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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

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

Comments
Over on the newer Wigner thread , I wrote this:
I like these remarks from Wigner’s article, which I think are somewhat compatible with some of the things I have been trying to say.
However, the point which is most significant in the present context is that all these laws of nature contain, in even their remotest consequences, only a small part of our knowledge of the inanimate world. All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present, except that some aspects of the present state of the world, in practice the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction. … [This] is in consonance with this, first, that the laws of nature can be used to predict future events only under exceptional circumstances, when all the relevant determinants of the present state of the world are known. It is also in consonance with this that the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted. … The principal purpose of the preceding discussion is to point out that the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world. It should be mentioned, for the sake of accuracy, that we discovered about thirty years ago that even the conditional statements [of the motion of bodies] cannot be entirely precise: that the conditional statements are probability laws which enable us only to place intelligent bets on future properties of the inanimate world, based on the knowledge of the present state. They do not allow us to make categorical statements, not even categorical statements conditional on the present state of the world.
I’d like to come back to this thread and add some remarks in response to Wigner. However, to forestall a possible reaction, I’ll point out that I am not “championing” Wigner, nor being in agreement with everything he says. In fact, as bornagain77 pointed out (as did other articles about Wigner I’ve read since yesterday), Wigner himself changed his mind about a number of important issues later in life. (BA posted seven long posts about Wigner and related items on the other thread, if you want to get caught up.) Also, I’m not claiming that Wigner is a definitive expert on this subject. He is an important physicist who also considered more philosophical issues such as the nature of nature and of consciousness, and he thought and wrote well, but many of his equally competent peers had different ideas. With all those disclaimers said … Wigner wrote,
However, the point which is most significant in the present context is that all these laws of nature contain, in even their remotest consequences, only a small part of our knowledge of the inanimate world. All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present, … [although] in practice the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction. …
That is, our “laws of nature” are always abstractions which isolate a few details of our experience as the content of our abstraction from the incredibly rich amount of detail present in the actual physical world. We choose details that are associated with changes that we want to use to make predictions: in part we create descriptions of nature for their utility to us. Furthermore, we constantly test the validity and usefulness of our abstractions making predictions based on them, seeing if the predictions are born out, and then revising them accordingly. As Wigner says,
The exploration of the conditions which do, and which do not, influence a phenomenon is part of the early experimental exploration of a field. It is the skill and ingenuity of the experimenter which show him phenomena which depend on a relatively narrow set of relatively easily realizable and reproducible conditions.
I found it interesting that Wigner points out that the predictive power of the laws of nature are only effective when we know the initial conditions to which they apply (he expands on this more in another article I read), one of their most important uses has been us in the creation of machines:
It is also in consonance with this that the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted
Putting these two ideas together, I find it easy to see that all of our knowledge of the world consists of abstractions which, no matter how we refine them, cannot and do not encompass the incredible detail of the world, including the constantly changing initial in any causal chain which make predictions about the future even probable only in highly constrained cases. Wigner makes a similar conclusion, I think, when he writes,
The principal purpose of the preceding discussion is to point out that the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world.
I think, therefore, that Wigner’s statement are, to some degree, compatible with the view that our abstractions, including all mathematical descriptions of aspect of the world, are mental constructions which exist, collectively, in the minds of human beings and in the symbol systems we have created to share our concepts with each other. This is of course not to deny that there is an unreasonable effectiveness of math in abstracting and describing the physical world. That is the thesis of the essay. But I don’t think that thesis is inconsistent with the view I expressed in the preceding paragraph. I think this idea is reinforced by this:
It should be mentioned, for the sake of accuracy, that we discovered about thirty years ago that even the conditional statements [of the motion of bodies] cannot be entirely precise: that the conditional statements are probability laws which enable us only to place intelligent bets on future properties of the inanimate world, based on the knowledge of the present state. They do not allow us to make categorical statements, not even categorical statements conditional on the present state of the world.
As we have explored more deeply into the basis of the physical world we have encountered the quantum world that was the heart of Wigner’s work as physicist. At that level, many argue that all we have are our mathematical abstractions, and we don’t even know whether it makes sense to say they are abstractions of anything. They are useful human creations which can predict certain end results, as conditional and probable, and if sufficient initial conditions are known, without knowing what the actual nature of the reality leading up to those end results is.hazel
January 21, 2019
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H, pardon but the use of the just linked in this context is a clear example of loaded unjustified projection. I repeat, when an actual demonstration is on the table, dismissive objections to the contrary avail nothing. KFkairosfocus
January 21, 2019
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A companion to 238: More wisdomhazel
January 21, 2019
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EG, when demonstrative warrant is on the table, opinions to the contrary avail nothing. And, demonstrative warrant is on the table as has long since been shown. But then, that is part of the point of the OP. KFkairosfocus
January 20, 2019
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hazel:
First, I think perhaps too much has been made of the distinction between using the words invention and discovery in the area of pure math.
