Patrick May’s Mathgrrl hoax seven years ago. He publicly took credit, I can freely make the identification. The modus operandi is of course instructive. Those who chose that path must know that they permanently tainted the atmosphere, and frankly, I suspect that was part of the intent. Beyond, lie full orbed agit prop operations, lawfare and deplatforming censorship, delegitimisation of elections and referenda etc. For example, like him or lump them, Mr Bush Jr, Mr Obama and Mr Trump all won successive legitimate electoral mandates and Brexit as well as rejection of Scottish secession are referendum results — those are bare facts. Those who are now playing the dangerous power games across our civilisation do not seem to care that they are pushing our civilisation ever closer to the crumbling edge of a cliff. In the case of the USA, it is already at cold civil war. A warning, with nukes potentially on the loose as happened when the USSR collapsed: did some, or materials and know how get into Iran? Coming back to UD, the lesson is, if you intervene in a discussion in a context like UD, make a substantial contribution, by direct argument, citation or linking.

  186. 186
    kairosfocus says:

    PPPS: BB is popping up on multiple threads, using similar approaches. That further goes to the pattern. S/he would be well advised to make studied, substantial contributions and to ponder the weak argument correctives. I further draw this out to sensitise to the patterns and tactics we too often have to deal with.

  187. 187
    kairosfocus says:

    the log in system just ate a comment, another defect of the new system.

  188. 188
    kairosfocus says:

    BB,

    I summarise what was lost.

    IIf you trace back you will see logical demonstration of the way natural counting numbers etc are inevitable parts of any possible world. That is already decisive, there are abstracta in any possible world, in the fabric for them to have distinct identity.

    In absence of cogent refutation, that stands, where such things are inherently abstract. the question is how they come to be, and that is something that is at root of reality.

    As abstracta are mental, that points to contemplation of mind at root of reality.

    To come up with an alternative, one has to actually lay it out and justify on comparative difficulties. Harder to do than to say.

    When it comes to nominalism, it is sufficiently shown that it must implicitly appeal to the reality of the abstracta and universals it would deny. It collapses.

    KF

  189. 189

    KF,

    This is why I don’t allow those types of behaviors (cheerleading, sniping, etc.) in my threads. It derails meaningful debate and conversations. IMO, it’s just easier to use editorial control to delete. IMO, “exposing them” is just giving them the reaction and attention they crave. I mean, once a person outs themselves as a cheerleader/sniper with nothing of substance to contribute, why not just ban them completely?

  190. 190
    kairosfocus says:

    Not without due warning.

  191. 191
    kairosfocus says:

    Also note, the OP is about fallacies.

  192. 192
    ET says:

    Strange that we are still waiting for any evidence that mathematics was invented and not discovered. On the other hand there has been evidence presented for mathematics being discovered- because the universe was designed using mathematics.

    hazel’s quote-mine in 177 does not help.

  193. 193
    kairosfocus says:

    ET, obviously, numeral systems [binary, octal, decimal, duodecimal, hexadecimal, sexagesimal come to mind], symbols, choice of parameters to use as main constants, a lot of modelling, even precise formulation of axioms etc are culturally conditioned and invented. However, they are created in response to structures, quantities, relationships, facts etc that are embedded in reality. In effect you can’t have a study without a substance and that substance needs to be in key parts anchored in reality. KF

  194. 194
    kairosfocus says:

    H, 177 cites me clipping Wigner:

    >>mathematics is the science>> = study

    >> of skillful operations>> = relationships, so what is relating to what

    >> with concepts and rules invented just for this purpose. >> –> that act on what and why are they going to be acceptable . . . accountable before core facts and coherence etc

    >>The principal emphasis is on the invention of concepts.>> –> In the study. What is the substance being studied, starting with N? Besides, concepts are at least as often recognised as invented.

    KF

  195. 195
    ET says:

    kairosfocus:

    ET, obviously, numeral systems [binary, octal, decimal, duodecimal, hexadecimal, sexagesimal come to mind], symbols, choice of parameters to use as main constants, a lot of modelling, even precise formulation of axioms etc are culturally conditioned and invented.

    I do not think that is true. I think we are just conduits for information that is available for conscious agents to tap in to. All the information flows from The Source, ie The Intelligent Designer. Information can neither be created or destroyed (by us).

    That is my PoV from reading the people I have quoted and especially from Srinivasa Ramanujan.

  196. 196
    kairosfocus says:

    ET, I had several Indian profs. It was interesting to see how they wrote Indo-Arabic numerals, especially the cipher or zero, like a tiny “o” almost a dot. We might argue that binary digits are natural, but decimals are not. Duodecimals and sexagesimals, too. BTW, we see traces of that in time keeping, the foot and in angle measures. KF

  197. 197
    ET says:

    That is very interesting, kairosfocus. Thanks for sharing.

  198. 198
    kairosfocus says:

    Forgot, my HP 50 offers the choice between commas and dots for the decimal marker. And, there is a choice of suspended vs on the line dots for writing. Do you close or open the 4? Loop the 2? Put the 7 half below the baseline? Use a crossbar on the 7? How do you write deltas? Do you use Newton’s dashes and dots or dy/dx? Do you do a dot dx when you write an integral? How do you write zeta? Nu? Etc . . .

  199. 199

    KF,

    A similar case can be made for an appeal to consequences. It depends on the consequence. For example, I described your appeal to consequences as fallacious – perhaps, technically, it is, but the consequence in question would invalidate the entire line of reasoning if it was, in fact the only possible or most likely consequence. So, it was something that did require rebutting.

  200. 200
    kairosfocus says:

    WJM, there is a difference between I don’t like consequences or logical consequents and either a slippery slope with a ratchet heading for a cliff or a reduction to manifest absurdity or self-referential incoherence. For case in point p => q, But if there is excellent reason for ~q then ~p. So, what is the quality of your ~q, and what alternatives to p do you have that are credible and don’t end up at q? KF

  201. 201
    StephenB says:

    SB: By concept, I mean an abstract idea that can be defined and known for what it is. If you mean something else, then we are not talking about the same thing.”
    Hazel:

    How about the concept of “tree”? Does your sentence above apply?

    Absolutely. You know what a tree is because you understand the universal concept of treeness. Without the intellectual concept of a universal, you would have only your sense impressions of a particular tree, which cannot inform you that all trees have something in common—their category of being. Your experience of an individual tree, its color, size, shape and so on, are the product of your sense impressions and cannot, without the aid of a universal concept, inform you about what it is you are experiencing.

  202. 202
    hazel says:

    Thanks, Stephen. This leaves me with questions. As you said earlier, right triangles have a precise definition so that everyone who knows that definition will have the same concept – a universal concept of right triangle.

    But “tree” doesn’t have a universally shared precise definition. I have created an understanding of “treeness” based on my various experiences of trees where I live, but someone who lives in the Amazon has had significantly different experiences of trees, so his concept and mine are not identical. Therefore, I don’t see what “universal concept” of tree would mean.

    So, it’s not the a matter of the sensory experience of a single tree, as that is not enough to create an abstraction, I don’t think. But my abstract idea of “tree”, formed from the experience of many trees, some of which are very different than other trees (or might not even be clearly a tree, as opposed to a bush or shrub, for instance), will not be the same as the guy in the Amazon.

    So in what sense is there a “universal concept” of tree when there is no logically precise definition, as there is with a right triangle, and the sensory experiences which inform our creation of the mental concept “tree” can be very different than someone else’s?

    Thoughts on these questions?

  203. 203
    kairosfocus says:

    H, someone from Amazonia taken to say Finland will readily recognise trees there as trees. But in any case, we are dealing with numbers, shapes and the like. KF

  204. 204
    hazel says:

    Kf, Stephen and I are talking about more than “numbers, shapes and the like”

    But you write, “Someone from Amazonia taken to say Finland will readily recognise trees there as trees.”

    I think it is not that simple. For instance, I have some Lonicera morrow in my backyard. This is technically a shrub, but the average person would might easily call it a small tree. Similarly, my guess is that Amazon natives have a variety of words to cover really big trees, small understory trees, shrubby trees, etc. I seriously doubt we and they and someone from Finland have categories that overlap exactly.

    There is not, I’m willing to bet, a universal clearcut category that all human beings, no matter where they live, would agree upon is a “tree”, as opposed to a “bush” or a”shrub”, for instance. The lack of a simple and clearcut definition of ”treeness” and the wide variety of experiences we have with different things that grow up out of the ground is why the concept of tree varies among people, and that there is no universal concept of “treeness”.

  205. 205
    Ed George says:

    Sorry, but Satan made me do it.

    https://m.youtube.com/watch?v=69iB-xy0u4A

    The discussion was getting too serious. 🙂

  206. 206
    hazel says:

    Sorry, Ed, I’ll try to lighten up! 🙂

  207. 207
    ET says:

    I love how hazel sets up a straw man, rips it down and thinks something has been accomplished.

    If you asked the average person, in their language, to point out a tree, they could do it.

    If you asked the average person, in their language, to point out a bush, they could do it.

    If you asked the average person, in their language, to point out a shrub, they could do it

    NEXT

  208. 208
    Ed George says:

    ET

    If you asked the average person, in their language, to point out a tree, they could do it.

    Before the last century there were many people who have never seen a tree.

  209. 209
    Ed George says:

    Hazel

    Sorry, Ed, I’ll try to lighten up! ????

    All it will take is for KF to bring me a shrubbery. 🙂

  210. 210
    ET says:

    Ed George:

    Before the last century there were many people who have never seen a tree.

    So what? Those people are dead and will not be participating. Do you have another fallacious argument to bring up? This is the thread. 😛

  211. 211
    kairosfocus says:

    H, many concepts turn on archetypes or reference standards and degree of family resemblance. In a good fraction of such cases borders are fuzzy and there is no one size fits all and only precising definition, e.g. what is life. This does not mean that the core concepts do not exist and it does not imply that a Yanomami girl taken to Minnesota will not recognise that she sees trees in her new home. However, such is a further set of tangents in a context where even the exchanges on Mathematics are in significant part though they do help to concretely exemplify — this moves towards ostensive definitions — ways in which arguments can fail or can be pulled away from on tangents, or how actual warrant may be shunted aside, etc. All of this gives us a storm in a teacup scale view of some fairly serious trends in our civilisation. In my native land, a saying runs: fire deh ‘pon mus-mus tail, but him think seh a cool breeze dere. The mouse may not perceive its real peril and may severely misunderstand dangers. This is why we really do need to get our thinking straightened out. KF

  212. 212
    Ed George says:

    ET

    So what? Those people are dead and will not be participating. Do you have another fallacious argument to bring up? This is the thread.

    I thought we were talking about the universality of “treeness”. How can it be universal if there were thousands of people over hundreds of generation who never saw a tree? Or bush? Or shrubbery?

  213. 213
    hazel says:

    Hey Ed, how about the aliens who send us the first 100 prime numbers to prove there is intelligent life elsewhere, but live on a planet where there is nothing resembling a tree at all. Prime number may be a universal mathematical concept, but “tree” really can’t be in the same way.

  214. 214
    ET says:

    Ed George:

    I thought we were talking about the universality of “treeness”.

    Such a thing doesn’t include dead people.

    How can it be universal if there were thousands of people over hundreds of generation who never saw a tree?

    We can’t ask them so that is a straw man.

    If you asked the average person, in their language, to point out a tree, they could do it.

    If you asked the average person, in their language, to point out a bush, they could do it.

    If you asked the average person, in their language, to point out a shrub, they could do it

    Ed wants to have a séance…

  215. 215
    ET says:

    hazel:

    Hey Ed, how about the aliens who send us the first 100 prime numbers to prove there is intelligent life elsewhere, but live on a planet where there is nothing resembling a tree at all.

    Hey hazel, you really like erecting straw men. Do you live near a hay field? How are your allergies?

  216. 216
    Ed George says:

    ET

    If you asked the average person, in their language, to point out a tree, they could do it.

    Universality does not deal with averages, it deals with totality. There are people alive today who have never seen a tree.

  217. 217
    ET says:

    Ed George:

    Universality does not deal with averages, it deals with totality.

    It depends on the context, Ed. But then again you always have a problem with that word is used.

  218. 218
    ET says:

    The uncanny coincidence of “Ed George” and “acartia bogart” showing up on two different discussion forums just to spew some nonsense in my direction, is mind boggling. It never fails. Every time acartia is there, Ed is here.

    I know I have a response from Ed when I get one from acartia and vice versa. Amazing coincidences- daily even and even more than once a day.

    And all after acartia boasted of having pretend pro-ID sock puppets here…

  219. 219
    StephenB says:

    Hazel:

    Thanks, Stephen. This leaves me with questions. As you said earlier, right triangles have a precise definition so that everyone who knows that definition will have the same concept – a universal concept of right triangle.

    The key idea is that the universal notion of a right triangle—the concept—tells us what it is. The sense experience of a particular right triangle (such as one in which the right angle is on the left side and is the size of a house) does not qualify as a concept because it doesn’t inform us about what we are experiencing. Since not everyone knows the meaning of a right triangle, we must define it to make things clear. We don’t need to define tree as a universal concept because everyone already knows what it is.

    But “tree” doesn’t have a universally shared precise definition. I have created an understanding of “treeness” based on my various experiences of trees where I live, but someone who lives in the Amazon has had significantly different experiences of trees, so his concept and mine are not identical. Therefore, I don’t see what “universal concept” of tree would mean.

    Your experience of trees is different from everyone else’s because each particular tree is different in some way from every other tree. However, your conception of a tree (the universal concept) is the same as everyone else’s. Similarly, and for the same reason, your conception of a dog or a cat is exactly the same as my conception, even though our experience with particular dogs or cats is varied. For that reason, you can say that a dog is not a cat [law of identity] and anyone who says that it is has made a false statement. Our reasoning process is, in large part, based on the existence of, and our understanding of, identities (essences)

    The knowing process begins with sensory experience (particular), but it ends in the mental concept of a universal produced through a process of abstraction. It is a matter of recognizing that all the trees that have been experienced belong to the same category of being. Even if one thing (shrub or bush) seems to resemble another thing in some way (tree), the very fact that you can make such distinctions indicates that all three categories of being exist and can be universally understood as such.

  220. 220
    hazel says:

    Stephen writes,

    However, your conception of a tree (the universal concept) is the same for everyone.

    I don’t think I see it that way, but there probably isn’t anything more to say about the issue. Thanks for the input, though.

  221. 221
    math guy says:

    h@213 says
    “how about the aliens who send us the first 100 prime numbers to prove there is intelligent life elsewhere, but live on a planet where there is nothing resembling a tree at all. Prime number may be a universal mathematical concept, but “tree” really can’t be in the same way.”

    First, thank you for conceding that “prime number” may be a universal mathematical concept. Have you considered that there might actually be a precise specification for “tree” ? For instance certain DNA, AA, and/or metabolic pathways could be necessary and sufficient conditions for “tree” . Such a specification could be beyond the intellectual grasp of any given human (sans computer), but Goedel’s theorems imply that there are precisely defined mathematical results beyond the intellectual grasp of any given human.

    In fact, the proof of the Four Color Theorem is a likely candidate. Most people can grasp the conclusion of the theorem but the only known proofs require brute force resolution of thousands of cases impossible to verify without a computer. The analogy to “tree” could be similar. We all recognize “tree” but are incapable of expressing or understanding its precise (hypothetical) recipe.

  222. 222
    math guy says:

    My point being that the existence of abstract universals does not necessarily depend on every human, most humans, or even a single human being able to recognize such a universal.

    “If a tree falls in the forest with no humans nearby, did it make a sound?”

    Before Haken and Appel wrote their computer assisted proof, was the Four Color Theorem false?

