That is evidence that mathematics, and what fishfry calls "formal abstractions", are not separate, or independent from the world, as fishfry argues. They are not ideal perfections, separate Forms, but they must share in the imperfections of the material world, as only being useful in that world, if they are part of that world. The proposition that they could have independence from the material world is a false premise. So to create a formalism and present it as free from the negative influence of content, is to present a smoke and mirrors illusion, because the most one can do in this respect, is hide that negative influence.

For mathematicians, the words abstract and pure are frequently taken to mean the same thing. Here are comments from the Wiki page:

Pure [abstract] mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. . . .


. . . It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians. — Wikipedia

As a working math person I rarely thought about these philosophical issues, But "formal abstractions" in foundations appears to be a very sophisticated game. Whether this game is entirely separated from the material world might depend on how one interprets "separate".

These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. . . . — Wikipedia

I have a number of issues with this passage. Any concept which originates in real world concerns cannot be said to have been produced from an appeal to aesthetic beauty. And if we suppose that there are some of each, wouldn't it be the ones which deal with real world concerns which get accepted into the community. So, as much as I see the claim that "pure mathematicians" are motivated by aesthetic beauty, as opposed to real world concerns, I don't see that any such concepts as being produced purely for aesthetic beauty actually exist in mathematics.

The next is with the phrase " the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles". What we are discussing is the production of the basic principles themselves. If "pure mathematics" simply involves working out the logical consequences of already established principles, then it is not really relevant to what we were discussing, which is the derivation of those basic principles. The question is whether those principles ought to be derived from pure imagination, or ontology.

So, as much as I see the claim that "pure mathematicians" are motivated by aesthetic beauty, as opposed to real world concerns, I don't see that any such concepts as being produced purely for aesthetic beauty actually exist in mathematics. — Metaphysician Undercover

I can attest this (my) mathematical concept was produced purely for aesthetic beauty. The fact you don't see a string of symbols is immaterial. It is found on a mathematics page in Wikipedia.

The question is whether those principles ought to be derived from pure imagination, or ontology — Metaphysician Undercover

Is there such a thing as "pure imagination" that does not arise ultimately from observations and experiences in the physical world?

Is there such a thing as "pure imagination" that does not arise ultimately from observations and experiences in the physical world? — jgill

That's a good question. But if all such principles can be said to have empirical causes, then how can you say that you have a mathematical concept which was produced purely for aesthetic beauty? If there are experiential concerns which enter into your conception then how can you say that your intention of aesthetic beauty is pure?

This is essentially the freewill vs determinism question. To make your conception pure, you'd need the capacity to mentally make a clean break between past and future. It is the assumed continuity between past and future, which forces real world concerns into our thinking. We cannot escape the reality of what we experience as what has just happened, and how this bears on what is about to happen. But if we make a break between past and future, then past experience has no necessary bearing on what we produce for the future because what has just happened will not influence our thinking about what is about to happen. Then your goal for the future, a creation of pure aesthetic beauty, could be completely free from the notion that past occurrences put a necessary constraint on your future production, and you could draw from your past experiences, in complete freedom. The creation of your aesthetic beauty could be done purely without any real world concerns, i.e. knowing that the past has no necessary causal relation with the future, allowing you complete freedom from real world pressure.

This is the issue of inherent order, which we've discussed for months now, in a nutshell. If there is order which inheres within a thing, then that order puts a necessary constraint on future possibilities of order, due to the continuity (causal relation) which we assume to exist between past and future. Logically, we want to start with the assumption of unlimited possibility, to give us the capacity to understand any possible ordering. So, we start with the premise fishfry suggested of "no inherent order". But this is not a real representation of the necessity imposed by inherent order. To remove the necessity of inherent order in a more realistic way, I think, requires that we make a clean break between past and future, annihilating the supposition of continuity, thus allowing that the order which inheres within a thing has no real bearing on the thing's future order. This would allow for the true possibility of any order, it doesn't start with the premise of no inherent order, but it rejects the order which is imposed by the supposition of continuity.