Well, it matters. I understand that it may bother some people that we are just discoverers. But hey, the discovery, the journey, is clearly not for everyone and is very important in itself. People like Srinivasa Ramanujan should be heralded, studied and perhaps even followed- methodologically with resect to mathematical discovery- if it leads to a higher understanding and more discoveries. Being able to tap into the universal information, including mathematics, channel it, understand it and be able to make use of it is a great feat. A great individual achievement, if that is what you seek. [O]ur creative, rational mind can think about a logical system, get new ideas that might be interesting, and develop terms and symbols that let us then explore the consequences of what we have developed because that is what they were intelligently designed to do. We are designed for scientific discovery. And given everything we need to discover. It is a real-life scavenger hunt of sorts.ET
January 20, 2019
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Thanks, Ed. It's been useful to me to respond to your questions with some more expanded thoughts.hazel
January 20, 2019
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Hazel@244&246, thank you for the response. For the most part, my view aligns with yours, although you present it much better than I do.Ed George
January 20, 2019
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H, Next, observe your remark:
I think perhaps too much has been made of the distinction between using the words invention and discovery in the area of pure math. From a theoretical point of view, once the counting numbers have been established, the idea of starting with one number and then counting some additional amount from there is a logical and virtually inevitable idea.
Now, what does "established" mean? Normally, it means or suggests set up by some power brokers or innovators or the like. But what has been demonstrated is that once a distinct world is possible, it embeds in its framework the counting numbers as effectively a corollary of the principle of identity. From this, Z, Q, R and C etc follow. This is one way to see that there is an embedding of intelligible rational principles and entities of structure and quantity in the framework for any possible world. Thus, a substantial body of core Mathematical facts are discovered as necessary aspects of reality rather than invented through our studies. That is, they are antecedent to discovery and analysis. For particular instance, there are properties of certain numbers such that they cannot be broken up into smaller groups evenly without remainder, save by ones. We recognised this and have called such primes. Further to this, any other number will be capable of being grouped by prime factor-sized groups, which is a powerful tool in many areas of our study of numbers. That property of primeness clearly is antecedent to our study and is a discovery not an invention. In short, your remarks come close to begging the first question. Likewise, the even grouping concept implies repeated addition by n-scale groups, 1's, 2's, 3's etc. Thus multiplication and division are directly associated with the property that distinct entities can be grouped. Where, addition is a direct property where groups may be clustered to form a larger one, which may be counted up. Subtraction is then instantly also present as removal of a group. All of this is routinely explored in elementary school. So, while we study the four rules and invent symbols, the underlying intelligible structures and quantities inhere to the nature of groups of discrete entities, concrete or abstract. Where, the four rules with additional structured operations that build on them are a key part of the core of operations, functions and relationships. The dual character of naturally occurring substance and culturally influenced study remains. Going on:
our creative, rational mind can think about a logical system, get new ideas that might be interesting, and develop terms and symbols that let us then explore the consequences of what we have developed. We are creatively both inventing and discovering as we go along.
That is, we may study intelligible properties of our world and other possible worlds. That was never in question, the issue is that it is demonstrated that there are intelligible, rational principles of structure and quantity that on distinct identity are inextricably embedded in the framework for any possible world. That is, these are necessarily occurring entities. Further to this, being closely tied to the logic of being, such things then become manifest in the objects and processes we study in science. In such light, "we have developed" is again perilously close to begging the question and dismissing that which was demonstrated from the outset. Indeed, it is that demonstration which you reacted to originally, seemingly taking exception to the consequences of the demonstration. You continue to EG, as though the other side of many implicit questions were never on the table, with substantial reasons offered in support -- which you have never cogently, substantially answered:
In the beginning, as with the primes, the creation of new concepts may appear obvious: more a discovery than an invention. But later in the development of math, people had to think long and hard about the issue they were trying to grasp, and explore different ways as to how to formalize symbolic tools to open new understandings. Newton’s invention of calculus is a case in point, along with Leibnitz’s notation for the derivative. Similarly, I think it’s more reasonable to say that Descartes invented coordinate geometry than to say he discovered it.