  223. 223
    kairosfocus says:

    H & SB:

    First, as a handy source, Wikipedia on trees:

    In botany, a tree is a perennial plant with an elongated stem, or trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are usable as lumber or plants above a specified height. Trees are not a taxonomic group but include a variety of plant species that have independently evolved a woody trunk and branches as a way to tower above other plants to compete for sunlight. Trees tend to be long-lived, some reaching several thousand years old. In wider definitions, the taller palms, tree ferns, bananas, and bamboos are also trees. Trees have been in existence for 370 million years. It is estimated that there are just over 3 trillion mature trees in the world.[1]

    A tree typically has many secondary branches supported clear of the ground by the trunk. This trunk typically contains woody tissue for strength, and vascular tissue to carry materials from one part of the tree to another. For most trees it is surrounded by a layer of bark which serves as a protective barrier. Below the ground, the roots branch and spread out widely; they serve to anchor the tree and extract moisture and nutrients from the soil. Above ground, the branches divide into smaller branches and shoots. The shoots typically bear leaves, which capture light energy and convert it into sugars by photosynthesis, providing the food for the tree’s growth and development.

    Trees usually reproduce using seeds. Flowers and fruit may be present, but some trees, such as conifers, instead have pollen cones and seed cones. Palms, bananas, and bamboos also produce seeds, but tree ferns produce spores instead . . . . Although “tree” is a term of common parlance, there is no universally recognised precise definition of what a tree is, either botanically or in common language.[2] In its broadest sense, a tree is any plant with the general form of an elongated stem, or trunk, which supports the photosynthetic leaves or branches at some distance above the ground.[3] Trees are also typically defined by height,[4] with smaller plants from 0.5 to 10 m (1.6 to 32.8 ft) being called shrubs,[5] so the minimum height of a tree is only loosely defined.[4] Large herbaceous plants such as papaya and bananas are trees in this broad sense.[2][6]

    A commonly applied narrower definition is that a tree has a woody trunk formed by secondary growth, meaning that the trunk thickens each year by growing outwards, in addition to the primary upwards growth from the growing tip.[4][7] Under such a definition, herbaceous plants such as palms, bananas and papayas are not considered trees regardless of their height, growth form or stem girth. Certain monocots may be considered trees under a slightly looser definition;[8] while the Joshua tree, bamboos and palms do not have secondary growth and never produce true wood with growth rings,[9][10] they may produce “pseudo-wood” by lignifying cells formed by primary growth.[11]

    Aside from structural definitions, trees are commonly defined by use; for instance, as those plants which yield lumber.[12]

    In short, there is a difference between how an individual or culture forms a particular conception of trees through examples and family resemblance and the underlying archetype accessed thereby which allows us to recognise and agree that some new entity X is a tree. Where of course in many Sci Fi “universes” encounters with fictional worlds often lead to identification of new trees.

    Treeness is so commonly recognised that it is a commonplace metaphor. We speak of tree diagrams, forked branches in a road and the like.

    Now, too, a subtlety lurks, leading to errors of reasoning. It is obvious that we see a contrast between how we may have a precise definition of a triangle and how we may have never seen a tree (I suppose Eskimos beyond the tree line, desert dwellers and possibly people who are so urbanised that they have only an extreme form of concrete jungle to refer to — though, such will doubtless encounter wood.)

    We see here the apparent injection of subjectivism and relativism (as well as linked aspects of constructivism), likely intended as a challenge to undermine the concept of a distinct identity with core characteristics, some in common with other entities, some particular to the given class. Definitions, of course, are seen as imposed by the community of relevant experts, who may consult with experience and community usage or views — hence, dictionaries and their encyclopedia cousins. Individual or community experience it is suggested, leads to particular conceptions, which need not overlap. The underlying thesis is, we are dealing with notions we form as concepts and labels we impose individually and collectively. Nominalism, without having to call and defend that failed name.

    Immediately, notice how we have an argument here, implying a generally known duty to truth, right reason, prudence, justice, fairness etc. Thus, to logic, first principles of right reason, and to truth as a relationship between what we assert and reality that is independent of and ontologically prior to our opinions about it. Which brings up propositions as asserted truth/falsity claims. Where, all of these are essentially abstract. Illustrating, the primary objection to nominalism: it instantly radically undermines itself by being forced to assume, use and refer to the reality of precisely the abstracta, universals, categories/classes, archetypes, in-common and distinct characteristics, identity, etc it would deny and dismiss.

    So, I again put up the challenge: state the definition and core case for nominalism without so appealing to the abstracta one wishes to rule out: ________

    I confidently predict, it cannot be done, as we are here dealing with pervasive first principles of rationality which we cannot prove as to try to prove must use same; but which we must presume true in order to function as rationally and responsibly free thinking creatures.

    I also predict, this core point will most likely again be side-steped and studiously ignored by objectors.

    Thus, we see how the argument tends to deadlock.

    Going back to triangles and trees.

    We must distinguish between how we — creatures of bounded rationality who are error-prone but can know some things (e.g. “error exists”) to utter certainty, who are morally struggling (facing the existential is-ought gap) and too often become polarised, ideologically blinded and ill-willed (but can deal in good faith) — experience, infer and opine and that which is objective knowledge. Where, knowledge can be taken in the softer, broader sense: well-warranted, reliable, credibly true belief . . . which is subject to rational and/or empirical inspection and in principle correction. Where, degree of credibility of truth comes in degrees, from that delivered by inductive cases which cannot confer incorrigible certainty (common in science and day to day life) to that which is self-evidently, incorrigibly, undeniably true.

    For example, we experience ourselves as conscious, and that bare fact is incorrigibly true — we cannot be in error of the fact, once we experience it, support by experience and self-reflective observation is here certain. Likewise, it is undeniably true that error exists (proposition E) as that which intends to refer to truth but fails. Simply contrasting E with ~E leads to: ~E = it is an error to assert, “error exists.” That error exists is undeniable, self-evidently true. It forms a case of truth to absolute certainty which is knowable to us. Worldviews, ideologies, arguments and notions that pivot on discrediting or dismissing truth, knowledge to utter certainty, reducing truth to opinion, relativising or subjectivising knowledge etc are all swept away by this demonstrative test case. They are all failed, grand fallacies that unfortunately haunt and mislead our world today.

    We exorcise thee, we exorcise thee, we exorcise thee and banish thee to the ash-heap of corrected errors. Begone, grand fallacies, begone!

    So, now, we have already established that for a distinct world to be, the set N of counting numbers must exist, that set being {0,1,2,3 . . . }, which is in turn a necessary entity, both the numbers within and the collecting set being objective. Where, what a counting number is, can be generally accessed as an abstract, rationally discernible entity. As opposed to a mere label. Well do I recall my second father [God rest him] painting on a fence as I conversed about what I had been learning and showed a picture I had drawn. He was painting in yellow and dotted in the missing Sun. Soon, the topic was, counting. So, count, The tens came up, I was helped to see twen-ty, thir-ty . . seven-ty . . . Then, I said, “ten-ty.” I recognised the archetype, but did not have the culturally recognised label: “one hundred.” I further generalised two things: X-hundreds and on hitting the tenth in succession, expect a new label, here, thousands.

    I was accessing archetypes and culturally influenced, English Language decimal system numerals, labels.

    Years later, I encountered the approximation pi = 22/7, and was mis-taught. I recall writing out the recurring decimal many times, noting the repeating block that emerges. Subsequently, Geometry class taught me a better understanding. Eventually, I would learn the transcendental, irrational nature and how a transfinite power series approaches it in the limit. We cannot fully write it as a decimal but we know it is definite, it is there.

    It is locked into properties of circles, abstract figures in an abstract space that we represent with our sketches. Where, it is manifest in gear-trains, starting with those I saw in toy cars and fishing reels. Bicycle chain drives, too. Embedded, necessarily, in any possible world.

    Trees are different.

    These are radically contingent beings that exist in at least one possible world-state, but will not exist in at least one possible world. Indeed, did not exist on our planet at some point in the past. However, they do form, reflecting certain patterns that lead to the framework as described. We may imperfectly grasp the concept, must recognise its fuzzy borders, and yet we see there is an underlying archetype in a branching tree-chain (see, metaphorical usage): entity –> . . .plant –> . . . tree –> . . . guava trees, coconut trees, tree ferns . . . Where, a pear has woody tissue in it (the gritty substance we notice), bamboo grasses also form a variety of wood, bananas have tree forms similar enough that if a coconut palm is a tree, so is the banana tree. There is something suspiciously like wood in key parts of a banana plant — e.g. the stem that suspends the bunch, which we routinely cut with a machete but can saw off instead. Other plants, such as the pineapple, do not form trees. Mushrooms are not trees. Bamboo forms a woody stem (sufficient to create timber) but is clearly a grass, not a typical tree. Bananas have an unusual stem, which is not particularly woody, more like compressed leaves but I don’t doubt it could be turned into a timber — or a rope. Coconut logs can be used as timber and the woody stems and “bones” of their fronds — leaves — definitely are — improvised cricket bats and kite frameworks come to mind. In the Caribbean, one then graduates to bamboo kites. I remember being intrigued to see kites sold in shops that used more typical timbers. Papaya stems are clearly woody but not particularly strong — likely, they could make strings and ropes. Elephant grass is disappointingly weak for making fishing poles etc. Tree ferns I have seen across the Caribbean definitely have a sturdy-looking woody character, I have never handled their timber . . .h’mm, I just saw Amazon selling tree fern wood panels for use in specialty plant growing. I believe they are regarded as rare and are protected. I would suggest that various shrubs are miniature, naturally occurring perennials that may take tree form but are small — I have harvested wood and used it from such, much as from an immature full-size tree. Some plants may form vines or trees depending, and more.

    The tree form entity is abstract, and in our world, we first encountered roots, trunks, branches or fronds etc in plants. The labels were taken from those cases and have been extended to many other things, such as classification trees.

    So, there are relevant archetypes and we may distinguish contexts and cases that show themselves to be sufficiently analogous.

    KF

  224. 224
    kairosfocus says:

    MG, excellent input. KF

  225. 225
    hazel says:

    Math guy writes, “First, thank you for conceding that “prime number” may be a universal mathematical concept.”

    I have never denied anything that would imply that other intelligent beings in the universe wouldn’t discover the same fundamental math that we have developed, so I don’t think I’ve conceded anything by pointing out that aliens could broadcast their existence by sending a sequence of the first 100 primes.

    But that is different than saying that math concepts live in some Platonic world that is independent of minds of individual intelligent beings.

    Primes are “universal” in the sense that any intelligent being in the universe who started with the very basics of number theory (the unit and its successors) would develop them and discover properties about them. However, that doesn’t mean, in my view, that therefore they are “universal” in a Platonic sense.

    The issue still is where do universals or abstraction exist. I think they exist in the minds of individual human beings (or other intelligent beings) and in the shared symbol systems that we have developed, but not independently in some eternal realm.

    I think this is a philosophical belief that really can’t be resolved, and that neither Platonism nor my view can conclusively be said to be correct. The nature of my mind (this was the topic of the thread with Gpuucio) is unknown. I tend to accept what I can experience – my mind – and am reluctant to accept things I can’t experience – some universal mind or other home of concepts divorced from individual minds.

    So we should be clear about what I have “conceded” and what I have not.

  226. 226
    ET says:

    hazel:

    But that is different than saying that math concepts live in some Platonic world that is independent of minds of individual intelligent beings.

    No one is saying that. With ID we still have the Mind of the Intelligent Designer- the Mind that caused all this to be. The Mind that used the Mathematics to Design the universe. Ie, The Source.

  227. 227
    Ed George says:

    Hazel@225, for prime numbers to exist we first have to define a rule (only divisible by one and itself). And to set that rule we first have to develop the concept of division. This suggests that prime numbers, and everything that falls from them, are the consequence of us setting specific rules. Not that prime numbers are somehow inherent to the fabric of the universe. Does this make sense?

  228. 228
    ET says:

    Ed, we discovered primes by observing there are numbers only divisible by 1 an itself.

  229. 229
    hazel says:

    Hi Ed. Your questions touch on the big issues that we’ve been discussing for several threads. I don’t intend to re-argue the points with others, but I’d be glad to summarize what I think for you.

    Two issues: First, I think math concepts exist within the symbolic systems we have developed, and ultimately exist in our minds, which have the rational ability to comprehend the symbols and manipulate them with logic. I think we use those concepts to describe the world, but I don’t think the concepts themselves are embedded in the world.

    Once human beings began to develop the number system, starting with the idea of the unit one to describe objects with distinct identity, other basic concepts such as multiples, factors, and ultimately primes, were both logically and experientially inevitable next steps.

    As Wigner said in a quote I liked offered by kf above, “mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts.” We invented the word “prime” and its definition to represent a certain condition of a number not being a multiple of any number less than itself and one.

    However, as I have repeatedly said, once certain definitions are made and concepts established, the power of logic leads us to discover vast amounts of mathematical facts that are not inventions: they are logical discoveries.

    Math can be used to describe the world, but such descriptions are always provisional and subject to revision. One of my first posts in this series of threads quoted Einstein as saying, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” I think this is an important truth.

    P.S. Speaking of discoveries, I was thinking about right triangles recently, after a brief exchange with Stephen, and figured something out: even though I’m sure this is already know, for me it was a logical discovery, and fun to do. This is what motivates students who love math: the satisfaction of using your logical powers to go from given and know information to proving things that, to you, were not previously known to be certainly true.

    Anyway, a theorem that is not obvious, and isn’t relevant to the real world at all.

    Given any odd number n. Find n^2, and divide it into two consecutive integers x and x + 1.

    Then n, x, and x +1 will be a Pythagorean triple, such that n^2 + x^2 = (x + 1)^2.

    For example, let n = 17. n^2 = 289, so x =144 and x + 1 = 145.

    17^2 + 144^2 = 145^2 ? 289 + 20736 = 21025. It works 🙂

    I leave the proof to the interested reader.

  230. 230
    kairosfocus says:

    F/N: We may be getting to the nub:

    H: saying that math concepts live in some Platonic world that is independent of minds of individual intelligent beings . . .

    H’mm, I wonder if this reflects a subtle quasi-spatial understanding of “world”? Where, an abstract entity is normally something contemplated by a mind. Where, it bears repeating to note that a computational substrate is exactly what a free, responsible, rationally contemplative mind is not: blindly mechanical. Where, the logical mind to ponder as candidate would be the same as that behind the cosmos.

    Of course, such is beyond what has been shown repeatedly — structures and quantities are inherently and inextricably bound up in the framework or fabric for a world to be. It is reasonable to ask how that could be as a matter of logic of being, and rationally contemplative mind comes up. Where such a mind behind reality could very well contemplate all that we see is necessarily in the fabric of any world as intelligible rational principles that help to frame it.

    This brings us to an at last substantial contribution by EG:

    for prime numbers to exist we first have to define a rule (only divisible by one and itself). And to set that rule we first have to develop the concept of division. This suggests that prime numbers, and everything that falls from them, are the consequence of us setting specific rules. Not that prime numbers are somehow inherent to the fabric of the universe.

    ET is right in reply: “we discovered primes by observing there are numbers only divisible by 1 an itself.”

    Where, grade school math will remind us of the importance of factorisation and prime factors. Every natural number is prime in itself or else is a product of primes. Where, being able to be grouped in uniform sized bundles without remainder is a highly relevant property, e.g. for tiling a floor, etc.

    The fact that EG saw only that we composed a statement and that we created a procedure is in itself instructive, pointing to the way we are thinking. Beyond, was what was missed: division is a label for a natural process of splitting up a collection of discrete entities into groups of the same cardinality, e.g. pennies or tiles or peas etc. Further beyond was the discovery that some quantities of such units cannot be split into smaller groups without remainder, other than splitting up by ones. Onward, we can discover that any number that can be shared up in evenly sized groups, can ultimately be split into prime number sized groups. Where, of course, the first such case, 2, then divides numbers into odds and evens.

    Since, we likely explored this in primary school, the question is why we seem to consistently miss the discovery element and highlight the culturally influenced creations that respond to facts that are discovered.

