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

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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

170 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
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    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.

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