If there is order which inheres within a thing, then that order puts a necessary constraint on future possibilities of order, due to the continuity (causal relation) which we assume to exist between past and future. — Metaphysician Undercover

Interesting ideas, but a tad too ethereal for me. I prefer the solidity of pure mathematics. :cool:

Thanks for the reading material fishfry, I've read through the SEP article a couple times, and the other partially, and finally have time to get back to you. — Metaphysician Undercover

Glad you found some of this helpful. You talked about a lot of things here I'm not qualified to comment on, but I wanted to go to the end and respond to this first.

Actually, I think that often when one stops replying to the other it is because they get an inkling that the other is right. — Metaphysician Undercover

Another reason is that the conversation has gone past the point where there's any light being generated relative to the word count. And the fact that you think I'm arguing right or wrong is telling. I'm not arguing my point of view is right, I'm not even arguing a point of view. I'm telling you how modern math works. It's like this, if you don't mind a Galilean dialog.

Me: Here is how the knight moves in chess. Two squares in the horizontal or vertical direction, and one in the vertical or horizontal, respectively.

You: But that's wrong! Knights in the real world carry lances and save damsels. You can't just make up rules that don't match the real world.

Me: This is not about the real world, I'm just explaining the rules of this game.

You: You insist that you're right, but you're wrong.

Me: The rules of a game can't be right or wrong, they're just the way they are. I'm explaining, not advocating. I'm not claiming that this is the "right" way to define a knight move. I'm just telling you what the rules of chess are.

You: You're wrong, that's not how knights move.

Me. Ok, well let's agree to disagree. I prefer not to continue like this. Nice chatting with you.

You: That just shows that you know you're wrong.

Me: Oh brother.

I hope you see the parallel. I am NOT saying I'm "right" that math should be the way it is. I'm reporting to you from the front lines of math, about how things are. It's pointless for you to tell me that I'm wrong about how things are, because my report is objective. And it's pointless for you to tell me math "should" be other than it is, because I'm not the Lord High Commissioner of mathematics. I'm just telling you how the twentieth century went. It's the tediosity of holding up my end of this theater of the absurd that leads me to withdraw from the field of play.

So there's a matter of pride, where the person stops replying, and sticks to one's principles rather than going down the road of dismantling what one has already put a lot of work into, being too proud to face that prospect. — Metaphysician Undercover

That doesn't even make sense. I can take no pride in how math is, I was simply trained in universities in the modern style of the subject. Hilbert could take pride, he was one of the major instigators of the movement to abstraction. I'm only a very humble student. I'm putting my knowledge at your disposal, I'm not claiming modern math is right or wrong. I'm describing, not advocating.

I am not claiming the modern approach is right. I am only telling you how they do it. I'm telling you how the knight moves, I'm not claiming the rules of chess are "right" or "wrong." The fact that you don't see this emphasizes the futility of any time I spend typing here.

You, it appears, do not suffer from this issue of pride so much, because you keep coming back, and looking further and further into the issues. — Metaphysician Undercover

I keep thinking I might get through to you. I'm not trying to convince you the modern math approach is right. Why do you think I am?

I think Frege brings up similar issues to me. The main problem, relevant to what I'm arguing, mentioned in the referred SEP article, is the matter of content. — Metaphysician Undercover

Right. You are taking the Fregean point of view, and modern math the Hilbertian. But I don't believe I'm arguing the rightness of the modern abstract way; only trying to describe it to you.

The difference of opinion over the success of Hilbert’s consistency and independence proofs is, as detailed below, the result of significant differences of opinion over such fundamental issues as: how to understand the content of a mathematical theory, what a successful axiomatization consists in, what the “truths” of a mathematical theory really are, and finally, what one is really asking when one asks about the consistency of a set of axioms or the independence of a given mathematical statement from others.
— SEP — Metaphysician Undercover

All this is stipulated. I can't continue the convo since I haven't said anything new in weeks if not months.