    One answer is, likely, we were often taught in an authoritative, rote pattern, and were not invited to do even a guided exploration. Another is likely a penumbra of axiomatisation since C19. However, post Godel, we know that complex Mathematical domains have in them Math facts unreachable from finite sets of mutually consistent axioms. They are irreducible and facts stand independent of axiom systems.

    KF

    PS: Perhaps, this from Plato in The Laws, Bk X, may be food for thought:

    Ath. Nearly all of them, my friends, seem to be ignorant of the nature and power of the soul [[ = psuche], especially in what relates to her origin: they do not know that she is among the first of things, and before all bodies, and is the chief author of their changes and transpositions. And if this is true, and if the soul is older than the body, must not the things which are of the soul’s kindred be of necessity prior to those which appertain to the body?

    Cle. Certainly.

    Ath. Then thought and attention and mind and art and law will be prior to that which is hard and soft and heavy and light; and the great and primitive works and actions will be works of art; they will be the first, and after them will come nature and works of nature, which however is a wrong term for men to apply to them; these will follow, and will be under the government of art and mind.

    Cle. But why is the word “nature” wrong?

    Ath. Because those who use the term mean to say that nature is the first creative power; but if the soul turn out to be the primeval element, and not fire or air, then in the truest sense and beyond other things the soul may be said to exist by nature; and this would be true if you proved that the soul is older than the body, but not otherwise.

    [[ . . . .]

    Ath. . . . when one thing changes another, and that another, of such will there be any primary changing element? How can a thing which is moved by another ever be the beginning of change? Impossible. But when the self-moved changes other, and that again other, and thus thousands upon tens of thousands of bodies are set in motion, must not the beginning of all this motion be the change of the self-moving principle? . . . . self-motion being the origin of all motions, and the first which arises among things at rest as well as among things in motion, is the eldest and mightiest principle of change, and that which is changed by another and yet moves other is second.

    [[ . . . .]

    Ath. If we were to see this power existing in any earthy, watery, or fiery substance, simple or compound-how should we describe it?

    Cle. You mean to ask whether we should call such a self-moving power life?

    Ath. I do.

    Cle. Certainly we should.

    Ath. And when we see soul in anything, must we not do the same-must we not admit that this is life?

    [[ . . . . ]

    Cle. You mean to say that the essence which is defined as the self-moved is the same with that which has the name soul?

    Ath. Yes; and if this is true, do we still maintain that there is anything wanting in the proof that the soul is the first origin and moving power of all that is, or has become, or will be, and their contraries, when she has been clearly shown to be the source of change and motion in all things?

    Cle. Certainly not; the soul as being the source of motion, has been most satisfactorily shown to be the oldest of all things.

    Ath. And is not that motion which is produced in another, by reason of another, but never has any self-moving power at all, being in truth the change of an inanimate body, to be reckoned second, or by any lower number which you may prefer?

    Cle. Exactly.

    Ath. Then we are right, and speak the most perfect and absolute truth, when we say that the soul is prior to the body, and that the body is second and comes afterwards, and is born to obey the soul, which is the ruler?

    [[ . . . . ]

    Ath. If, my friend, we say that the whole path and movement of heaven, and of all that is therein, is by nature akin to the movement and revolution and calculation of mind, and proceeds by kindred laws, then, as is plain, we must say that the best soul takes care of the world and guides it along the good path. [[Plato here explicitly sets up an inference to design (by a good soul) from the intelligible order of the cosmos.]

  231. 231
    ET says:

    Math can be used to describe the world because the world was intelligently designed using mathematics.

  232. 232
    kairosfocus says:

    ET, in such parts as are contingent and fine tuned to support terrestrial planet, C-chemistry aqueous medium cell based life, that is utterly manifest. But there is more, Mathematical reality is demonstrably part of the very fabric of any possible world. Mathematical reality in core part is antecedent to our minds much less our studies and whatever cultural influences are involved. KF

  233. 233
    ET says:

    Can’t build a world without the maths…

  234. 234
    Ed George says:

    KF

    ET is right in reply: “we discovered primes by observing there are numbers only divisible by 1 an itself.”

    But we invented division. The fact that when we apply division we see certain “trends” does not mean that those “trends” are inherent in the universe. It just means that they are inherent (or consequences) of our invention. I have been out of this conversation for a while so I apologize if this has already been gone over.

  235. 235
    kairosfocus says:

    EG, no, stacking or tiling in patterns goes down to atomic, molecular and material structure. The property of being discrete units is natural, ponder peas in a pod. The onward one of separability in evenly grouped patterns goes beyond that to the realities and quantitative properties of numbers which lie in the numbers. KF

    PS: What would you make of Fibonacci numbers and linked properties commonly arising in nature?

  236. 236
    ET says:

    What is the evidence that we invented division?

  237. 237
    ET says:

    We invented mathematics if an only if materialism is true. However, if materialism was true we wouldn’t be here. There wouldn’t be any life at all.

  238. 238
  239. 239
    Ed George says:

    Hazel@238. 🙂 🙂 🙂 🙂 🙂

  240. 240
    StephenB says:

    Hazel, KF

    First, I appreciate KF’s comments on archetypes and their relevance to the discussion. Let me respond to some of his points in the context of Hazel’s objections. For me, the question on the table is whether or not trees qualify as a category of being and if we can know a tree for what it is. It is not a perfect parallel to mathematics for reasons that I will not get into at this time.

    Nominalists say that we cannot know a tree as a tree; we simply call it a “tree” for the sake of convenience. In their minds, we cannot know anything for what it is, only how it appears to us. Under that assumption, then, major concepts could be understood in a variety of ways. However, knowledge is not about perceptions or appearances; it is about the known reality behind perceptions and appearances. In that sense, there are no differences in what we know about the essence of a tree.

    It doesn’t matter that some people have not experienced a tree. In such a case, they simply don’t know what a tree is until someone informs them or until they have experienced enough subcategories of trees (Oak tree, Maple tree, etc.) to abstract the universal reality behind the general category (tree). Once that happens, they will understand what a tree is just as we do.

    Nor does it matter that some other entities, such as shrubs or bushes, may resemble a tree in some way. This muddies the waters because the difference between the two is often defined by what is done to them (pruning, cutting, etc.) There is a substantial difference between those two entities and a tree and anyone with sufficient experience knows that difference. If not, they can learn about it. We can learn about all kinds of things that we have not experienced.

    The process of abstraction often requires enough to time to grasp the universal concept (or principle) from particular examples, and sometimes, many examples are required. Under the heading of “tree,” Wikipedia defines the word to mean this: “A perennial plant with an elongated stem, or trunk, supporting branches and leaves in most species.” That is not a bad definition. Yes, it does go on to say that *“Trees are not a taxonomic group,”* but I would question that assessment. Indeed, Wikipedia itself seems to contradict the point under the heading “Categories: trees,” describing categories and subcategories of trees, as well as the genus and species of trees.

    In any case, we need the general (genus) to understand the particular (species), and we also need the particular to understand the general. For most us, the particular is the first experience. So, yes, if one person has experienced only oak trees and another has experienced only maple trees, there will be some variation in their perceptions, but that is because, in both cases, there is the false perception that only their experienced subcategory exists, which means that they do not have the variety of experiences necessary to know that a general category of trees exists. Concepts, as I define them, are about knowledge, not perceptions.

    Until one understands the relationship between the general category (tree) and the sub category (Oak tree, Maple tree, etc.), there is no knowledge of a tree’s essence or whatness – only the experience of a particular kind of tree, which does not suffice as knowledge. Perception does not equal conception. I submit, therefore, that the perceptions of trees can differ, but the knowledge of a tree’s “whatness” cannot. Obviously, we cannot know everything there is to know about a tree, or even a small part of it, but we can know what a tree is in terms of the same shared concept.

  241. 241
    kairosfocus says:

    H,

    perhaps, it is time to talk about yet another fallacious tendency, loaded projection. For cause, I disagree that we are persistently, even possibly willfully misunderstanding what you have advocated; which is what your linked suggests. Instead, there is a disagreement pivoting on a demonstration that leaves you uncomfortable and which you seem to wish to set aside as if it were irrelevant. In fact, that demonstration is the crux of the issue and the rhetorical pattern of responses is instructive, albeit less than happy.

    Again, there is a fundamental issue that has come up over several threads of argument, the nature of Mathematics i/l/o a demonstration that there are rational principles and associated facts of structure and quantity embedded in the framework for any world to exist. That suffices to demonstrate that there is a core substance of structure and quantity that is in reality antecedent to our coming along and creating a culturally influenced study of the logic of structure and quantity. So, there is a both-and; something you have in effect formally agreed with. But then, again and again you have come along and argued as if the first part vanishes, emphasising individual and cultural aspects to the point that it is evident that something is profoundly disturbing to you about the other aspect

    Further to this, you have stated a strong disagreement, not only with classical Platonism and apparently with the classic view by theologians that roughly speaking the world of forms is the mind of God, but also with what is called modern mathematical platonism which effectively holds that key abstracta have reality independent of our opinions and studies. They are discovered not invented. And here I am specifically not claiming that all abstract entities or clusterings are like this, just some relevant ones tracing to N and extensions therefrom.

    You have championed nominalism. To this, I have given a two-level response. Level one, knowing the ideological dominance of evolutionary materialism and its fellow travellers as well as of associated subjectivism and relativism, I have shown how the core materialism cannot warrant nominalism, starting with the non-rationality of computational substrates. That in my view is necessary regardless of your particular views precisely due to ideological dominance.

    Level 2, I have spoken to how statements of nominalism are inextricably entangled in implicit appeals to abstracta and universals. Indeed, I have boiled it down to the challenge of stating and arguing for it without so implicitly appealing. No-one has taken this up, unsurprisingly as truth is an abstract relationship, implication, entailment and evidential support are the same, any number of universals will stubbornly insist on popping up in the arguments and more. I point out that this is exactly how first principles of rationality operate, we cannot reason without them, we therefore cannot prove them, we prove from them, we must take them as givens.

    Thus, we have good reason to retain the conclusions that have been on the table.

    And, these pivotal issues are not matters for clashing opinion, they hold adequate warrant.

    KF

  242. 242
    kairosfocus says:

    SB, treeness actually extends beyond biology, in terms of a branching structural pattern. Confining to biological cases, there are clearly exemplars and counter examples that suffice to form a core conception on common characteristics, a growth habit that is functional and often leads to timber or the like as a result, as strength is needed to elevate leaves or fronds and to withstand strong winds. I suggest shrubs and bushes are in effect miniature forms, though the height etc differences may be significant. Plants that may grow as bushes or shrubs or as climbing vines seem to be an interesting bridging case. KF

    PS: The Kantian ugly gulch seems to pop up in all sorts of places. F H Bradley’s corrective that to claim the un-knowability of things in themselves is to imply knowledge of things in themselves will bear pondering.

  243. 243
    hazel says:

    kf writes, “You have championed nominalism.”

    No, I have not “championed” it. I liked a sentence I read about it on Wikipedia. When you declared that it led to, or implied, physicalism, I specifically wrote at 138, “kf, I really don’t know why you think nominalism implies physicalism, but if that is the case, then nominalism doesn’t describe my position. ”

    Your persistence in the idea that I am “championing” nominalism is an example of the problem I addressed at 238.

  244. 244
    hazel says:

    Hi Ed. At 234, you wrote,

    But we invented division. The fact that when we apply division we see certain “trends” does not mean that those “trends” are inherent in the universe. It just means that they are inherent (or consequences) of our invention. I have been out of this conversation for a while so I apologize if this has already been gone over.

    I’d like to expand on some points that I briefly made at 229. ( think I’ll break them up into two posts.)

    First, I think perhaps too much has been made of the distinction between using the words invention and discovery in the area of pure math. From a theoretical point of view, once the counting numbers have been established, the idea of starting with one number and then counting some additional amount from there is a logical and virtually inevitable idea. However, to develop the idea and share it with others we invent a word “addition” and a symbol “+” to represent the concept. So the development involves both some recognition of something new that we can do (a discovery) and the invention of some terms and symbols with which to manipulate our new concept.

    Then the process is repeated: we note multiples occur by repeated addition of the same number: if you count by 3’s you get a sequence of the multiples of 3. Then we define multiplication as a shorthand for repeated addition, again a creative recognition of a new pattern and the invention of new terms and symbols, so now we have a new concept. Eventually this leads to the idea of primes, and a whole world of new discoveries opens up.

    All of the preceding has been about pure math: our creative, rational mind can think about a logical system, get new ideas that might be interesting, and develop terms and symbols that let us then explore the consequences of what we have developed. We are creatively both inventing and discovering as we go along.

    In the beginning, as with the primes, the creation of new concepts may appear obvious: more a discovery than an invention. But later in the development of math, people had to think long and hard about the issue they were trying to grasp, and explore different ways as to how to formalize symbolic tools to open new understandings. Newton’s invention of calculus is a case in point, along with Leibnitz’s notation for the derivative. Similarly, I think it’s more reasonable to say that Descartes invented coordinate geometry than to say he discovered it.

    But as I am trying to make clear, I don’t think there is a clearcut distinction between what is invented and what is discovered in math, which is why I like words such as create and develop, which express the interplay, perhaps, between the two ideas that is always going on..

    Also, to be clear, and to try to avoid misunderstanding, given a set of developed concepts, the stream of logical consequence that follow are discovered. Logical consequences are entailed in the beginning set of accepted know concepts and facts, and then we explore those consequence streams and discover all sorts of new stuff. For instance, we have discussed two facts that are not obvious at all: for any prime p, p^2 -1 is divisible by 24, and for any odd number n, dividing n^2 into x and x + 1 produces a Pythagorean triple n, x, x + 1. Those are discoveries within the realm of pure math. We didn’t invent those.

  245. 245
    StephenB says:

    KF @242: Interesting. Thanks for the comment.

  246. 246
    hazel says:

    The second idea I want to discuss is the relationship between pure math and the physical world, and about the idea of distinct identity. I hope, Ed, that you find my remarks somewhat useful in respect to the questions you asked at 227 and 234.

    Without a doubt, pure math starts with the idea of a distinct unit, which we name “one”. We then define successors so as to create the counting numbers, and with suitable development as outlined in my last post at 244, a vast amount of pure math follows.

    How does this apply to the physical world? My basic position is that we build mathematical models of the world: abstractions which describe general aspects of the world. We then draw logical conclusions within our model about what we would expect to find if our model were true. Then we re-examine the physical world to see if it behaves as our model predicted. If it does, that supports our model and makes it worthwhile to further develop our model. If the expectations of our model are not empirically confirmed, we revise our model.

    The model and other concepts we use to describe the physical world are part of our mind, which uses the symbolic systems of language and math to both formalize our own understanding and to share it with others.

    So we have two things: the physical world and our understanding of it. Our understandings are abstractions that we use to describe aspects of the world. We can’t describe every detail of the world (the map is not the territory) we have to selectively generalize about the parts we have focussed our attention on.

    All this takes place in all human beings all the time: it’s part of being a human being with, among other things, rational cognitive skills. There is a constant interplay between the empirical data we get about the physical world through our senses and the understandings/concepts/abstractions/models we create in our minds, and share with each other through language, written and verbal, including the language of math.

    So let’s talk about the simplest model of all: counting things is the physical world. The simplest idea is that of one-to-one correspondence. We have words for one, two, three, etc. We count three pebbles by aligning the words with the objects. (Children start getting this about the age of three or four).

    This is a model. We experience the three pebbles as distinct entities: they retain their separateness from other things in the world. Therefore, the concept of a “unit” applies to them. Also the concept of addition applies to them, because when we put them together, they retain their individuality. The model works, and continues to work as we divide them into groups to represent multiplication. There is a very solid correspondence between our mathematical model and how objects like pebbles behave.