Appearing obvious is a pivotal rhetorical construct here. Its effective function is that it furthers the dismissal of the demonstration of necessary structural and quantitative entities embedded in the fabric of any possible world antecedent to our recognition and study. Such entities are enough to ground the claim that there is significant discovery in Mathematical study, regardless of what we may otherwise invent. However, it is appropriate to again point out that rates of change and accumulations of change including growths, flows and spatial motion (speed, velocity, acceleration, time, impulse, jerk, work, power as rate of work, fluxes in the sense of energy and fields etc) are naturally occurring structures and quantities that were found important objects, processes etc worthy of study. Yes, Newton et al (including classical antecedents over a thousand years earlier) did develop approaches, insights, terminology and paradigmatic results, but that study does not sweep away the naturally occurring substance that they examined and discovered properties of. Sweeping ahead to non-standard analysis, I suggest that infinitesimals and transfinites . . . domain of hyperreal and surreal numbers great and small . . . are reasonably viewed as extensions of the sets already highlighted, e.g. recognition that naturals continue endlessly beyond any given finite value leads to recognition of another class of quantity, the transfinite. Now, we see a caveat with a regrettable marker that allows a loaded projection to any counter-argument:
to be clear, and to try to avoid misunderstanding, given a set of developed concepts, the stream of logical consequence that follow are discovered. Logical consequences are entailed in the beginning set of accepted know concepts and facts, and then we explore those consequence streams and discover all sorts of new stuff. For instance, we have discussed two facts that are not obvious at all: for any prime p, p^2 -1 is divisible by 24, and for any odd number n, dividing n^2 into x and x + 1 produces a Pythagorean triple n, x, x + 1. Those are discoveries within the realm of pure math. We didn’t invent those.
What is at stake, first, is that it was demonstrated that there are embedded abstract entities and principles of structure and quantity in any possible world that are antecedent to any exploration or discovery or invention on our part. That demonstration has never been refuted, though it has often been rhetorically sidelined, dismissed as allegedly irrelevant or a misunderstanding of the views being put forward etc. Next, it was thus shown that there are embedded core mathematical facts that formed a body of knowledge which shaped and constrained formation of axiomatic systems starting with Geometry and proceeding on to the modern axiomatisations. For simple example finding twelve segment ropes implies knowledge of key Pythagorean results long before Elements was composed. The underlying body of such facts and knowledge bases is sufficient to show that there are material bodies of structure and quantity embedded in this and other possible worlds, which we discover in our studies rather than invent. Onward theorems or model results are secondary to the primary point. You then proceed to say further to EG (as opposed to the parties you actually have been debating with all along) -- and I enfold annotations:
>>pure math starts with the idea of a distinct unit, which we name “one”. We then define successors>> a: we observe, having demonstrated, that nullity, unity and duality are implicit in the possibility of any distinct world. Thus, we see a successive cumulation of distinct discrete quantities, thus justifying use of the von Neumann succession principle to lay out the Naturals, which we may label as we please. b: The above therefore again suppresses material demonstration that demonstrates mathematical entities antecedent to our studies. >>so as to create the counting numbers,>> c: create is here synonymous with invent; the question is begged in the teeth of an existing demonstration that has been repeatedly pointed out. >> and with suitable development as outlined in my last post at 244, a vast amount of pure math follows.>> d: Thus, the demonstrated embedded structure and quantity antecedent to our study is material in its import. >>How does this apply to the physical world? My basic position is that we build mathematical models of the world: abstractions which describe general aspects of the world.>> e: Yes, we build abstract, logic model worlds, which are constrained by the embedded facts antecedent to our study and their logic of being import. f: Describing general aspects of the world avoids acknowledging that a body of core abstract structure and quantity is necessarily embedded in the fabric of any possible world, thus our experienced one. g: Thus, it implies the main point, while the acknowledgement of its material relevance is sidestepped. >>We then draw logical conclusions within our model about what we would expect to find if our model were true.>> h: Yes, and this requires that the logic of being import of the logic of structure and quantity extends from the abstract model world to the physical one. i: This occurs in two ways, necessary entities and properties extend to any world and when certain properties are archetypally present in the model world and our own, they will be relevant. Circularity properties extend to gear trains and other approximately round entities, etc. j: As has been repeatedly pointed out by the other side but is not acknowledged in how it is now being presented. The rhetorical effect in wider context is to pummel a gagged strawman. k: And this is occurring when I am present and thread owner and where others are likewise present. Now imagine what happens when the other side is locked out, silenced, demonised, stereotyped, tainted and scapegoated, expelled, censored or de-platformed, etc. l: Resemblance to trends with the ID debates and wider cultural conflict is not coincidental. >>Then we re-examine the physical world to see if it behaves as our model predicted. If it does, that supports our model and makes it worthwhile to further develop our model.>> n: That is, modern induction is argument by support and linked inference to best current explanation. >> If the expectations of our model are not empirically confirmed, we revise our model.>> O: provisionality as a scientific ideal. >>The model and other concepts we use to describe the physical world are part of our mind,>> p: They are abstracta, raising issues of the nature, power and relevance of such abstract entities and clusterings including universals. >> which uses the symbolic systems of language and math>> q: Abstracta again. >> to both formalize our own understanding and to share it with others.>> r: That is, we assert propositions, infer logical relationships, argue to warrant and to explain, thus study. Where the demonstrated embedded entities are material and are antecedent to our studies. >>So we have two things: the physical world and our understanding of it.>> s: Yes, the world which embeds abstracta, and our studies that build on such. >>Our understandings are abstractions that we use to describe aspects of the world>> t: Yes, we do abstract through inference, pattern-recognition, symbolic representation etc. However, the elephant in the room is the demonstration of abstract necessary entities involving N, Z, Q, R, C etc antecedent to our studies. >>. We can’t describe every detail of the world (the map is not the territory) we have to selectively generalize about the parts we have focussed our attention on. >> u: We have bounded rationality