In critical analysis, we have the classical distinction of form and content. You can find very good examples of this usage in early Marx. Content is the various ideas themselves, which make up the piece, and form is the way the author relates the ideas to create an overall structured unity. — Metaphysician Undercover

Karl or Groucho? You seem to still want to argue that I'm "wrong" somehow, or explain your point of view. I now understand your point of view. There's nothing I can do about it, I can't imagine doing math the way you suggest, not because one couldn't, but because Frege lost the 20th century and I know no other way.

Hilbert appears to be claiming to remove content from logic, — Metaphysician Undercover

From math, not from logic. Surely you don't claim there's content in logic "P implies Q" and "Q" imply P. There's no content there but that's one of the most ancient logical forms there is.

to create a formal structure without content. In my opinion, this is a misguided adventure, because it is actually not possible to pull it off, in reality. — Metaphysician Undercover

You say modern math is misguided as if you want me to defend the opposite proposition. I'm not defending anything. I'm just describing to you how modern math is. I'm not defending it. I'm reporting to you on what I know about it. And for what it's worth, if it's misguided, the abstract point of view not only won the 20th century, math got even more abstract in the second half of the 20th century and into the present.

This is because of the nature of human thought, logic, and reality. Traditionally, content was the individual ideas, signified by words, which are brought together related to each other, through a formal structure. Under Hilbert's proposal, the only remaining idea is an ideal, the goal of a unified formal structure. So the "idea' has been moved from the bottom, as content, to the top, as goal, or end. This does not rid us of content though, as the content is now the relations between the words, and the form is now a final cause, as the ideal, the goal of a unified formal structure. The structure still has content, the described relations. — Metaphysician Undercover

I'm not arguing the point.

Following the Aristotelian principles of matter/form, content is a sort of matter, subject-matter, hence for Marx, ideas, as content, are the material aspect of any logical work. This underlies Marxist materialism — Metaphysician Undercover

Over my head, way out of my bailiwick. Can't respond.

However, in the Aristotelian system, matter is fundamentally indeterminate, making it in some sense unintelligible, producing uncertainty. Matter is given the position of violating the LEM, by Aristotle, as potential is is what may or may not be. Some modern materialists, dialectical materialists, following Marx's interpretation of Hegel prefer a violation of the law of non-contradiction. — Metaphysician Undercover

This is all very impressive-sounding but is an alien language to me. I can't respond, I have no stake in the matter and no understanding of what you're talking about.

So the move toward formalism by Frege and Hilbert can be seen as an effort to deal with the uncertainty of content, Uncertainty is how the human being approaches content, as a sort of matter, there is a fundamental unintelligibility to it. Hilbert appears to be claiming to remove content from logic, to create a formal structure without content, thus improving certainty. In my opinion, what he has actually done is made content an inherent part of the formalized structure, thus bringing the indeterminacy and unintelligibility, which is fundamental to content, into the formal structure. The result is a formalism with inherent uncertainty. — Metaphysician Undercover

Ok. I'm sure you have the core of a nice essay there, but why are you telling it to me? And again, what do you mean "remove content from logic?" When was logic EVER about content? P and Q, remember?

I believe that this is the inevitable result of such an attempt. The reality is that there is a degree of uncertainty in any human expression. Traditionally, the effort was made to maintain a high degree of certainty within the formal aspects of logic, and relegate the uncertain aspects to a special category, as content. — Metaphysician Undercover

You're just going on without me. I don't relate any of this to anything we've been discussing.