    (FWIW, when teaching children about numbers, and lots of other math, it’s important to give them lots of experience with concrete situations, so they can build strong concepts that have a depth of understanding.)

    There are, however, some additional issues here. One is that right from the beginning treating the three pebbles are units is an abstraction that ignores a huge amount of uniqueness (color, shape, size, etc.) and models just the fact that the object can be moved around while maintaining its separateness. Right from the beginning we have an abstract mathematical model-physical world relationship in just considering a pebble representing the concept one.

    This is easier to see if we think about things that don’t behave like pebbles. Clouds are an example. You can’t count clouds. They might look like one cloud from one view, and not another: how do you determine if a cloud is a continuous whole? They don’t retain their separate identity when brought together. We can’t apply the model of the counting numbers to clouds.

    So, to summarize this point, counting objects, when applied to objects which have a distinct identity and retain that identity when moved around, is applying a mathematical model to a phenomena in the physical world.

    One last point: kf wrote, “ stacking or tiling in patterns goes down to atomic, molecular and material structure. The property of being discrete units is natural …”

    Actually, I think at the quantum level we are perhaps finding this not be true. But that is another subject.

  247. 247
    kairosfocus says:

    H, I have first repeatedly pointed out that a dominant form of nominalism is rooted in or a fellow traveller of evolutionary materialistic scientism. Accordingly, I began with this and then continued to the wider case. I observe that you are now specifically distancing yourself. Previous remarks and arguments you raised several times (readers, scroll up to see) led to the response I made that you have championed it in thread. In particular, it appeared that you championed conceptualism, which seems to be an unstable position which I cited on above. KF

  248. 248
    kairosfocus says:

    H,

    Next, observe your remark:

    I think perhaps too much has been made of the distinction between using the words invention and discovery in the area of pure math. From a theoretical point of view, once the counting numbers have been established, the idea of starting with one number and then counting some additional amount from there is a logical and virtually inevitable idea.

    Now, what does “established” mean? Normally, it means or suggests set up by some power brokers or innovators or the like. But what has been demonstrated is that once a distinct world is possible, it embeds in its framework the counting numbers as effectively a corollary of the principle of identity. From this, Z, Q, R and C etc follow. This is one way to see that there is an embedding of intelligible rational principles and entities of structure and quantity in the framework for any possible world. Thus, a substantial body of core Mathematical facts are discovered as necessary aspects of reality rather than invented through our studies. That is, they are antecedent to discovery and analysis. For particular instance, there are properties of certain numbers such that they cannot be broken up into smaller groups evenly without remainder, save by ones. We recognised this and have called such primes. Further to this, any other number will be capable of being grouped by prime factor-sized groups, which is a powerful tool in many areas of our study of numbers. That property of primeness clearly is antecedent to our study and is a discovery not an invention.

    In short, your remarks come close to begging the first question.

    Likewise, the even grouping concept implies repeated addition by n-scale groups, 1’s, 2’s, 3’s etc. Thus multiplication and division are directly associated with the property that distinct entities can be grouped. Where, addition is a direct property where groups may be clustered to form a larger one, which may be counted up. Subtraction is then instantly also present as removal of a group. All of this is routinely explored in elementary school.

    So, while we study the four rules and invent symbols, the underlying intelligible structures and quantities inhere to the nature of groups of discrete entities, concrete or abstract. Where, the four rules with additional structured operations that build on them are a key part of the core of operations, functions and relationships. The dual character of naturally occurring substance and culturally influenced study remains.

    Going on:

    our creative, rational mind can think about a logical system, get new ideas that might be interesting, and develop terms and symbols that let us then explore the consequences of what we have developed. We are creatively both inventing and discovering as we go along.

    That is, we may study intelligible properties of our world and other possible worlds. That was never in question, the issue is that it is demonstrated that there are intelligible, rational principles of structure and quantity that on distinct identity are inextricably embedded in the framework for any possible world. That is, these are necessarily occurring entities. Further to this, being closely tied to the logic of being, such things then become manifest in the objects and processes we study in science. In such light, “we have developed” is again perilously close to begging the question and dismissing that which was demonstrated from the outset. Indeed, it is that demonstration which you reacted to originally, seemingly taking exception to the consequences of the demonstration.

    You continue to EG, as though the other side of many implicit questions were never on the table, with substantial reasons offered in support — which you have never cogently, substantially answered:

    In the beginning, as with the primes, the creation of new concepts may appear obvious: more a discovery than an invention. But later in the development of math, people had to think long and hard about the issue they were trying to grasp, and explore different ways as to how to formalize symbolic tools to open new understandings. Newton’s invention of calculus is a case in point, along with Leibnitz’s notation for the derivative. Similarly, I think it’s more reasonable to say that Descartes invented coordinate geometry than to say he discovered it.

    Appearing obvious is a pivotal rhetorical construct here. Its effective function is that it furthers the dismissal of the demonstration of necessary structural and quantitative entities embedded in the fabric of any possible world antecedent to our recognition and study. Such entities are enough to ground the claim that there is significant discovery in Mathematical study, regardless of what we may otherwise invent. However, it is appropriate to again point out that rates of change and accumulations of change including growths, flows and spatial motion (speed, velocity, acceleration, time, impulse, jerk, work, power as rate of work, fluxes in the sense of energy and fields etc) are naturally occurring structures and quantities that were found important objects, processes etc worthy of study. Yes, Newton et al (including classical antecedents over a thousand years earlier) did develop approaches, insights, terminology and paradigmatic results, but that study does not sweep away the naturally occurring substance that they examined and discovered properties of. Sweeping ahead to non-standard analysis, I suggest that infinitesimals and transfinites . . . domain of hyperreal and surreal numbers great and small . . . are reasonably viewed as extensions of the sets already highlighted, e.g. recognition that naturals continue endlessly beyond any given finite value leads to recognition of another class of quantity, the transfinite.

    Now, we see a caveat with a regrettable marker that allows a loaded projection to any counter-argument:

    to be clear, and to try to avoid misunderstanding, given a set of developed concepts, the stream of logical consequence that follow are discovered. Logical consequences are entailed in the beginning set of accepted know concepts and facts, and then we explore those consequence streams and discover all sorts of new stuff. For instance, we have discussed two facts that are not obvious at all: for any prime p, p^2 -1 is divisible by 24, and for any odd number n, dividing n^2 into x and x + 1 produces a Pythagorean triple n, x, x + 1. Those are discoveries within the realm of pure math. We didn’t invent those.

    What is at stake, first, is that it was demonstrated that there are embedded abstract entities and principles of structure and quantity in any possible world that are antecedent to any exploration or discovery or invention on our part. That demonstration has never been refuted, though it has often been rhetorically sidelined, dismissed as allegedly irrelevant or a misunderstanding of the views being put forward etc. Next, it was thus shown that there are embedded core mathematical facts that formed a body of knowledge which shaped and constrained formation of axiomatic systems starting with Geometry and proceeding on to the modern axiomatisations. For simple example finding twelve segment ropes implies knowledge of key Pythagorean results long before Elements was composed.

    The underlying body of such facts and knowledge bases is sufficient to show that there are material bodies of structure and quantity embedded in this and other possible worlds, which we discover in our studies rather than invent. Onward theorems or model results are secondary to the primary point.

    You then proceed to say further to EG (as opposed to the parties you actually have been debating with all along) — and I enfold annotations:

    >>pure math starts with the idea of a distinct unit, which we name “one”. We then define successors>>

    a: we observe, having demonstrated, that nullity, unity and duality are implicit in the possibility of any distinct world. Thus, we see a successive cumulation of distinct discrete quantities, thus justifying use of the von Neumann succession principle to lay out the Naturals, which we may label as we please.

    b: The above therefore again suppresses material demonstration that demonstrates mathematical entities antecedent to our studies.

    >>so as to create the counting numbers,>>

    c: create is here synonymous with invent; the question is begged in the teeth of an existing demonstration that has been repeatedly pointed out.

    >> and with suitable development as outlined in my last post at 244, a vast amount of pure math follows.>>

    d: Thus, the demonstrated embedded structure and quantity antecedent to our study is material in its import.

    >>How does this apply to the physical world? My basic position is that we build mathematical models of the world: abstractions which describe general aspects of the world.>>

    e: Yes, we build abstract, logic model worlds, which are constrained by the embedded facts antecedent to our study and their logic of being import.

    f: Describing general aspects of the world avoids acknowledging that a body of core abstract structure and quantity is necessarily embedded in the fabric of any possible world, thus our experienced one.

    g: Thus, it implies the main point, while the acknowledgement of its material relevance is sidestepped.

    >>We then draw logical conclusions within our model about what we would expect to find if our model were true.>>

    h: Yes, and this requires that the logic of being import of the logic of structure and quantity extends from the abstract model world to the physical one.

    i: This occurs in two ways, necessary entities and properties extend to any world and when certain properties are archetypally present in the model world and our own, they will be relevant. Circularity properties extend to gear trains and other approximately round entities, etc.

    j: As has been repeatedly pointed out by the other side but is not acknowledged in how it is now being presented. The rhetorical effect in wider context is to pummel a gagged strawman.

    k: And this is occurring when I am present and thread owner and where others are likewise present. Now imagine what happens when the other side is locked out, silenced, demonised, stereotyped, tainted and scapegoated, expelled, censored or de-platformed, etc.

    l: Resemblance to trends with the ID debates and wider cultural conflict is not coincidental.

    >>Then we re-examine the physical world to see if it behaves as our model predicted. If it does, that supports our model and makes it worthwhile to further develop our model.>>

    n: That is, modern induction is argument by support and linked inference to best current explanation.

    >> If the expectations of our model are not empirically confirmed, we revise our model.>>

    O: provisionality as a scientific ideal.

    >>The model and other concepts we use to describe the physical world are part of our mind,>>

    p: They are abstracta, raising issues of the nature, power and relevance of such abstract entities and clusterings including universals.

    >> which uses the symbolic systems of language and math>>

    q: Abstracta again.

    >> to both formalize our own understanding and to share it with others.>>

    r: That is, we assert propositions, infer logical relationships, argue to warrant and to explain, thus study. Where the demonstrated embedded entities are material and are antecedent to our studies.

    >>So we have two things: the physical world and our understanding of it.>>

    s: Yes, the world which embeds abstracta, and our studies that build on such.

    >>Our understandings are abstractions that we use to describe aspects of the world>>

    t: Yes, we do abstract through inference, pattern-recognition, symbolic representation etc. However, the elephant in the room is the demonstration of abstract necessary entities involving N, Z, Q, R, C etc antecedent to our studies.

    >>. We can’t describe every detail of the world (the map is not the territory) we have to selectively generalize about the parts we have focussed our attention on. >>

    u: We have bounded rationality

    This is enough to make the substantial concerns plain, and to show how things have got so tangled up.

    KF

    PS: The blanket appeal to the quantum world fails. At macro level, tilings and groupings are facts of life, as are constraints on packing. Going to materials, molecules and solids or liquids, packing and clusters are major phenomena, the crystal structure of metals and semiconductors being a major part of why they act as they do. Where liquids, glasses etc take properties from the degree to which that packing does not obtain. Semiconductors with controlled dopants are again relevant. Of course, the informational structure of R/DNA and the chain and fold aspect of proteins is also relevant.

  249. 249
    Ed George says:

    Hazel@244&246, thank you for the response. For the most part, my view aligns with yours, although you present it much better than I do.

  250. 250
    hazel says:

    Thanks, Ed. It’s been useful to me to respond to your questions with some more expanded thoughts.

  251. 251
    ET says:

    hazel:

    First, I think perhaps too much has been made of the distinction between using the words invention and discovery in the area of pure math.

    Well, it matters.

    I understand that it may bother some people that we are just discoverers. But hey, the discovery, the journey, is clearly not for everyone and is very important in itself. People like Srinivasa Ramanujan should be heralded, studied and perhaps even followed- methodologically with resect to mathematical discovery- if it leads to a higher understanding and more discoveries.

    Being able to tap into the universal information, including mathematics, channel it, understand it and be able to make use of it is a great feat. A great individual achievement, if that is what you seek.

    [O]ur creative, rational mind can think about a logical system, get new ideas that might be interesting, and develop terms and symbols that let us then explore the consequences of what we have developed because that is what they were intelligently designed to do.

    We are designed for scientific discovery. And given everything we need to discover. It is a real-life scavenger hunt of sorts.

  252. 252
    kairosfocus says:

    EG, when demonstrative warrant is on the table, opinions to the contrary avail nothing. And, demonstrative warrant is on the table as has long since been shown. But then, that is part of the point of the OP. KF

  253. 253
    hazel says:

    A companion to 238: More wisdom

  254. 254
    kairosfocus says:

    H, pardon but the use of the just linked in this context is a clear example of loaded unjustified projection. I repeat, when an actual demonstration is on the table, dismissive objections to the contrary avail nothing. KF

  255. 255
    hazel says:

    Over on the newer Wigner thread , I wrote this:

    I like these remarks from Wigner’s article, which I think are somewhat compatible with some of the things I have been trying to say.

    However, the point which is most significant in the present context is that all these laws of nature contain, in even their remotest consequences, only a small part of our knowledge of the inanimate world. All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present, except that some aspects of the present state of the world, in practice the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction. …

    [This] is in consonance with this, first, that the laws of nature can be used to predict future events only under exceptional circumstances, when all the relevant determinants of the present state of the world are known. It is also in consonance with this that the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted. …

    The principal purpose of the preceding discussion is to point out that the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world.

    It should be mentioned, for the sake of accuracy, that we discovered about thirty years ago that even the conditional statements [of the motion of bodies] cannot be entirely precise: that the conditional statements are probability laws which enable us only to place intelligent bets on future properties of the inanimate world, based on the knowledge of the present state. They do not allow us to make categorical statements, not even categorical statements conditional on the present state of the world.

    I’d like to come back to this thread and add some remarks in response to Wigner.

    However, to forestall a possible reaction, I’ll point out that I am not “championing” Wigner, nor being in agreement with everything he says. In fact, as bornagain77 pointed out (as did other articles about Wigner I’ve read since yesterday), Wigner himself changed his mind about a number of important issues later in life. (BA posted seven long posts about Wigner and related items on the other thread, if you want to get caught up.)

    Also, I’m not claiming that Wigner is a definitive expert on this subject. He is an important physicist who also considered more philosophical issues such as the nature of nature and of consciousness, and he thought and wrote well, but many of his equally competent peers had different ideas.

    With all those disclaimers said …

    Wigner wrote,

    However, the point which is most significant in the present context is that all these laws of nature contain, in even their remotest consequences, only a small part of our knowledge of the inanimate world. All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present, … [although] in practice the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction. …

    That is, our “laws of nature” are always abstractions which isolate a few details of our experience as the content of our abstraction from the incredibly rich amount of detail present in the actual physical world. We choose details that are associated with changes that we want to use to make predictions: in part we create descriptions of nature for their utility to us.

    Furthermore, we constantly test the validity and usefulness of our abstractions making predictions based on them, seeing if the predictions are born out, and then revising them accordingly. As Wigner says,

    The exploration of the conditions which do, and which do not, influence a phenomenon is part of the early experimental exploration of a field. It is the skill and ingenuity of the experimenter which show him phenomena which depend on a relatively narrow set of relatively easily realizable and reproducible conditions.

    I found it interesting that Wigner points out that the predictive power of the laws of nature are only effective when we know the initial conditions to which they apply (he expands on this more in another article I read), one of their most important uses has been us in the creation of machines:

    It is also in consonance with this that the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted

    Putting these two ideas together, I find it easy to see that all of our knowledge of the world consists of abstractions which, no matter how we refine them, cannot and do not encompass the incredible detail of the world, including the constantly changing initial in any causal chain which make predictions about the future even probable only in highly constrained cases.

    Wigner makes a similar conclusion, I think, when he writes,

    The principal purpose of the preceding discussion is to point out that the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world.