This is enough to make the substantial concerns plain, and to show how things have got so tangled up. KF PS: The blanket appeal to the quantum world fails. At macro level, tilings and groupings are facts of life, as are constraints on packing. Going to materials, molecules and solids or liquids, packing and clusters are major phenomena, the crystal structure of metals and semiconductors being a major part of why they act as they do. Where liquids, glasses etc take properties from the degree to which that packing does not obtain. Semiconductors with controlled dopants are again relevant. Of course, the informational structure of R/DNA and the chain and fold aspect of proteins is also relevant.kairosfocus
January 19, 2019
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H, I have first repeatedly pointed out that a dominant form of nominalism is rooted in or a fellow traveller of evolutionary materialistic scientism. Accordingly, I began with this and then continued to the wider case. I observe that you are now specifically distancing yourself. Previous remarks and arguments you raised several times (readers, scroll up to see) led to the response I made that you have championed it in thread. In particular, it appeared that you championed conceptualism, which seems to be an unstable position which I cited on above. KFkairosfocus
January 19, 2019
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The second idea I want to discuss is the relationship between pure math and the physical world, and about the idea of distinct identity. I hope, Ed, that you find my remarks somewhat useful in respect to the questions you asked at 227 and 234. Without a doubt, pure math starts with the idea of a distinct unit, which we name “one”. We then define successors so as to create the counting numbers, and with suitable development as outlined in my last post at 244, a vast amount of pure math follows. How does this apply to the physical world? My basic position is that we build mathematical models of the world: abstractions which describe general aspects of the world. We then draw logical conclusions within our model about what we would expect to find if our model were true. Then we re-examine the physical world to see if it behaves as our model predicted. If it does, that supports our model and makes it worthwhile to further develop our model. If the expectations of our model are not empirically confirmed, we revise our model. The model and other concepts we use to describe the physical world are part of our mind, which uses the symbolic systems of language and math to both formalize our own understanding and to share it with others. So we have two things: the physical world and our understanding of it. Our understandings are abstractions that we use to describe aspects of the world. We can’t describe every detail of the world (the map is not the territory) we have to selectively generalize about the parts we have focussed our attention on. All this takes place in all human beings all the time: it’s part of being a human being with, among other things, rational cognitive skills. There is a constant interplay between the empirical data we get about the physical world through our senses and the understandings/concepts/abstractions/models we create in our minds, and share with each other through language, written and verbal, including the language of math. So let’s talk about the simplest model of all: counting things is the physical world. The simplest idea is that of one-to-one correspondence. We have words for one, two, three, etc. We count three pebbles by aligning the words with the objects. (Children start getting this about the age of three or four). This is a model. We experience the three pebbles as distinct entities: they retain their separateness from other things in the world. Therefore, the concept of a “unit” applies to them. Also the concept of addition applies to them, because when we put them together, they retain their individuality. The model works, and continues to work as we divide them into groups to represent multiplication. There is a very solid correspondence between our mathematical model and how objects like pebbles behave. (FWIW, when teaching children about numbers, and lots of other math, it’s important to give them lots of experience with concrete situations, so they can build strong concepts that have a depth of understanding.) There are, however, some additional issues here. One is that right from the beginning treating the three pebbles are units is an abstraction that ignores a huge amount of uniqueness (color, shape, size, etc.) and models just the fact that the object can be moved around while maintaining its separateness. Right from the beginning we have an abstract mathematical model-physical world relationship in just considering a pebble representing the concept one. This is easier to see if we think about things that don’t behave like pebbles. Clouds are an example. You can’t count clouds. They might look like one cloud from one view, and not another: how do you determine if a cloud is a continuous whole? They don’t retain their separate identity when brought together. We can’t apply the model of the counting numbers to clouds. So, to summarize this point, counting objects, when applied to objects which have a distinct identity and retain that identity when moved around, is applying a mathematical model to a phenomena in the physical world. One last point: kf wrote, “ stacking or tiling in patterns goes down to atomic, molecular and material structure. The property of being discrete units is natural ...” Actually, I think at the quantum level we are perhaps finding this not be true. But that is another subject.hazel
January 19, 2019
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KF @242: Interesting. Thanks for the comment.StephenB
January 19, 2019
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Hi Ed. At 234, you wrote,
But we invented division. The fact that when we apply division we see certain “trends” does not mean that those “trends” are inherent in the universe. It just means that they are inherent (or consequences) of our invention. I have been out of this conversation for a while so I apologize if this has already been gone over.