Think of the classical distinction between the truth of premises, and the validity of the logic. We can know the validity of the logic with a high degree of certainty, that is the formal aspect. But the premises (or definitions, as argued by Frege) contain the content, the material element where indeterminateness, unintelligibility and incoherency may lurk underneath. We haven't got the same type of criteria to judge truth or falsity of premises, that we have to judge the validity of the logic. There is a much higher degree of uncertainty in our judgement of truth of premises, than there is of the validity of logic. So we separate the premises to be judged in a different way, a different system of criteria, knowing that uncertainty and unsoundness creeps into the logical procedures from this source. — Metaphysician Undercover

My point was that, having learned about the Frege-Hilbert dispute, I see that you have been arguing Frege's view and I Hilbert's. And in math, Hilbert won the 20th century. This is a matter of fact. There is nothing to argue and no right or wrong. But your discourse in this present post is alien to me, I have no idea what it's about. I am sure you are making interesting points, but they're lost on me.

Now, imagine that we remove this separation, between the truth of the premises, and the validity of the logic, because we want every part of the logical procedure to have the higher degree of certainty as valid logic has. However, the reality of the world is such that we cannot remove the uncertainty which lurks within human ideas, and thought. All we can do is create a formalism which lowers itself, to allow within it, the uncertainties which were formerly excluded, and relegated to content. Therefore we do not get rid of the uncertainty, we just incapacitate our ability to know where it lies, by allowing it to be scattered throughout the formal structure, hiding in various places, rather than being restricted to a particular aspect, the content. — Metaphysician Undercover

Modern math is what it is, and nothing you say changes that, nor am I defending it, only reporting on it. You think it's bad and wrong, ok, I'm no longer arguing the point with you if I ever was.

I will not address directly, Hilbert's technique, described in the SEP article as his conceptualization of independence and consistency, unless I read primary sources from both Hilbert and Frege. — Metaphysician Undercover

Sprichst du Deutsch?

Geometers, and mathematicians have taken a turn away from accepted philosophical principles. This I tried to describe to you in relation to the law of identity. So there is no doubt, that there is a division between the two. Take a look at the Wikipedia entry on "axiom" for example. Unlike mathematics, in philosophy an axiom is a self-evident truth. — Metaphysician Undercover

In math, axioms used to be self-evident truths, and now they're more or less arbitrary assumptions to get a given theory off the ground. No truth is claimed. I had no idea philosophers were still clinging to the old concept of axioms. No wonder they're so far behind in understanding modern math.

Principles in philosophy are grounded in ontology, but mathematics has turned away from this. — Metaphysician Undercover

It was forced on math by the discovery of non-Euclidean geometry. Once mathematicians discovered the existence of multiple internally consistent but mutually inconsistent geometries, what else could they do but give up on truth and focus on consistency?

I'm curious to hear your response to this point. What were they supposed to do with non-Euclidean geometry? Especially when 70 years later it turned out to be of vital importance in physics?

One might try to argue that it's just a different ontology, but this is not true. There is simply a lack of ontology in mathematics, as evidenced by a lack of coherent and consistent ontological principles. — Metaphysician Undercover

You say that like it's a bad thing! It was forced on math by non-Euclidean geometry. Physics is about ontology now. But of course even contemporary physics has abandoned ontology, and if you say that's a bad thing, I'd be inclined to agree with you.

You might think that this is all good, that mathematics goes off in all sorts of different directions none of which is grounded in a solid ontology, but I don't see how that could be the case. — Metaphysician Undercover

It's not good or bad, it is simple inevitable. What should math do? Abolish Eucidean or non-Euclidean geometry? On what basis?