    I think, therefore, that Wigner’s statement are, to some degree, compatible with the view that our abstractions, including all mathematical descriptions of aspect of the world, are mental constructions which exist, collectively, in the minds of human beings and in the symbol systems we have created to share our concepts with each other.

    This is of course not to deny that there is an unreasonable effectiveness of math in abstracting and describing the physical world. That is the thesis of the essay. But I don’t think that thesis is inconsistent with the view I expressed in the preceding paragraph.

    I think this idea is reinforced by this:

    It should be mentioned, for the sake of accuracy, that we discovered about thirty years ago that even the conditional statements [of the motion of bodies] cannot be entirely precise: that the conditional statements are probability laws which enable us only to place intelligent bets on future properties of the inanimate world, based on the knowledge of the present state. They do not allow us to make categorical statements, not even categorical statements conditional on the present state of the world.

    As we have explored more deeply into the basis of the physical world we have encountered the quantum world that was the heart of Wigner’s work as physicist. At that level, many argue that all we have are our mathematical abstractions, and we don’t even know whether it makes sense to say they are abstractions of anything. They are useful human creations which can predict certain end results, as conditional and probable, and if sufficient initial conditions are known, without knowing what the actual nature of the reality leading up to those end results is.

  256. 256
    Ed George says:

    KF

    EG, when demonstrative warrant is on the table, opinions to the contrary avail nothing.

    But who decides that “demonstrative warrant” is on the table? I think it has been mentioned before, but just because you have declared “demonstrative warrant” doesn’t make it so.

  257. 257
    kairosfocus says:

    EG [attn, H],

    that you still ask WHO decides that demonstrative warrant on the table is the fatal tell. Leff’s fallacy of the grand sez who.

    Let me therefore cite Aristotle in The Rhetoric, Bk I Ch 2 [yes, 2300+ years ago — this is Alexander the Great’s tutor: Socrates –> Plato –> Aristotle –> Alexander], on the three levers of persuasion, pathos, ethos, logos:

    Of the modes of persuasion furnished by the spoken word there are three kinds. The first kind depends on the personal character of the speaker [ethos]; the second on putting the audience into a certain frame of mind [pathos]; the third on the proof, or apparent proof, provided by the words of the speech itself [logos]. Persuasion is achieved by the speaker’s personal character when the speech is so spoken as to make us think him credible . . . Secondly, persuasion may come through the hearers, when the speech stirs their emotions. Our judgements when we are pleased and friendly are not the same as when we are pained and hostile . . . Thirdly, persuasion is effected through the speech itself when we have proved a truth or an apparent truth by means of the persuasive arguments suitable to the case in question . . . . [Aristotle, The Rhetoric, Book I, Ch. 2. Cf. summary with scholarly observations at http://plato.stanford.edu/entr.....-rhetoric/ and http://www.public.iastate.edu/.....index.html for a hypertext version of the book]

    Appeals to our passions, perceptions and felt reactions are of no more weight than the soundness of underlying judgements. Those to the credibility of an authority or presenter hold no more weight than the merits of the underlying case. It is therefore to the merits of fact and logic that we must always go. And in context, your heaping praise on H’s recent summary while studiously side-stepping the corrective at 248 above is a further tell. You are clearly mostly cheerleading and sniping rather than making consistent substantial contributions.

    Now, in 248, I only alluded to the demonstration on the table, so let me supply the lack once again, before challenging you to warrant your objection.

    1: Consider reality, and within it some distinct entity, say a bright red ball on a table, B. Thus the rest of reality is the complement to B, ~B. Reality, R = {B|~B}

    2: Immediately, B is itself (distinctly identifiable i/l/o its core, distinguishing characteristics), this is the fundamental law of thought, Law of Identity, which sets up the dichotomy and its corollaries.

    3: Clearly, no x in R can be B AND ~B. Law of non-contradiction, a corollary.

    4: Likewise, any x in R must be in B or in ~B, not between them or separate from them: B X-OR ~B, law of the excluded middle. The second corollary.

    5: Now, ponder a possible world, W, a sufficiently complete description of a possible [coherent!] state of affairs in reality,i.e. in this or any other world that could be or is.

    6: So far, we have set up a framework for discussion, including pointing out the key first principles of right reason that we must use so soon as we type out a message using distinct characters, etc. These are not provable, they are inevitable, inescapable and thus have a right to be presumed first truths of right reason.

    7: Now, W, holds distinct identity, it is a particular possible world, different from all others. That is, if claimed entities W1 and W2 are not discernibly different in any respect, they are just different labels for the same thing W.

    8: Notice, all along we are trafficking in statements that imply or assert that certain things are so or are not so, i.e. propositions and that relationship of accurate description of reality that we term truth.

    9: All of these are not merely concrete particulars or mere labels, they are abstracta which are inevitable in reasoning. Indeed, the relationship of intentionality implicit in attaching a name is an abstractum, too.

    10: Now, W is one of infinitely many possible states of affairs, and shares many attributes in common with others. So, we mark the in-common [genus] and the distinct [differentia].

    11: So, we freely identify some unique aspect of W, A. W, then is: W = {A|~A}.

    12: But already, we see rationally discernible abstract entities, principles and facts or relationship, quantity and structure; i.e. the SUBSTANCE of Mathematics. Namely,

    13: first, that which is in W but external to A and ~A is empty, as is the partition: nullity.

    14: Likewise, A is a distinct unit, as is ~A [which last is obviously a complex unity]. This gives us unity and duality.

    15: So, simply on W being a distinct possible world, we must have in it nullity, unity and duality. These are abstract structural and quantitative properties embedded in the framework for W.

    16: This is, strictly, already enough for the claim that there is an abstract substance of mathematical character that is necessarily embedded in any possible world, which is itself an abstract entity, being a collection of propositions. In at least one case such are actualised, i.e. it is possible to have an accurate summary of our world.

    17: However, much more is necessarily present, once we see the force of the von Neumann succession of ordinals (which substantiates Peano’s succession), actually presenting the natural counting numbers starting from the set that collects nothing, which is itself an undeniable abstract entity:

    {} –> 0
    {0} –> 1
    {0,1) –> 2
    {0,1,2} –> 3
    . . .
    {0,1,2,3 . . . } –> w [first transfinite ordinal]
    etc, without limit

    18: We here have N. Define for some n in N, that -n is such that n + (-n) = 0, and we equally necessarily have Z. Again, rooted in the distinct identity of a world, we are studying, exploring, discovering, warranting (as opposed to proving), not creating through our culturally influenced symbolism and discussion.

    19: Similarly, identify the ratio n:m, and we attain the rationals, Q.

    20: Use power series expansions to capture whole part + endless sum of reducing fractions converging on any given value such as pi or e or phi etc, and we have the reals, R, thus also the continuum. Where, from Z on, we have has entities with magnitude and direction, vectors.

    21: Now, propose an operation i*, rotation pivoting on 0 through a right angle. This gives us i*R, an orthogonal axis with continuum, and where for any r in R+, i*r is on the new [y] axis.

    22: Now too, go i*i*r, and we find -r. That is we have that i = sqrt(-1), which here has a natural sense as a vector rotation. Any coordinate in the xy plane as described is now seen as a position vector relative to the origin.

    23: We have abstract planar space, thus room for algebraic and geometrical contemplation of abstract, mathematically perfect figures. For instance consider the circle r^2 = x^2 + y^2, centred on o.

    24: In its upper half let us ponder a triangle standing at -r [A] and r [B] with third vertex at C on the upper arc. This is a right angle triangle with all associated spatial properties, starting with angle sum triangle and Pythagorean relationships, trig identities etc. Between these two figures and extensions, the world of planar figures opens up.

    25: Extend rotations to ijk unit vectors and we are at 3-d abstract “flat” space. All of this, tracing to distinct identity.

    26: We may bring in Quaternions and Octonions, the latter now being explored as a context for particle physics.

    27: The Wigner Math-Physics gap is bridged, at world-root level.

    28: Similarly, we have established a large body of intelligible, rational entities and principles of structure and quantity implicit in distinct identity. Such are the substance we discover by exploration (which is culturally influenced) rather than invent.

    29: Where it is an obvious characteristic of invention, that it is temporally bound past-wards, Until some time t, entity e did not exist. Then, after t, having been created, it now exists.

    30: The above abstracta are implicit in the distinct identity of a world and so have existed so long as reality has. That is, without past bound. (It can readily be shown that if a world now is, some reality always was.)

    I have expanded in more details.

    KF

  258. 258
    kairosfocus says:

    H, argumentum ad quantum is common and too often goes beyond what is warranted. A key result is, once things are scaled to macro levels, Q-results must reduce to classical ones. Further to this, we have the issue of observable empirical reliability as the context of science. The formal quantum result of position-momentum or energy-time uncertainty etc at macro scale have effectively no practical import. I note too that the Weak Argument correctives address other sides of the appeal to quantum results, including how first principles of reason are not obviated by Q-th. KF

    PS: Abstract entities are those things which are not concrete and tangible. truth is intangible, a relationship between description and its meaning referred to states of actual affairs, for example. Sets and other collectives are abstract and can collect abstracta including subsets as above. Numbers, which we can take as effectively defined per von Neumann and extensions, are abstracta. The counting trick exploits the ordinal succession, which is another abstract entity. And so forth. Abstract does not imply imaginary, fictional, not real.

  259. 259
    hazel says:

    kf writes, “Abstract does not imply imaginary, fictional, not real.”

    I agree. I have never said abstractions are imaginary or fictional. Abstractions are real. The question is where does their reality lie?, and the answer I am offering is that they exist in our minds.

    The things that abstractions are abstractions of can be physical real, but the abstraction never encapsulates all of reality: that is one of the main points of Wigner, and mine, above.

  260. 260
    hazel says:

    re kf’s response to Ed’s question at 256:

    Kd writes, “It is therefore to the merits of fact and logic that we must always go.”

    Yes, but still individual people still have to judge whether adequate “merits of fact and logic” have been presented, which gets us back to Ed’s original question. Kf claims that he has put “demonstrative warrant on the table”, and that therefore “opinions to the contrary avail nothing”, apparently dismissing arguments against his position or for any other position as mere opinions.

    But who judges whether kf has in fact put “demonstrative warrant on the table” is still an issue. For the person who puts the facts and logic on the table to be the person who then judges they are impeccably sound is circular. Of course we are inclined to think that the facts and logic we ourselves are presenting are sound, but it is other’s judgment of that that eventually leads to the acceptance of that soundness.

    If I offer a proof of something in math, I expect that others will review my work, and either agree, or perhaps find a flaw in my reasoning. Given that the subjects we are talking about are philosophical about math, and not math itself, our “facts and logic” includes propositions which are themselves a matter of perspective, not subject to proof: we all have some philosophical opinions that provide some framework for the facts and logic we assess.

    So for kf to say of the issues we have been discussing, “This is my philosophical perspective, and these are my supporting facts and logic: I think they are solid and I present them for your consideration” is reasonable. For him to say that “I have put ‘demonstrative warrant on the table’, and that therefore ‘opinions to the contrary avail nothing’,” is not reasonable.

  261. 261
    ET says:

    The evidence says there is demonstrative warrant on the table

  262. 262
    Ed George says:

    ET

    The evidence says there is demonstrative warrant on the table

    That statement asserts that the person making it will not accept any argument to the contrary. A more appropriate statement would be “The evidence suggests there is demonstrative warrant on the table.”

  263. 263
    ET says:

    Ed George:

    That statement asserts that the person making it will not accept any argument to the contrary.

    No, it does not. My you are one desperate person.

  264. 264
    kairosfocus says:

    EG, it is obvious that the answer to a demonstration is a counter demonstration. That you resort to opinion games suggests that you do not have such, but do not wish to acknowledge the probative force of an argument that literally starts with the first principles of reason and being. Instead you try to create a loaded projection of closed mindedness. That speaks, volumes. KF

  265. 265
    kairosfocus says:

    H, once demonstrative warrant is on the table, indeed OPINIONS to the contrary avail nothing. A counter demonstration or a p => q, ~q so ~p that does not end in absurdity would be a different story. But the mater on the table starts with first principles of reason and being which are for the most part self-evident. this meaning that they are true, seen as true by one with the experience and insight to understand, are seen as necessarily so on pain of immediate, patent absurdity on the attempted denial. See how far you get trying to deny the principle of distinct identity while relying on said distinction to communicate using distinct symbols. Note, the identity of any world leads to numerical and structural consequences, where numbers are inherently abstract, and where perforce, such antedates our existence and cannot be a creation. The resistance without counter-demonstration, but including loaded suggestions is a tell. KF

  266. 266
    Ed George says:

    KF

    EG, it is obvious that the answer to a demonstration is a counter demonstration.

    No, it is not obvious. Another answer could be that the premises and assumptions used for your demonstration are flawed, which has been pointed out repeatedly through this 250+ comment thread. I have also mentioned that Hazel and I agree with you more than we disagree with you. It is just that we are approaching the issue from different perspectives.

    Just for the record my views align largely with those that Hazel has presented much better than I can.

    KF, on a side note, might I suggest that you put a leash on ET? He often has very thought provoking things to say but intersperses them with things like comment 263. Nobody has treated him or anyone else like he treats those he disagrees with. Being the thread owner, you are free to allow whatever behavior you like, but allowing it reflects poorly on the argument you are trying to make.

  267. 267
    ET says:

    Ed George:

    No, it is not obvious.

    It is to anyone who understands evidence

    Another answer could be that the premises and assumptions used for your demonstration are flawed,…

    Too subjective to be of any use.

    Nobody has treated him or anyone else like he treats those he disagrees with.

    Demonstrably FALSE. And AGAIN, it has nothing to do with mere disagreement.

    And you still don’t have an argument. But your whining is very telling.

  268. 268
    ET says:

    Perhaps if Ed George wouldn’t make so many bald assertions he wouldn’t feel attacked when he his soundly dismissed.

    Perhaps if Ed George didn’t think his bald assertions were actual arguments and evidence, he wouldn’t feel attacked when he is corrected.

    The common denominator? Ed George and bald assertions

  269. 269
    Ed George says:

    EG

    Another answer could be that the premises and assumptions used for your demonstration are flawed,…

    ET in response

    Too subjective to be of any use.

    The assumptionsand premises we use to make arguments is a highly subjective process. Absolutely necessary, but subjective none the less.

    For example, in analytical chemistry we assume that the procedure we use accounts for all possible interferences. However, we also know that, at a fundamental level, this is not true. That is why we always report our analytical results with measurement uncertainty.

  270. 270
    ET says:

    Whatever, Ed. In science and math the only way to go is through evidence. But I understand why you wouldn’t want to go that route.

  271. 271
    kairosfocus says:

    EG, again, we are dealing with warrant on logic, facts, first principles. I am sure you have seen logical cases in school and know that p => q, p so q is a logically valid, chainable pattern of argument. It is also the case that ~q => ~p in such a case. The challenge then is, is the denial of q absurd and/or is the denial of p. If you are a serious participant in this discussion, you will know that a common proof pattern is to reduce an argument to absurdity. That is what has always been on the table and it is one reason why the injection of relativistic or subjectivist rhetoric pivoting on the Leff grand sez who fallacy fails. When a demonstration on good premises is on the table, it settles a matter by force of logic. If you wish instead to reject the conclusion to deny the premises, you face two potential points of absurdity: ~q may be false, and ~p may be false. In the case on the table, the start point is the principle of distinct identity and its corollaries. No one can use language or reason coherently without using the LOI. In your case, if that is what you want to try, notice that you are forced to rely on distinct alphanumeric characters arrayed in distinct locations, in strings. You literally cannot make your points without implicitly relying on the premises, which is exactly why LOI and corollaries, including the natural numbers (which then ground Z, Q, R, C etc) are embedded in any distinct world and are necessary, abstract entities. Neither in this thread nor elsewhere have you or any other objector managed to provide a successful ~q, where the attempt will run into the absurdity of relying on distinct identity to argue for its denial. I strongly suggest that you reconstruct your reasoning on a sounder footing and refrain from further projection of closed mindedness. KF

  272. 272
    hazel says:

    I have never denied the validity of logic, kf, and I agree that “No one can use language or reason coherently” without it. Several time I have pointed to the rational ability of our minds to logically manipulate concepts as a key component in human understanding. This is not in question. I’m guessing Ed would agree with me about this, but he can speak up if not.