I’d like to expand on some points that I briefly made at 229. ( think I’ll break them up into two posts.) First, I think perhaps too much has been made of the distinction between using the words invention and discovery in the area of pure math. From a theoretical point of view, once the counting numbers have been established, the idea of starting with one number and then counting some additional amount from there is a logical and virtually inevitable idea. However, to develop the idea and share it with others we invent a word “addition” and a symbol “+” to represent the concept. So the development involves both some recognition of something new that we can do (a discovery) and the invention of some terms and symbols with which to manipulate our new concept. Then the process is repeated: we note multiples occur by repeated addition of the same number: if you count by 3’s you get a sequence of the multiples of 3. Then we define multiplication as a shorthand for repeated addition, again a creative recognition of a new pattern and the invention of new terms and symbols, so now we have a new concept. Eventually this leads to the idea of primes, and a whole world of new discoveries opens up. All of the preceding has been about pure math: our creative, rational mind can think about a logical system, get new ideas that might be interesting, and develop terms and symbols that let us then explore the consequences of what we have developed. We are creatively both inventing and discovering as we go along. In the beginning, as with the primes, the creation of new concepts may appear obvious: more a discovery than an invention. But later in the development of math, people had to think long and hard about the issue they were trying to grasp, and explore different ways as to how to formalize symbolic tools to open new understandings. Newton’s invention of calculus is a case in point, along with Leibnitz’s notation for the derivative. Similarly, I think it’s more reasonable to say that Descartes invented coordinate geometry than to say he discovered it. But as I am trying to make clear, I don’t think there is a clearcut distinction between what is invented and what is discovered in math, which is why I like words such as create and develop, which express the interplay, perhaps, between the two ideas that is always going on.. Also, to be clear, and to try to avoid misunderstanding, given a set of developed concepts, the stream of logical consequence that follow are discovered. Logical consequences are entailed in the beginning set of accepted know concepts and facts, and then we explore those consequence streams and discover all sorts of new stuff. For instance, we have discussed two facts that are not obvious at all: for any prime p, p^2 -1 is divisible by 24, and for any odd number n, dividing n^2 into x and x + 1 produces a Pythagorean triple n, x, x + 1. Those are discoveries within the realm of pure math. We didn’t invent those.hazel
January 19, 2019
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kf writes, "You have championed nominalism." No, I have not "championed" it. I liked a sentence I read about it on Wikipedia. When you declared that it led to, or implied, physicalism, I specifically wrote at 138, "kf, I really don’t know why you think nominalism implies physicalism, but if that is the case, then nominalism doesn’t describe my position. " Your persistence in the idea that I am "championing" nominalism is an example of the problem I addressed at 238.hazel
January 19, 2019
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SB, treeness actually extends beyond biology, in terms of a branching structural pattern. Confining to biological cases, there are clearly exemplars and counter examples that suffice to form a core conception on common characteristics, a growth habit that is functional and often leads to timber or the like as a result, as strength is needed to elevate leaves or fronds and to withstand strong winds. I suggest shrubs and bushes are in effect miniature forms, though the height etc differences may be significant. Plants that may grow as bushes or shrubs or as climbing vines seem to be an interesting bridging case. KF PS: The Kantian ugly gulch seems to pop up in all sorts of places. F H Bradley's corrective that to claim the un-knowability of things in themselves is to imply knowledge of things in themselves will bear pondering.kairosfocus
January 19, 2019
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H, perhaps, it is time to talk about yet another fallacious tendency, loaded projection. For cause, I disagree that we are persistently, even possibly willfully misunderstanding what you have advocated; which is what your linked suggests. Instead, there is a disagreement pivoting on a demonstration that leaves you uncomfortable and which you seem to wish to set aside as if it were irrelevant. In fact, that demonstration is the crux of the issue and the rhetorical pattern of responses is instructive, albeit less than happy. Again, there is a fundamental issue that has come up over several threads of argument, the nature of Mathematics i/l/o a demonstration that there are rational principles and associated facts of structure and quantity embedded in the framework for any world to exist. That suffices to demonstrate that there is a core substance of structure and quantity that is in reality antecedent to our coming along and creating a culturally influenced study of the logic of structure and quantity. So, there is a both-and; something you have in effect formally agreed with. But then, again and again you have come along and argued as if the first part vanishes, emphasising individual and cultural aspects to the point that it is evident that something is profoundly disturbing to you about the other aspect Further to this, you have stated a strong disagreement, not only with classical Platonism and apparently with the classic view by theologians that roughly speaking the world of forms is the mind of God, but also with what is called modern mathematical platonism which effectively holds that key abstracta have reality independent of our opinions and studies. They are discovered not invented. And here I am specifically not claiming that all abstract entities or clusterings