I know you keep saying this, but you've provided no evidence, or proof. Suppose we want to say something insightful about the world. So we start with what you call a "formal abstraction", something produced from imagination, which has absolutely nothing to do with the world. Imagine the nature of such a statement, something which has nothing to do with the world. How do you propose that we can use this to say something about the world. It doesn't make any sense. Logic cannot proceed that way, there must be something which relates the abstraction to the world. But then we cannot say that the abstraction says nothing about the world. If the abstraction is in some way related to the world, it says something about the world. If it doesn't say anything about the world, then it's completely independent from any descriptions of the world, so how would we bring it into a system which is saying something about the world? — Metaphysician Undercover

As evidence I give you "The unreasonable effectiveness of math etc."

it is not the case that Frege and I do not get the method of abstraction. Being philosophers, we get abstraction very well, it is the subject matter of our discipline. You do not seem to have respect for this. — Metaphysician Undercover

You've given me not the slightest evidence that you have any idea how math works. And a lot of evidence to the contrary.

This is the division of the upper realm of knowledge Plato described in The Republic. Mathematicians work with abstractions that is the lower part of the upper division, philosophers study and seek to understand the nature of abstractions, that is the upper half of the upper division. — Metaphysician Undercover

To the extent that philosophers can't deal with mathematical practice as it is, they have no claim on such exaltation.

Really, it is people like you, who want to predicate to abstraction, some sort of idealized perfection, where it is free from the deprivations of the world in which us human beings, and our abstractions exist, who don't get abstraction. — Metaphysician Undercover

I wish. I predicated nothing. I only struggled to learn what I was taught, and I'm reporting back to you how the subject works. "People like me." Jeez man what are you going on about?

The knight moves the way it does. Or as Galileo would have said: "Yet it moves."

It might be fair to say that for Hilbert the syntax of logic does not include content. But Hilbert did not consider content irrelevant for mathematics. Indeed, Hilbert was very concerned with what he called the 'contentual' aspect of mathematics. Consistency and independence are syntactical, so Hilbert emphasized that they can be regarded irrespective of content, but mathematics also includes concerns other than consistency and independence, and Hilbert was keenly interested in the contentual, particularly finitary mathematics and application of mathematics to the sciences. Moreover, the relational structures themselves are an aspect of meaning. It is often said that Hilbert claimed that "mathematics is just a meaningless game of manipulating symbols". But I have not found any reference to a piece of writing or speech in which Hilbert said such a thing. In context of evaluating correctness of the application of the axioms and rules, Hilbert said that they may as well pertain to beer mugs as to anything else, but he did not say that therefore mathematics is devoid of content, as indeed he made clear that he thought quite the contrary.

/

In mathematical logic, of course, any set of formal sentences is an axiomatization of a formal theory. But many (maybe the preponderance of) mathematicians regard the axioms for particular fields of study not to be merely arbitrary, but rather as meaningful and true.

But many (maybe the preponderance of) mathematicians regard the axioms for particular fields of study not to be merely arbitrary, but rather as meaningful and true. — TonesInDeepFreeze

True. The degree of meaningfulness in direct proportion to the number of hours spent working in that discipline. In real and complex analysis one takes (as non-arbitrary?) and axiomatic the definitions of convergence , continuity, etc. given by several mathematicians, including Cauchy and Weierstrass (my math genealogy ancestor - along with 36K other descendants). :cool:

Weierstrass (my math genealogy ancestor — jgill

That's pretty cool!

I'm not arguing my point of view is right, I'm not even arguing a point of view. I'm telling you how modern math works. It's like this, if you don't mind a Galilean dialog. — fishfry

This is not true. You've been making arguments about "pure math", and "pure abstractions". So it is you who is making a division between the application of mathematics, "how modern math works", and pure mathematics, and you've been arguing that pure mathematics deals with pure abstractions. You've argued philosophical speculation concerning the derivation of mathematical axioms through some claimed process of pure abstraction, totally removed from any real world concerns, rather than the need for mathematics to work. So your chess game analogy is way off the mark, because what we've been discussing here, is the creation of the rules for the game, not the play of the game. And, in creating the rules we must rely on some criteria.