    However, the existence of logic in our reasoning does not mean that the abstractions to which we apply that logic are “embedded in any distinct world and are necessary, abstract entities.”

    That is the point of disagreement. That is a philosophical premise, but it does not follow by any necessary logical chain from the presence of our use of logic in our reasoning process. Yes, starting with the idea of distinct identity as formalized in the unit amount and using the fundamental laws of logic, we have mathematics.

    But the relationship between the concepts of math in our minds and their application to the physical world is precisely the philosophical issue under discussion, and it in itself can not be resolved by pure logic alone. Hence different perspectives.

  273. 273
    kairosfocus says:

    H, on the table, again, is an argument that starts from distinct identity and shows that N will therefore be embedded in the fabric of any distinctly identifiable world W, with further things flowing therefrom. Such will clearly be antecedent to our existence much less our mathematical explorations. Where, numbers are necessarily abstract entities that through the associated logic of being, will affect many physical realities. I have often put up, gear trains, which inevitably pivot on pi. The issue is not so much, we can reason and invent, but there are rational principles and abstract entities demonstrated to be framework to any possible world that manifest structure and quantity, the substance of mathematics. KF

    PS: It seems from arguments made that EG objects to the probative force of arguments on first principles of reason and being. Indeed, it seems manifest that he just tried to debate the point that the answer to a demonstration, patently, is a counter-demonstration.

  274. 274
    hazel says:

    kf writes,

    N will therefore be embedded in the fabric of any distinctly identifiable world W, with further things flowing therefrom. Such will clearly be antecedent to our existence much less our mathematical explorations.

    You have not shown that these statements logically follow from the existence of pure mathematics. N may apply to certain aspects of the world as we experience it, but that is different than being embedded in it.

    I understand that your position has a long and important place in philosophy, but it is not “demonstrated warrant on the table.” It is one of a number of philosophical perspectives that have been offered by substantial philosophers over the centuries.

  275. 275
    kairosfocus says:

    H, there is a demonstration on the table; the last being an updated summary given various points made previously, e.g. I took time to explicitly go back to first principles of reason. Note, I started from the import of a distinct possible world existing, which leads to N, from which Z, Q, R, C and related matters follow. That is, we start from the root of being. Instead of dismissing, kindly address. KF

  276. 276
    hazel says:

    I did address.

    I hereby invoke 238 and 253.

  277. 277
    kairosfocus says:

    H, cf 257 with 248 (with 247), also 240, 241 and far more. KF

    PS: Note onward OP on how such embedding of structure and quantity leads to interesting physical results: https://uncommondescent.com/physics/logic-first-principles-8-bridging-the-wigner-math-physics-gap-with-help-from-phase-configuration-state-space/

  278. 278
    ET says:

    hazel:

    But the relationship between the concepts of math in our minds and their application to the physical world is precisely the philosophical issue under discussion, …

    And here I thought that the scientific discussion was about whether or not mathematics was invented or discovered by us- and whether or not mathematics permeates the universe because the universe was designed using mathematics.

  279. 279
    kairosfocus says:

    ET, notice the persistent context shift from substance of structure and quantity manifest in any possible, distinct world and our ideas about it. Of course, from the outset I have stressed that dual character by contrasting substance demonstrably embedded in the logic of being of any possible world and our culturally influenced study. When a demonstration is on the table mere opinions to the contrary avail nothing. A counter demonstration is called for. Persistent absence may be connected to how such an argument must implicitly rely on the key premise of my argument, distinct identity. LOI is truly the start-point of reasoning. KF

  280. 280

    Ed George and Hazel,

    Until you can logically refute the validity of KF’s argument for (1) the warrant he has described including necessary first principles and (2) the reason no other warrant will do, OR until you provide an alternate system of warrant that supports an alternate view on the issues under discussion, you have ceded the point. Simply disagreeing or claiming there may other arguments or systems of sufficient, credible warrant is not a refutation nor is it providing a reasoned alternative.

    That’s the point of what KF is saying – there is no other credible system of warrant, period, and that necessary system of warrant has logical ramifications that must be accept or reason must be abandoned. Just saying that “other philosophers disagree” provides you no cover. You either make your case (counter-argument) or you have ceded the point. “Other people disagree with you” and “This is problem that can’t be solved” are not counter-arguments.

  281. 281
    Ed George says:

    WJM

    Until you can logically refute the validity of KF’s argument for (1) the warrant he has described including necessary first principles and (2) the reason no other warrant will do, OR until you provide an alternate system of warrant that supports an alternate view on the issues under discussion, you have ceded the point.

    So, the fact that his assumptions and premises are flawed are irrelevant?

  282. 282
    Ed George says:

    KF, given your lack of response to my comment at 266 I can only conclude that you support and condone ET’s continued insults. That is very telling.

  283. 283
    kairosfocus says:

    EG,

    with all due respect, you have continued to argue by assertions instead of engaging substantially, starting with the significance of the law of identity and the associated observation that in absence of distinguishing characteristics, entities W1 and W2 are in effect different labels for W, the same entity.

    Now, you have in effect asserted errors on my part on your own authority, ipse dixit. In this context, that fails, you need to provide warrant. And if you dispute LOI, you need to do so without using distinction of alphanumeric characters to make your point.

    Now, you point to 266, by implication demanding a point by point response — where, you need to note those I gave above. At any rate:

    [EG, 266:] >>KF EG, it is obvious that the answer to a demonstration is a counter demonstration.>>

    a: this was a response to your earlier remark, and in the context of a serious discussion it is correct. If you object to a demonstration, provide a rebuttal, or stand as refuted.

    >>No, it is not obvious.>>

    b: This reveals dismissal of a commonplace principle of serious discussion, it falls of its own weight.

    >> Another answer could be that the premises and assumptions used for your demonstration are flawed, which has been pointed out repeatedly through this 250+ comment thread.>>

    c: repeated, insubstantial assertions do not constitute warrant.

    d: Kindly observe my response at 271:

    EG, again, we are dealing with warrant on logic, facts, first principles. I am sure you have seen logical cases in school and know that p => q, p so q is a logically valid, chainable pattern of argument. It is also the case that ~q => ~p in such a case. The challenge then is, is the denial of q absurd and/or is the denial of p. If you are a serious participant in this discussion, you will know that a common proof pattern is to reduce an argument to absurdity. That is what has always been on the table and it is one reason why the injection of relativistic or subjectivist rhetoric pivoting on the Leff grand sez who fallacy fails. When a demonstration on good premises is on the table, it settles a matter by force of logic. If you wish instead to reject the conclusion to deny the premises, you face two potential points of absurdity: ~q may be false, and ~p may be false. In the case on the table, the start point is the principle of distinct identity and its corollaries. No one can use language or reason coherently without using the LOI. In your case, if that is what you want to try, notice that you are forced to rely on distinct alphanumeric characters arrayed in distinct locations, in strings. You literally cannot make your points without implicitly relying on the premises, which is exactly why LOI and corollaries, including the natural numbers (which then ground Z, Q, R, C etc) are embedded in any distinct world and are necessary, abstract entities. Neither in this thread nor elsewhere have you or any other objector managed to provide a successful ~q, where the attempt will run into the absurdity of relying on distinct identity to argue for its denial. I strongly suggest that you reconstruct your reasoning on a sounder footing and refrain from further projection of closed mindedness.

    >> I have also mentioned that Hazel and I agree with you more than we disagree with you.>>

    e: agreement or disagreement are immaterial, substantial warrant for claims is required, which has not been provided across many weeks on multiple threads.

    f: For record, I take from 257, what you or another party would need to refute:

    1: Consider reality, and within it some distinct entity, say a bright red ball on a table, B. Thus the rest of reality is the complement to B, ~B. Reality, R = {B|~B}

    2: Immediately, B is itself (distinctly identifiable i/l/o its core, distinguishing characteristics), this is the fundamental law of thought, Law of Identity, which sets up the dichotomy and its corollaries.

    3: Clearly, no x in R can be B AND ~B. Law of non-contradiction, a corollary.

    4: Likewise, any x in R must be in B or in ~B, not between them or separate from them: B X-OR ~B, law of the excluded middle. The second corollary.

    5: Now, ponder a possible world, W, a sufficiently complete description of a possible [coherent!] state of affairs in reality,i.e. in this or any other world that could be or is.

    6: So far, we have set up a framework for discussion, including pointing out the key first principles of right reason that we must use so soon as we type out a message using distinct characters, etc. These are not provable, they are inevitable, inescapable and thus have a right to be presumed first truths of right reason.

    7: Now, W, holds distinct identity, it is a particular possible world, different from all others. That is, if claimed entities W1 and W2 are not discernibly different in any respect, they are just different labels for the same thing W.

    8: Notice, all along we are trafficking in statements that imply or assert that certain things are so or are not so, i.e. propositions and that relationship of accurate description of reality that we term truth.

    9: All of these are not merely concrete particulars or mere labels, they are abstracta which are inevitable in reasoning. Indeed, the relationship of intentionality implicit in attaching a name is an abstractum, too.

    10: Now, W is one of infinitely many possible states of affairs, and shares many attributes in common with others. So, we mark the in-common [genus] and the distinct [differentia].

    11: So, we freely identify some unique aspect of W, A. W, then is: W = {A|~A}.

    12: But already, we see rationally discernible abstract entities, principles and facts or relationship, quantity and structure; i.e. the SUBSTANCE of Mathematics. Namely,

    13: first, that which is in W but external to A and ~A is empty, as is the partition: nullity.

    14: Likewise, A is a distinct unit, as is ~A [which last is obviously a complex unity]. This gives us unity and duality.

    15: So, simply on W being a distinct possible world, we must have in it nullity, unity and duality. These are abstract structural and quantitative properties embedded in the framework for W.

    16: This is, strictly, already enough for the claim that there is an abstract substance of mathematical character that is necessarily embedded in any possible world, which is itself an abstract entity, being a collection of propositions. In at least one case such are actualised, i.e. it is possible to have an accurate summary of our world.

    17: However, much more is necessarily present, once we see the force of the von Neumann succession of ordinals (which substantiates Peano’s succession), actually presenting the natural counting numbers starting from the set that collects nothing, which is itself an undeniable abstract entity:

    {} –> 0
    {0} –> 1
    {0,1) –> 2
    {0,1,2} –> 3
    . . .
    {0,1,2,3 . . . } –> w [first transfinite ordinal]
    etc, without limit

    18: We here have N. Define for some n in N, that -n is such that n + (-n) = 0, and we equally necessarily have Z. Again, rooted in the distinct identity of a world, we are studying, exploring, discovering, warranting (as opposed to proving), not creating through our culturally influenced symbolism and discussion.

    19: Similarly, identify the ratio n:m, and we attain the rationals, Q.

    20: Use power series expansions to capture whole part + endless sum of reducing fractions converging on any given value such as pi or e or phi etc, and we have the reals, R, thus also the continuum. Where, from Z on, we have has entities with magnitude and direction, vectors.

    21: Now, propose an operation i*, rotation pivoting on 0 through a right angle. This gives us i*R, an orthogonal axis with continuum, and where for any r in R+, i*r is on the new [y] axis.

    22: Now too, go i*i*r, and we find -r. That is we have that i = sqrt(-1), which here has a natural sense as a vector rotation. Any coordinate in the xy plane as described is now seen as a position vector relative to the origin.

    23: We have abstract planar space, thus room for algebraic and geometrical contemplation of abstract, mathematically perfect figures. For instance consider the circle r^2 = x^2 + y^2, centred on o.

    24: In its upper half let us ponder a triangle standing at -r [A] and r [B] with third vertex at C on the upper arc. This is a right angle triangle with all associated spatial properties, starting with angle sum triangle and Pythagorean relationships, trig identities etc. Between these two figures and extensions, the world of planar figures opens up.

    25: Extend rotations to ijk unit vectors and we are at 3-d abstract “flat” space. All of this, tracing to distinct identity.

    26: We may bring in Quaternions and Octonions, the latter now being explored as a context for particle physics.

    27: The Wigner Math-Physics gap is bridged, at world-root level.

    28: Similarly, we have established a large body of intelligible, rational entities and principles of structure and quantity implicit in distinct identity. Such are the substance we discover by exploration (which is culturally influenced) rather than invent.

    29: Where it is an obvious characteristic of invention, that it is temporally bound past-wards, Until some time t, entity e did not exist. Then, after t, having been created, it now exists.

    30: The above abstracta are implicit in the distinct identity of a world and so have existed so long as reality has. That is, without past bound. (It can readily be shown that if a world now is, some reality always was.)

    >>It is just that we are approaching the issue from different perspectives.>>

    g: Difference of perspectives translates to divergence of presuppositions and first principles or truths. In this case, the pivotal truth, distinct identity and its law of being import that a distinct possible world must have some distinguishable feature A, are things that are at root literally undeniable. The attempt to deny will inevitably resort to using such, just to post text.

    >>Just for the record my views align largely with those that Hazel has presented much better than I can.>>

    h: In short, you are playing the cheerleading sidekick.

    i: I have responded substantially, and H has not had a material counter. WJM’s summary applies.

    >>KF, on a side note, might I suggest that you put a leash on ET? He often has very thought provoking things to say but intersperses them with things like comment 263. Nobody has treated him or anyone else like he treats those he disagrees with.>>

    j: ET has been banned for cause before, if he merits it, those who hold that power will not hesitate to do so again.

    k: Meanwhile I have RW to attend to and insomnia power only goes so far. I have already recently cautioned ET, which you have overlooked.

    l: Here is 263:

    Ed George:

    That statement [by KF, to effect that a demonstration requires a counter demonstration and that mere opinions to the contrary are of no avail] asserts that the person making it will not accept any argument to the contrary.

    No, it does not. My you are one desperate person.

    m: You have made an unjustified, loaded insinuation of closed mindedness in reply to a simple statement that reasoned argument of substantial character obviously must be answered by substantial argument on the merits. That’s why I disagree and dismissive remarks targetting first principles of reason are not enough.

    n: This is an ad hominem, subtle form, and it has been sustained in teeth of repeated correction.

    o: The first part of ET’s reply is exactly correct, your assertion of closed mindedness on my part is an unjustified false accusation, a loaded projection which may well fail the mirror principle test for defence mechanisms.

    p: Where, that sort of tactic is in fact a common device of the concern troll, again another point where you would be well advised to adjust your behaviour.

    q: ET then characterises you as desperate. In the context of weeks of sustained dismissive remarks which have not been substantiated and given repeated endorsement of failed arguments, that may well be fair, if harsh comment. Harshness being mitigated by persistence.

    r: You would be well advised to note that WJM’s policy of deleting insubstantial persistent dismissive, distractive commentary is under consideration. Part of why I am tolerating is that this thread is in root about fallacies, and cases in point end up inadvertently supporting the main point.

    >> Being the thread owner, you are free to allow whatever behavior you like, but allowing it reflects poorly on the argument you are trying to make.>>

    r: Again, a resort to emotive appeals of distractive character.

    In the end, it is clear that you are unable or unwilling to substantially engage the technical issue imported into the thread from previous discussions. You have projected an unjustified personal attack, accusing me of the fallacy of the closed mind when I have repeatedly invited a substantial counter argument, with but little response on the merits.

    You have provided examples of subtler fallacies, in effect, which is a useful if inadvertent service.

    KF

  284. 284

    Ed George said:

    So, the fact that his assumptions and premises are flawed are irrelevant?