are like this, just some relevant ones tracing to N and extensions therefrom. You have championed nominalism. To this, I have given a two-level response. Level one, knowing the ideological dominance of evolutionary materialism and its fellow travellers as well as of associated subjectivism and relativism, I have shown how the core materialism cannot warrant nominalism, starting with the non-rationality of computational substrates. That in my view is necessary regardless of your particular views precisely due to ideological dominance. Level 2, I have spoken to how statements of nominalism are inextricably entangled in implicit appeals to abstracta and universals. Indeed, I have boiled it down to the challenge of stating and arguing for it without so implicitly appealing. No-one has taken this up, unsurprisingly as truth is an abstract relationship, implication, entailment and evidential support are the same, any number of universals will stubbornly insist on popping up in the arguments and more. I point out that this is exactly how first principles of rationality operate, we cannot reason without them, we therefore cannot prove them, we prove from them, we must take them as givens. Thus, we have good reason to retain the conclusions that have been on the table. And, these pivotal issues are not matters for clashing opinion, they hold adequate warrant. KFkairosfocus
January 19, 2019
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Hazel, KF First, I appreciate KF’s comments on archetypes and their relevance to the discussion. Let me respond to some of his points in the context of Hazel’s objections. For me, the question on the table is whether or not trees qualify as a category of being and if we can know a tree for what it is. It is not a perfect parallel to mathematics for reasons that I will not get into at this time. Nominalists say that we cannot know a tree as a tree; we simply call it a “tree” for the sake of convenience. In their minds, we cannot know anything for what it is, only how it appears to us. Under that assumption, then, major concepts could be understood in a variety of ways. However, knowledge is not about perceptions or appearances; it is about the known reality behind perceptions and appearances. In that sense, there are no differences in what we know about the essence of a tree. It doesn’t matter that some people have not experienced a tree. In such a case, they simply don’t know what a tree is until someone informs them or until they have experienced enough subcategories of trees (Oak tree, Maple tree, etc.) to abstract the universal reality behind the general category (tree). Once that happens, they will understand what a tree is just as we do. Nor does it matter that some other entities, such as shrubs or bushes, may resemble a tree in some way. This muddies the waters because the difference between the two is often defined by what is done to them (pruning, cutting, etc.) There is a substantial difference between those two entities and a tree and anyone with sufficient experience knows that difference. If not, they can learn about it. We can learn about all kinds of things that we have not experienced. The process of abstraction often requires enough to time to grasp the universal concept (or principle) from particular examples, and sometimes, many examples are required. Under the heading of “tree,” Wikipedia defines the word to mean this: “A perennial plant with an elongated stem, or trunk, supporting branches and leaves in most species.” That is not a bad definition. Yes, it does go on to say that *“Trees are not a taxonomic group,”* but I would question that assessment. Indeed, Wikipedia itself seems to contradict the point under the heading “Categories: trees,” describing categories and subcategories of trees, as well as the genus and species of trees. In any case, we need the general (genus) to understand the particular (species), and we also need the particular to understand the general. For most us, the particular is the first experience. So, yes, if one person has experienced only oak trees and another has experienced only maple trees, there will be some variation in their perceptions, but that is because, in both cases, there is the false perception that only their experienced subcategory exists, which means that they do not have the variety of experiences necessary to know that a general category of trees exists. Concepts, as I define them, are about knowledge, not perceptions. Until one understands the relationship between the general category (tree) and the sub category (Oak tree, Maple tree, etc.), there is no knowledge of a tree’s essence or whatness – only the experience of a particular kind of tree, which does not suffice as knowledge. Perception does not equal conception. I submit, therefore, that the perceptions of trees can differ, but the knowledge of a tree’s “whatness” cannot. Obviously, we cannot know everything there is to know about a tree, or even a small part of it, but we can know what a tree is in terms of the same shared concept.StephenB
January 19, 2019
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Hazel@238. :) :) :) :) :)Ed George
January 19, 2019
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Pertinent wisdomhazel
January 19, 2019
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We invented mathematics if an only if materialism is true. However, if materialism was true we wouldn't be here. There wouldn't be any life at all.ET
January 19, 2019
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What is the evidence that we invented division?ET
January 19, 2019
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EG, no, stacking or tiling in patterns goes down to atomic, molecular and material structure. The property of being discrete units is natural, ponder peas in a pod. The onward one of separability in evenly grouped patterns goes beyond that to the realities and quantitative properties of numbers which lie in the numbers. KF PS: What would you make of Fibonacci numbers and linked properties commonly arising in nature?kairosfocus
January 19, 2019
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KF
ET is right in reply: “we discovered primes by observing there are numbers only divisible by 1 an itself.”