    I must have missed where you (or someone) showed that his premises are flawed. Would you care to direct me to that (or those) posts?

  285. 285

    Hazel said:

    I agree. I have never said abstractions are imaginary or fictional. Abstractions are real. The question is where does their reality lie?, and the answer I am offering is that they exist in our minds.

    Until you define “mind” and what it means to “exist in mind”, you are begging the question.

  286. 286
    ET says:

    For Ed George

    –> No need to rub in salt. KF

  287. 287
    kairosfocus says:

    H, the above and the newer thread provide answers. The demonstration, on logic. Cases up to and beyond a Mobius strip, physically. Twisted paper loops that separate differently if you cut in the middle vs 1/3 way across are not figments of our conceptual imagination. They are telling us that logic of being structural and quantitative factors directly constrain space and objects in it, independent of our opinions. KF

    PS: I will shortly add the 1/2 way across vs 1/3 way across Mobius strip cut vid to the OP. Why not carry out the physical exercises? U/D: added.

  288. 288
    hazel says:

    William, you’ve said this before. “Exist in mind” means it’s something I’m aware of in my consciousness, or have access to in my consciousness at relevant times. “Existing in mind” is an experiential fact, not something to be defined.

    I can’t explain what consciousness actually is, or how things exist in my mind, or how the mind interfaces with my body, and neither, I think, can you or anybody else.

  289. 289
    hazel says:

    kf, you are wasting your time posting all these examples. We all know that math can describe the world, and that modeling relationships between pure math and actual events in the physical world exist, and that there is a mystery in Wigner’s words, as the unreasonable effectiveness of this relationship.

    The only issue that I am addressing is your claim that the abstractions which exist in our minds as parts of pure math are embedded in the physical world. The physical world contains physical things which behave in certain ways that can be described by our mathematical abstractions, but those abstractions don’t themselves exist in the physical world.

    This is a philosophical point of view, and your position is also. Your Platonic premise that the abstractions exist antecedent to the physical world, are embedded in it, and constrain it may be true, but it is not demonstrably true. Other premises lead to different conclusions, and may also be true. The history of philosophy shows clearly that there is no definitive way to resolve this issue.

    So no further amount of examples of neat math and neat examples in the physical world will add any more weight to your perspective.

  290. 290
    ET says:

    hazel:

    We all know that math can describe the world, and that modeling relationships between pure math and actual events in the physical world exist, and that there is a mystery in Wigner’s words, as the unreasonable effectiveness of this relationship.

    Except there is only a mystery to those who deny the obvious.

    The only issue that I am addressing is your claim that the abstractions which exist in our minds as parts of pure math are embedded in the physical world.

    That sounds like a strawman. There is much that is embedded and then rest exists outside of that.

  291. 291
    kairosfocus says:

    H,

    in re 288: >>“Exist in mind” means it’s something I’m aware of in my consciousness, or have access to in my consciousness at relevant times. “Existing in mind” is an experiential fact, not something to be defined.>>

    1 – tantamount to, a label or awareness, or a concept not a description applicable to external reality, the Kantian gap again.

    2 –> This fails, as nominalism and Kantian gaps fail, also such are here sustained as opinions in the teeth of demonstrations not countered through showing invalid or unsound.

    3 — demonstration pivoting on inescapable first principles.

    289: >>you are wasting your time posting all these examples>>

    4 — as cases show direct physical instantiation of abstracta constraining extramental being, they are relevant.

    5 — have you set up and done the cut exercises on Mobius strips? How do these NOT show that space embeds key quantitative and structural properties that are independent of our opinions, i.e. objectively . . . and often amazingly . . . fact-on-the-ground present?

    6 — How is momentum or energy not an abstract quantity, showing also fundamental theorem of calculus through cumulative effects of force through space and time?

    7 — How is the capillary effect hyperbola not, again, an objective demonstration?

    8 – Likewise, the behaviour of a projectile.

    >>We all know that math can describe the world, and that modeling relationships between pure math and actual events in the physical world exist>>

    9 – It seems, your premises lock out something that is evident: structure and quantity are embedded in space, objects and dynamic processes.

    10 – Moreover, effects of energy and momentum conservation are palpably evident and observable, the cut Mobius strips are not in the mind, they can be held in hands, the projectile’s behaviour is observable and reliably predictable, hence gunlaying.

    11 — these call for bridging the Wigner Math-Physics gap and we have a demonstration on the table that the link comes through distinct identity and linked logic of being.

    12 — you need not accept, you are free; we are also free to draw the conclusion that by failing to provide serious counter warrant and sidestepping manifest demonstrations you are in the position of posing opinions against demonstrations from undeniable first principles AND disputing or dismissing significance of facts of observation.

    13 — Conclusion, the balance on merits is decisive and not favourable to the objections you have made.

    KF

  292. 292

    Hazel @ 288,

    All you are doing is rephrasing the same thing a different way. Until you have a model that explains/describes what it means to “exist in mind” or “exist in consciousness”, you’re not providing an alternative to platonic realism, you’re begging the question and ceding the point by default. Platonic realism provides a model of what it means to “exist in mind” which can then be logically examined back to necessary principles, experiential (empirical, first-person) evidence, and forward to conclusions – which both KF and I (and others) have done. You can either show how the model(s) on the table are logically flawed; provide your own competing model; or cede the point that you have no competing model and no argument to make against those already on the table.

    This is the WHOLE POINT of the discussion, and you’re avoiding providing us with the most necessary aspect of any counter-proposal or criticism.

  293. 293
    hazel says:

    Sorry, I can’t explain consciousness and don’t know what mind is other than my experience of it. If that disqualifies me in your eyes from disputing your philosophy, then there isn’t any sense in your paying any attention to me, it seems, I think I need a new meme.

  294. 294
    kairosfocus says:

    H, labelling what you disagree with as a “philosophy” does not change the warrant from literally first principles of reason that is on the table. That warrant that certain structural and quantitative entities are embedded in this or any other possible world, is not rooted in debatable presuppositions but in distinct identity of a possible world and what it entails. It is impossible to argue against distinct identity as to argue you must use it. The structural and quantitative consequences follow on inspecting what that means once we acknowledge that two different things W1 and W2 must differ in at least some aspect A, or else they are but labels for the same thing W. That is what you need to overturn. If in objecting, you mean that a valid deduction on true premises leads to true conclusions is to be dismissed, good luck. If you mean the reasoning does not follow, show why, though it will be hard to see how W = {A|~A} does not show two distinct units, or that outside A and ~A but in W there is no thing, or that inside a partitioning element there is likewise no thing. Thus, we see nullity, unity (including complex unity BTW) and duality. Thus, embedded quantitative aspects. From this, the von Neumann succession follows thence N, thereafter Z, Q, R, C etc. KF

  295. 295

    Hazel,

    Sorry, I can’t explain consciousness and don’t know what mind is other than my experience of it.

    You’re not being asked to know. You’re being asked to present a theoretical model that is subject to rational criticism that explains (describes) the existence of universal, discoverable abstract concepts.

    One such model describes “mind” as the experience of purely material forces, and in concept experienced “mentally” to be the product of material forces. (Materialism / Nominalism)

    Another model describes “mind” (or at least the universal aspects of it) to be (at least) the substrate upon which the physical universe exists. (Platonic Realism).

    You’ve rejected both models, if memory serves, offering no logical criticism or alternative model. Until you do so, “exists in my mind” cannot move the conversation forward because that is the beginning of the conversation. Yes, WE ALL KNOW these things “exist in my mind”; that is not being debated. What is being debated is (1) how do they exist in mind (theoretical model), and (2) what are the logical implications of any such model and (3) does that model comport with (explain, describe) what we actually experience (the universality of some things that exist in our mind, and the apparent capacity to discover those universals, and their apparent intimate and necessary relationship with the physical world)?

  296. 296
    hazel says:

    You got me, William. I don’t have an alternate model.

    I don’t know how abstract concepts exist in the mind. I don’t know if they are “universal” concepts in the sense of existing someplace other than in my mind and other people’s minds, and I don’t know how to explain the way the math in our minds has an”apparent intimate and necessary relationship with the physical world.”

    And neither do you. The difference is that you think you know, and I know I don’t. I think you are wrong about knowing. You have a theoretical model, and so does the naturalist, but neither is testable. I’m content to have the same attitude Wigner did when he said:

    The full meaning of life, the collective meaning of all human desires, is fundamentally a mystery beyond our grasp. … I have made peace with it. I even feel a certain honor to be associated with such a mystery.

    That expresses my feelings pretty well. YMMV, and I don’t expect you to like it, but I’d rather be honest to myself about what I know and don’t know then to attach a greater sense of certainty than is warranted to philosophical things that are more thought games than they are grounded in experience.

  297. 297

    Hazel,

    You don’t seem to understand some things I’ve explicitly stated. I’m not asking or expecting anyone here to “know” how things exist in the mind. The fact that you and I don’t know (if by “know” you are referring to certainty) is irrelevant. I never said that I know my model to be true. I don’t know that it is true; frankly, I don’t really care as long as it is useful. I do know that it is logically sound (at least, so far irrefutable via rational criticisms), and cogently explains (describes) the issues under examination and common experience.

    I also know that KF’s model is logically sound and although we disagree on whether or not the existence of an actual material world is necessary for the logic to remain sound, we present each other with logical arguments and criticisms in our discussion.

    Unless you provide a rational criticism of the models offered, or provide an alternative theoretical model that is subject to rational criticism and debate, all you can be doing here is offering personal exposition.

    Thought games? ROFL. I don’t waste my time on “thought games”. I’m a philosophical pragmatist. If there was no practical, experiential payoff for doing my best to understand the nature of mind, I wouldn’t bother.

    IMO, understanding mind is the single most important task one can embark on. It’s all we have to make decisions, sort our experience out and make sense out of our very existence. It’s the root of our very self-identity. If you want to throw your hands up and proclaim that it is some unsolvable mystery, fine, but don’t expect the rest of us to not call out your groundless, irrational and undefended exposition for what it is.

  298. 298
    hazel says:

    William, my view is useful to me. If that is the criteria, then let us each go our own way.
    Also, I agree about the importance of understanding mind, and with “It’s all we have to make decisions, sort our experience out and make sense out of our very existence. It’s the root of our very self-identity. ”

    But I’m not “throwing up my hands”: I’m making a considered decision to not think I know more than I do, and to work doing all the things you say above (decisions, making sense, etc.) with the knowledge I do have.

    I’ve said I think you and kf both think you know more than you really can: you’ve invented your own set of philosophical abstractions and a logic that holds them all together, but they are a self contained system without any definitive appeal to any experience that can validate them to others.

    So you live with your useful theory, and kf with his, and me with mine, including my willingness to live with not thinking I know more than I do.

  299. 299
    vividbleau says:

    Hazel
    “I’m making a considered decision to not think I know more than I do,”

    Sheesh WJM has already covered this ground several times ( in 295 and 297)why do you insist in bringing up that which is not an issue?

    Vivid

  300. 300
    ET says:

    Total nonsense- the naturalist does not have any models. The naturalist cannot explain the existence of life.

    And now hazel is telling us what we can and cannot know. How pathetic is that?

  301. 301

    Hazel,

    You seem to be intent on not understanding the nature of my conversation with you, and on misunderstanding my position (as Vividbleau points out @299 – BTW thanks VB, I was thinking maybe I somehow wasn’t being clear). Let’s look at what you say in 298 and see if we cannot reach an understanding.

    “William, my view is useful to me. If that is the criteria, then let us each go our own way.”

    The criteria for what? If you mean, the criteria for how one sorts their views, then yes. Each individual can apply whatever criteria they deem fit to acquire and hold beliefs and views. The individual doesn’t even have to defend or argue those views.

    However, the criteria for rationally discussing, debating and arguing those views is that one is willing to submit those views to rational criticism, part of which is a willingness to concede flaws in one’s own position and admit when they cannot offer any rational criticism against a particular counter-argument.

    Additionally, you keep returning to the theme of “let’s go our own way,” or something similar yet – so far – continue to engage. Why? Do you actually expect others to stop responding to your posts? Do you expect yourself to do so at some point in the future?

    Further, let’s look at a couple of statements you’ve made:

    Also, I agree about the importance of understanding mind, and with “It’s all we have to make decisions, sort our experience out and make sense out of our very existence. It’s the root of our very self-identity. ”

    ..and:

    That expresses my feelings pretty well. YMMV, and I don’t expect you to like it, but I’d rather be honest to myself about what I know and don’t know then to attach a greater sense of certainty than is warranted to philosophical things that are more thought games than they are grounded in experience.

    This, again, seems to be a recurring pattern … dismissing the whole exercise as either impenetrable (… no one can know) or ultimately unimportant and a kind of side-discussion you’ll suffer through as long as you feel like it, or, in contrast, an extremely important – if not THE single most important – introspection, argument, discussion and line of thought one can embark on that is inextricably embedded in the very nature of experience. It seems like you try to have it both ways at different times during the conversation – something I noticed some time back when, on one of my threads, I told you to either commit to the C/A/D or get off the thread.

    You said:

    I’m making a considered decision to not think I know more than I do,…

    Interesting, but is it true? Let’s look at recurring statement you make:

    @296:

    I don’t know how abstract concepts exist in the mind….And neither do you. The difference is that you think you know …

    How did you know that I don’t know before I agreed to it? Why is it that after I have agreed with you that I don’t know, you keep characterizing me as feeling like I “know” it?

    You have a theoretical model, and so does the naturalist, but neither is testable.

    How do you know they are not testable? You haven’t even asked me if my theory is testable. Did I not tell you that unless my theory had practical applications, I would have no use for it? Doesn’t that at least imply that it does have testable predictions?

    @289:

    Your Platonic premise that the abstractions exist antecedent to the physical world, are embedded in it, and constrain it may be true, but it is not demonstrably true. Other premises lead to different conclusions, and may also be true. The history of philosophy shows clearly that there is no definitive way to resolve this issue.

    Can you support your assertion that it is not demonstrably true? Wouldn’t an honest assessment of “what you know and what you do not know” mean admitting that you do not know if it is demonstrably true or not?

    I think at some point you admitted that you weren’t much of a philosopher. Are you a student of philosophy? I wonder how it is that you so easily state that the “history of philosophy shows clearly” that the issue cannot be resolved definitively. Is this something you know based on research into the matters and having a deep understanding of the various arguments; and, even if so (which I don’t think is the case), why would that mean that the issue cannot be resolved definitively, even if prior attempts have failed?

    This doesn’t really sound like a person that admits to herself what she doesn’t know. This pattern looks more like a person who admits what she doesn’t know when it suits her (apparently, when pressured to offer their own charactizations, definitions and models of what “exist in mind” means), but makes assertions about things as if she knows when it suits her – even assertions about what is going on in the minds of others when they have explicitly said otherwise.

    Another interesting assertion of knowledge on your part:

    But the relationship between the concepts of math in our minds and their application to the physical world is precisely the philosophical issue under discussion, and it in itself can not be resolved by pure logic alone. Hence different perspectives.

    How do you know it cannot be resolved by pure logic alone? (And, anyway, it’s not “pure logic alone”; the argument is also based on agreed mutual experience of the world and in mind.)

    I’d like to go back to an earlier comment of yours:

    Yes, but still individual people still have to judge whether adequate “merits of fact and logic” have been presented, which gets us back to Ed’s original question. Kf claims that he has put “demonstrative warrant on the table”, and that therefore “opinions to the contrary avail nothing”, apparently dismissing arguments against his position or for any other position as mere opinions.

    But who judges whether kf has in fact put “demonstrative warrant on the table” is still an issue. For the person who puts the facts and logic on the table to be the person who then judges they are impeccably sound is circular. Of course we are inclined to think that the facts and logic we ourselves are presenting are sound, but it is other’s judgment of that that eventually leads to the acceptance of that soundness.

    What, in your view, is the proper way to judge a logical argument or model? The reason KF says “opinions” are irrelevant is because they are. The only thing that matters in the debate is whether or not anyone can point out a flaw in the reasoning. Period. Your “opinion” on the matter is irrelevant. You or EG “not being convinced” is irrelevant. You not being able to accept the implications is irrelevant. You either point out the flaws in the logic, offer a sound alternative, or you have conceded that the argument on the table is valid. That doesn’t make it true. That doesn’t mean I or anyone presenting the argument know it is true. Whether it is actually true, and whether or not KF or I or others “know” it is true, is entirely irrelevant. The argument/model is either logically sound, or it is not. You either have a rational criticism or you do not. You either have a logically (and evidentially) sound alternative, or you do not.

    Saying “other philosophers disagree” is not pointing out the flaw or offering an alternative. Saying “no one can know that the model on the table is true” is irrelevant. Posting quotes by philosophers that are happy to accept not knowing is not a rational criticism or counter-argument.

    I AGREE that I do not know. I don’t hold ANY belief so dearly as to say “I know with certainty”. “Nobody can know with certainty” is denialism and sophistry and has no place in a rational discussion. The only thing I hold as an absolute certainty is “I experience” (whatever “I” am, and whatever “experience” is).

    Please abandon the rhetoric and engage the logic. If the logic premise is unsound, don’t just say “you disagree” or “others disagree” or “you can’t know that”; show how the premise is flawed. The same with the extending logic – show how it is flawed, don’t just claim it could be and that nobody really knows and others disagree.

    If you can’t do that, then the model/argument of platonic realism stands as valid until shown otherwise, whether or not it is actually true, and whether or not anyone “knows” it is true, whether or not anyone likes it or has this or that opinion of it.

    You’ve already admitted you don’t have an alternative model/argument, so that route doesn’t seem to be an available option.

  302. 302
    hazel says:

    william, you write, “Additionally, you keep returning to the theme of “let’s go our own way,” or something similar yet – so far – continue to engage. Why? Do you actually expect others to stop responding to your posts? Do you expect yourself to do so at some point in the future?”

    This is a key question: what good am I getting out of this, and what somewhat compulsive feelings keep me responding? I’m been thinking about this, and will try to deal with it.

  303. 303

    Hazel,

    To be clear, I don’t consider anything that goes on here a “win” or a “loss”. I contribute here in on certain subjects because I invite rational criticism. I want to discover flaws in my reasoning if they exist.

    It’s perfectly fine, IMO, to simply hold any view one wishes, whether or not they can be defended rationally – that’s our innate right via our free will. Usually, people can have all sorts of ill-considered views (not saying that yours fall in this category, just generally speaking) and lead a very successful and happy life. So, when it comes to practical use, even totally unexamined and irrational beliefs are usually adequate.

    IMO, the significant question is, can anything meaningful be gained if one develops a better model of mind, a model that makes some predictions that can (at least personally) be experimented with? What if a better model of mind is like a better theory of physics, and actually opens a door to a whole new world of practical application, much like quantum physics did?

    Food for thought.

  304. 304
    kairosfocus says:

    WJM (attn H):

    One of the themes that keeps surfacing is “certainty,” which sets up the issues: warrant, knowledge, reliability, credibility, and responsibility.

    Given that we ever so often use knowledge in a sense that is less than absolute, irrefutable certainty, as in science, I have put on the table that knowledge speaks of warranted, credibly true (so, reliable) belief. Obviously, this is normally provisional, but it leaves open cases where the degree of warrant and credibility are such that these claims are utterly certainly true and beyond doubt, save by the irresponsible. That is where self-evident truths and inescapable first principles of right reason live. Such includes mathematical truths of the order ||| + || –> |||||.

    Thus, we see that warrant comes in degrees and must attain to some degree of reliability that lends the credibility that leads to responsible belief. This may be less than certain, e.g. by and large scientific theories and models are not certain, they are open to correction on many grounds. Yes, Science is at a relatively low rung on the warrant ladder. Observations of science are another matter, they carry with them the credibility of witness, which can be morally certain.

    So, certainty is now on the table, and just like warrant (from which it derives) it comes in degrees depending on cases, context and subject matter.

    Moral certainty is a case where the grounds of warrant are sufficiently strong that one would be derelict of duty if one were to willfully treat something of that degree of credibility as though it were false, when something of great moment or value is on the table. For example, in a criminal case under Common Law jurisdictions, one must prove beyond reasonable doubt — this is a criterion of responsibility in the context of duty to justice. In commercial or civil matters, preponderance of evidence is a lower standard.

    Beyond that everyday sense of certainty, lie the cases where in effect there is reason to believe that the judgement that x is the case has passed beyond room for reasonable, responsible doubt and is utterly unlikely to be reversed; something is true and is so grounded that there is no real room for doubt, but is not a necessary truth — one that must be so in this and all other possible worlds. Then, there is self-evidence, where x is so, is seen to be so by one with enough experience to understand the claim properly, and is such that the denial is immediately, patently absurd. That error exists, is a case in point, the attempt to deny instantly exemplifies that error exists. Likewise, one cannot be deluded that s/he is conscious, as to doubt is an act of consciousness.

    Regrettably, we are so situated that it is impossible to build a whole worldview up from matters that are at least self-evidently so.

    However, this degree of warranted certainty (and what lies beyond) serves to provide yardsticks and plumblines to test our worldview cores. For example, that error exists is undeniably true and warranted to self-evident certainty. This confirms that truth beyond opinion exists. Likewise, that some truths are intelligible, accessible by reason. As we observe and experience that error exists means that observation and experience can access truth. Similarly, we have warrant to undeniable certainty, so certain knowledge exists. If certain knowledge exists, knowledge (embracing weaker senses) exists also.

    Further to this, beliefs, opinions, ideologies and worldviews that assume, argue, opine and assert that truth, or knowledge, or warrant or certain knowledge do not exist or that claims to such only serve “intolerance” and oppressor-classes — their name is Legion, are swept away wholesale as error. And yes, for cause I have the fell work of cultural marxism squarely in my sights, along with radical relativism and radical subjectivism.

    Moreover, having warranted this point to certainty, I freely hold there is demonstrative warrant and that for cause opinion and rhetorical objection to the contrary avail nothing. Though in a politically correct era, many will take the vapours and will be frightened that I have announced a policy of right wing, Christofascist totalitarianism dressed up in Torquemada’s robes. That is how far ever so many in our civilisation have been misled.

    That agit prop induced and/or mal-education induced reaction is unwarranted, the issue is to act responsibly and rationally in light of duty to truth, right reason, prudence, justice, etc.

    However, there are higher yet degrees of warrant and certainty of knowledge.

    Some truths are necessary, certain, intelligible and knowable to utter, incorrigible certainty and even absolute: the truth, the whole truth on a material matter, nothing but the truth on the material matter. Where, a necessary truth will be so in this or any other possible world. And what is more, many such truths are intelligible and warranted to similarly necessary certainty. Many core principles of reason and mathematics are of this order.

    For relevant example, for a distinct world to be possible of existence, it must have in it at least one feature [A] such that it is different from all other possible worlds. We may then freely dichotomise W: W = {A|~A}. This already indicates that rationally intelligible structure and quantity are present in the fabric for such a world, we may readily identify here duality, unity (and complex unity in the case ~A), also nullity. The von Neumann construction then gives muscle to Peano’s succession from unity, and we have the natural counting numbers. From this, we may further recognise Z, Q, R, C and more.

    Widening scope, and using reality in the widest sense, in reality (to include the case where there may be plural worlds as domains in reality) there will be some A, thus too ~A and a similar dichotomy obtains, R = {A|~A}. Instantly, A is itself i/l/o its core characteristics, perhaps a bright red ball on a table. This is the Law of Identity, LOI. Similarly, by the contrast and dichotomy, no x in R will be in A and in ~A, law of non contradiction, LNC. Thirdly (notice how counting numbers are implicit) any x in R will be in A or else in ~A, not in both or neither. Law of the excluded middle, LEM.

    These three are inescapably true. We cannot prove them by appealing to something deeper, as to try to prove abnnot but assume and implicitly use them. Likewise a claimed disproof or possible world in which they do not hold will on inspection be found to be implicitly using them. Such are the start-points for reasoning.

    And more,

    KF

  305. 305

    Nicely laid out, KF.

  306. 306
    StephenB says:

    Hazel

    The physical world contains physical things which behave in certain ways that can be described by our mathematical abstractions, but those abstractions don’t themselves exist in the physical world.

    The law like regularities in nature, which we describe (and perceive) as laws, are *non-physical realities* that govern the activity of matter in the physical world. Matter cannot empower matter to move or inform matter about how to behave, so something non-material must play that role. This should be obvious because there is nothing in the cause that could possibly produce the effect. Put another way, a cause cannot give what it does not have to give.

  307. 307
    Ed George says:

    StephenB

    Matter cannot empower matter to move or inform matter about how to behave, so something non-material must play that role. This should be obvious because there is nothing in the cause that could possibly produce the effect. Put another way, a cause cannot give what it does not have to give.

    I don’t think anyone is arguing that an effect does not require a cause. I take that as a self evident truth. But to suggest that mathematics is the cause does not make any sense. We employ mathematics to help us affect causes, but mathematics is not the cause. We use mathematics to help design the engine and the wing to manufacture a plane that is optimized for lift and fuel consumption. But this is because the math helps us model the physical aspects of the world around us. Not because the mathematics is inherent to the world.

  308. 308
    kairosfocus says:

    EG, how many times has it been noted to you or laid out before your eyes that the specific issue at stake is that demonstratively (per logic and first principles), there are quantities and structures necessarily in the fabric of any possible world which include N and linked entities? Moreover, key aspects involved are tied to the logic of being, starting with requisites of distinct identity. In that context of a demonstration on the table, repeating a dismissive talking point without demonstrative warrant avails nothing. KF

    PS: Have you made three loops of paper yet, one ordinary and two Mobius? Have you cut the ordinary loop and one Mobious going around the loop, in the middle of the strip? The other, at 1/3 the way across? Do you notice three sharply different results, each tied to structure and quantities expressed in three bodies in space? Did you notice, each is not a figment of imagination? That it is structure and quantity embedded in real objects that made clear differences? And notice, observing the differences is not explaining or analysing them, it is seeing concrete reality showing how the logic of structure and quantity embedded in objects has definite consequences.

  309. 309
    ET says:

    Ed George:

    But to suggest that mathematics is the cause does not make any sense.

    Perhaps not to you.

    We employ mathematics to help us affect causes, but mathematics is not the cause.

    Why not? You have to make a case. You don’t just get to baldly declare anything.

  310. 310
    StephenB says:

    SB: Matter cannot empower matter to move or inform matter about how to behave, so something non-material must play that role. This should be obvious because there is nothing in the cause that could possibly produce the effect. Put another way, a cause cannot give what it does not have to give.

    Ed George

    I don’t think anyone is arguing that an effect does not require a cause. I take that as a self evident truth. But to suggest that mathematics is the cause does not make any sense. We employ mathematics to help us affect causes, but mathematics is not the cause.

    Where in my comment did you find any suggestion that mathematics can cause anything to happen? Do you labor under the misconception that mathematical principles are the only abstract realities.. I was very clear in my statement that I was referring to non-physical forces that govern the behavior of matter, something that Hazel does not seem to acknowledge. Please read both my message and the message I am responding to so that you will understand the context of the interaction.

  311. 311

    Ed George said:

    But this is because the math helps us model the physical aspects of the world around us. Not because the mathematics is inherent to the world.

    I suggest you read again StephenB’s post at 306. Mathematics describes, with incredible precision, the behavior of things we observe in the physical world. We discover this mathematical behavior, we do not invent it. It doesn’t matter what language or symbols you use to describe this behavior because they would all be saying the exact same thing regardless of cultural influences and language. We would expect alien intelligences to have the equivalent of the same formulas (or better or worse versions, depending on the state of their physics).

    Physical objects do not cause their own behaviors, unless you want to give them free will. Something else is determining their behavior with mathematical precision. One might argue that the mass of an object, or the energy of an electron, or the curvature of space-time “causes” these behaviors, but that’s a version of begging the question and mistaking a model for the cause of the behavior in the model. What is causing the value of mass to interact with other mass in a particular, mathematical way? What is causing an electron’s energy to have the precise set of effects it has? What is causing the universal constants and forces to operate the way they do?

    As StephenB rightly pointed out, the properties and characteristics of what we call matter and energy cannot have been caused by matter and energy. Do they arbitrarily set their own values? If they did, why would they be consistent from one rock to another, from one electron to another? It is apparent (logically) that the apparently universal order of these things is set by by something else that is deeper – call it a higher order or a fundamental substrate, and that this substrate can usefully be called “mathematics”, although one might argue that “mathematics” is the tool by which universal mind generates physical behaviors. Since mathematics is “in mind”, I don’t see much distinction between saying mind causes those behaviors, or mathematics does.

    Which makes the case for Platonic Realism, and that it actually is mathematics (or some TOE algorithm employing mathematical principles) that actually determines the universal order we continue to discover. What we call the “abstract” would be the basis and foundation of what we call the physical world. QM theory research has consistently supported this for 150 years, which is why many theoreticians are now leaning towards information as the root of physics. Information is abstract, and only exists in mind.

    Until you can provide a theory of how matter and energy can cause their own properties, AND the universality of those properties, there is no logical materialist alternative to Platonic Realism.

  312. 312
    Ed George says:

    StephenB

    Where in my comment did you find any suggestion that mathematics can cause anything to happen?

    Then your argument should be with ET, not me. You and I obviously agree that mathematics is not a cause.

  313. 313
    StephenB says:

    Ed George

    Then your argument should be with ET, not me. You and I obviously agree that mathematics is not a cause.

    At the moment, my argument is with Hazel. You can confirm that fact by noticing that I was responding to her quote, which reflects the philosophy that nothing in nature is immaterial (or non material).

    I said nothing at all about mathematics and don’t intend to until everyone understands the broader point, which was well summarized by William J. Murray: the movement and behavior of matter, contrary to Hazel’s philosophy, can only be explained by a non-material cause.

    You have been silent on that matter, so I have no reason to believe that you agree with me in that context. Once we settle on the philosophical point, which is more important, we can discuss mathematics.

  314. 314
    ET says:

    The Intelligent Designer’s volition is the cause. But mathematics was used to intelligently design the physical world. That is the reason, ie cause, mathematics can be to used to effectively describe the physical world.

    My apologies. I forgot Ed was deficient in the use of dictionaries.

  315. 315
    kairosfocus says:

    SB, I think it is seriously arguable that the origin, continued existence and behaviour of matter in spacetime is credibly best explained on an immaterial source. Arguably, descent from a limitless past in finite successive causal-temporal stages is a supertask that cannot have been completed and in that context (DS et al notwithstanding) the claim of such an actually traversed limitless past does entail traversal of the transfinite in such stages. That is, it is reasonable to hold there was a finitely remote past origin of a matter-energy spacetime world, even if one tries to go beyond the singularity. By definition if matter came to be at a past terminus, it did not always exist and requires an adequate causal antecedent. Where, non-being (the true nothing, not some quantum foam or the like) patently has no causal capability. The best candidate adequate origin for our world which is fine tuned for C-Chem, aqueous medium, terrestrial planet cell based life and contains responsibly and rationally free morally governed creatures with conscience as witness and regulator of thought and deed, is the inherently good and wise creator God, a necessary and maximally great being. One worthy of our loyalty and of the responsible, rational service of doing the good in accord with our evident nature. That said, this is a distinct issue from a demonstration on first principles of reason that possible worlds necessarily manifest structure and quantity starting with the naturals. Where, it is a fallacy of distraction to suggest that an adequate answer to a demonstration is a contrary opinion in absence of sound counter demonstration. KF

Leave a Reply