But we invented division. The fact that when we apply division we see certain "trends" does not mean that those "trends" are inherent in the universe. It just means that they are inherent (or consequences) of our invention. I have been out of this conversation for a while so I apologize if this has already been gone over.Ed George
January 19, 2019
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Can't build a world without the maths...ET
January 19, 2019
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ET, in such parts as are contingent and fine tuned to support terrestrial planet, C-chemistry aqueous medium cell based life, that is utterly manifest. But there is more, Mathematical reality is demonstrably part of the very fabric of any possible world. Mathematical reality in core part is antecedent to our minds much less our studies and whatever cultural influences are involved. KFkairosfocus
January 19, 2019
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Math can be used to describe the world because the world was intelligently designed using mathematics.ET
January 19, 2019
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F/N: We may be getting to the nub:
H: saying that math concepts live in some Platonic world that is independent of minds of individual intelligent beings . . .
H'mm, I wonder if this reflects a subtle quasi-spatial understanding of "world"? Where, an abstract entity is normally something contemplated by a mind. Where, it bears repeating to note that a computational substrate is exactly what a free, responsible, rationally contemplative mind is not: blindly mechanical. Where, the logical mind to ponder as candidate would be the same as that behind the cosmos. Of course, such is beyond what has been shown repeatedly -- structures and quantities are inherently and inextricably bound up in the framework or fabric for a world to be. It is reasonable to ask how that could be as a matter of logic of being, and rationally contemplative mind comes up. Where such a mind behind reality could very well contemplate all that we see is necessarily in the fabric of any world as intelligible rational principles that help to frame it. This brings us to an at last substantial contribution by EG:
for prime numbers to exist we first have to define a rule (only divisible by one and itself). And to set that rule we first have to develop the concept of division. This suggests that prime numbers, and everything that falls from them, are the consequence of us setting specific rules. Not that prime numbers are somehow inherent to the fabric of the universe.
ET is right in reply: "we discovered primes by observing there are numbers only divisible by 1 an itself." Where, grade school math will remind us of the importance of factorisation and prime factors. Every natural number is prime in itself or else is a product of primes. Where, being able to be grouped in uniform sized bundles without remainder is a highly relevant property, e.g. for tiling a floor, etc. The fact that EG saw only that we composed a statement and that we created a procedure is in itself instructive, pointing to the way we are thinking. Beyond, was what was missed: division is a label for a natural process of splitting up a collection of discrete entities into groups of the same cardinality, e.g. pennies or tiles or peas etc. Further beyond was the discovery that some quantities of such units cannot be split into smaller groups without remainder, other than splitting up by ones. Onward, we can discover that any number that can be shared up in evenly sized groups, can ultimately be split into prime number sized groups. Where, of course, the first such case, 2, then divides numbers into odds and evens. Since, we likely explored this in primary school, the question is why we seem to consistently miss the discovery element and highlight the culturally influenced creations that respond to facts that are discovered. One answer is, likely, we were often taught in an authoritative, rote pattern, and were not invited to do even a guided exploration. Another is likely a penumbra of axiomatisation since C19. However, post Godel, we know that complex Mathematical domains have in them Math facts unreachable from finite sets of mutually consistent axioms. They are irreducible and facts stand independent of axiom systems. KF PS: Perhaps, this from Plato in The Laws, Bk X, may be food for thought:
Ath. Nearly all of them, my friends, seem to be ignorant of the nature and power of the soul [[ = psuche], especially in what relates to her origin: they do not know that she is among the first of things, and before all bodies, and is the chief author of their changes and transpositions. And if this is true, and if the soul is older than the body, must not the things which are of the soul's kindred be of necessity prior to those which appertain to the body? Cle. Certainly. Ath. Then thought and attention and mind and art and law will be prior to that which is hard and soft and heavy and light; and the great and primitive works and actions will be works of art; they will be the first, and after them will come nature and works of nature, which however is a wrong term for men to apply to them; these will follow, and will be under the government of art and mind. Cle. But why is the word "nature" wrong? Ath. Because those who use the term mean to say that nature is the first creative power; but if the soul turn out to be the primeval element, and not fire or air, then in the truest sense and beyond other things the soul may be said to exist by nature; and this would be true if you proved that the soul is older than the body, but not otherwise. [[ . . . .] Ath. . . . when one thing changes another, and that another, of such will there be any primary changing element? How can a thing which is moved by another ever be the beginning of change? Impossible. But when the self-moved changes other, and that again other, and thus thousands upon tens of thousands of bodies are set in motion, must not the beginning of all this motion be the change of the self-moving principle? . . . . self-motion being the origin of all motions, and the first which arises among things at rest as well as among things in motion, is the eldest and mightiest principle of change, and that which is changed by another and yet moves other is second. [[ . . . .] Ath. If we were to see this power existing in any earthy, watery, or fiery substance, simple or compound-how should we describe it? Cle. You mean to ask whether we should call such a self-moving power life? Ath. I do. Cle. Certainly we should. Ath. And when we see soul in anything, must we not do the same-must we not admit that this is life? [[ . . . . ] Cle. You mean to say that the essence which is defined as the self-moved is the same with that which has the name soul? Ath. Yes; and if this is true, do we still maintain that there is anything wanting in the proof that the soul is the first origin and moving power of all that is, or has become, or will be, and their contraries, when she has been clearly shown to be the source of change and motion in all things? Cle. Certainly not; the soul as being the source of motion, has been most satisfactorily shown to be the oldest of all things. Ath. And is not that motion which is produced in another, by reason of another, but never has any self-moving power at all, being in truth the change of an inanimate body, to be reckoned second, or by any lower number which you may prefer? Cle. Exactly. Ath. Then we are right, and speak the most perfect and absolute truth, when we say that the soul is prior to the body, and that the body is second and comes afterwards, and is born to obey the soul, which is the ruler? [[ . . . . ] Ath. If, my friend, we say that the whole path and movement of heaven, and of all that is therein, is by nature akin to the movement and revolution and calculation of mind, and proceeds by kindred laws, then, as is plain, we must say that the best soul takes care of the world and guides it along the good path. [[Plato here explicitly sets up an inference to design (by a good soul) from the intelligible order of the cosmos.]
kairosfocus
January 19, 2019
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Hi Ed. Your questions touch on the big issues that we’ve been discussing for several threads. I don’t intend to re-argue the points with others, but I’d be glad to summarize what I think for you. Two issues: First, I think math concepts exist within the symbolic systems we have developed, and ultimately exist in our minds, which have the rational ability to comprehend the symbols and manipulate them with logic. I think we use those concepts to describe the world, but I don’t think the concepts themselves are embedded in the world. Once human beings began to develop the number system, starting with the idea of the unit one to describe objects with distinct identity, other basic concepts such as multiples, factors, and ultimately primes, were both logically and experientially inevitable next steps. As Wigner said in a quote I liked offered by kf above, “mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts.” We invented the word “prime” and its definition to represent a certain condition of a number not being a multiple of any number less than itself and one. However, as I have repeatedly said, once certain definitions are made and concepts established, the power of logic leads us to discover vast amounts of mathematical facts that are not inventions: they are logical discoveries. Math can be used to describe the world, but such descriptions are always provisional and subject to revision. One of my first posts in this series of threads quoted Einstein as saying, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” I think this is an important truth. P.S. Speaking of discoveries, I was thinking about right triangles recently, after a brief exchange with Stephen, and figured something out: even though I’m sure this is already know, for me it was a logical discovery, and fun to do. This is what motivates students who love math: the satisfaction of using your logical powers to go from given and know information to proving things that, to you, were not previously known to be certainly true. Anyway, a theorem that is not obvious, and isn’t relevant to the real world at all. Given any odd number n. Find n^2, and divide it into two consecutive integers x and x + 1. Then n, x, and x +1 will be a Pythagorean triple, such that n^2 + x^2 = (x + 1)^2. For example, let n = 17. n^2 = 289, so x =144 and x + 1 = 145. 17^2 + 144^2 = 145^2 ? 289 + 20736 = 21025. It works :-) I leave the proof to the interested reader.hazel
January 19, 2019
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Ed, we discovered primes by observing there are numbers only divisible by 1 an itself.ET
January 19, 2019
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Hazel@225, for prime numbers to exist we first have to define a rule (only divisible by one and itself). And to set that rule we first have to develop the concept of division. This suggests that prime numbers, and everything that falls from them, are the consequence of us setting specific rules. Not that prime numbers are somehow inherent to the fabric of the universe. Does this make sense?Ed George
January 19, 2019
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hazel:
But that is different than saying that math concepts live in some Platonic world that is independent of minds of individual intelligent beings.
No one is saying that. With ID we still have the Mind of the Intelligent Designer- the Mind that caused all this to be. The Mind that used the Mathematics to Design the universe. Ie, The Source.ET
January 19, 2019
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