Nowhere do I dispute the obvious, that this is "how modern math works". That is not our discussion at all. What I dispute is the truth or validity of some fundamental principles (axioms) which mathematicians work with. This is why the game analogy fails, because applying mathematics in the real world, is by that very description, a real world enterprise, it is not playing a game which is totally unrelated to the world. So the same principle which makes playing the game something separate from a real world adventure, also makes it different from mathematics, therefore not analogous in that way

I have a proposal, a way to make your analogy more relevant. Let's assume that playing a game is a real world thing, *as it truly is something we do in the world, just like scientists, engineers and architects do real world things with mathematics. Then let's say that there are people who work on the rules of the game, creating the game and adjusting the rules whenever problems become evident, like too many stalemates or something like that. Do you agree that "pure mathematicians" are analogous to these people, fixing the rules? Clearly, the people fixing the rules are not in a bubble, completely isolated from the people involved in the real world play. Of course not, they are working on problems involved with the real world play, just like the "pure mathematicians" are working on problems involved with the application of math in science and engineering, etc.. Plato described this well, speaking about how tools are designed. A tool is actually a much better analogy for math than a game. The crafts people who use the tool must have input into the design of the tool because they know what is needed from the tool.

In conclusion, your claim that "pure mathematicians" are completely removed from the real world use of mathematics is not consistent with the game analogy nor the tool analogy. Those who create the rules of a game obviously have the real world play of the game in mind when creating the rules, so they have a purpose and those who design tools obviously have the real world use of the tool in mind when designing it, and the tool has a purpose. So if mathematics is analogous, then the pure mathematicians have the real world use of mathematics in mind when creating axioms, such that the axioms have a purpose.

Modern math is what it is, and nothing you say changes that, nor am I defending it, only reporting on it. — fishfry

The problem is that you have been "reporting" falsely. You consistently claimed, over and over again, that "pure mathematicians" work in a realm of pure abstraction, completely separated, and removed from the real world application of mathematics, and real world problems. That is the substance of our disagreement in this thread. My observations of things like the Hilbert-Frege discussion show me very clearly that this is a real world problem, a problem of application, not abstraction, which Hilbert was working on. And, the fact that Hilbert's principles were accepted and are now applied, demonstrates further evidence that Hilbert delivered a resolution to a problem of application, not a principle of pure abstraction.

It was forced on math by the discovery of non-Euclidean geometry. Once mathematicians discovered the existence of multiple internally consistent but mutually inconsistent geometries, what else could they do but give up on truth and focus on consistency?

I'm curious to hear your response to this point. What were they supposed to do with non-Euclidean geometry? Especially when 70 years later it turned out to be of vital importance in physics? — fishfry

I can't say that I see what I'm supposed to comment on. The geometry used is the one developed to suit the application, it's produced for a purpose. With the conflation of time and space, into the concept of an active changing space-time, Euclidean geometry which give principles for a static unchanging space, is inadequate. Hence the need for non-Euclidean geometry in modern physics.

It's not good or bad, it is simple inevitable. What should math do? Abolish Eucidean or non-Euclidean geometry? On what basis? — fishfry

This is why there is a need for solid ontological principles, an understanding of the real nature of time, the real nature of space. Only through such an understanding will the proper geometry be developed.
This is why it makes no sense to place the "pure mathematician" in a completely separate realm of "pure abstraction". The "pure mathematician" could dream up all sorts of different geometries, and have none of them any good for any real purpose, if the "pure mathematician" had absolutely no respect for the real nature of space.

As evidence I give you "The unreasonable effectiveness of math etc." — fishfry

Sorry fishfry, but this is evidence for my side of the argument. "The unreasonable effectiveness of math" is clear evidence that the mathematicians who dream up the axioms really do take notice, and have respect for real world problems. That's obviously why math is so effective. If the mathematicians were working in some realm of pure abstraction, with total disregard for any real world issues, then it would be unreasonable to think that they would produce principles which are extremely effective in the real world. Which do you think is the case, that mathematics just happens to be extremely effective in the real world, or that the mathematicians who have created the axioms have been trying to make it extremely effective?

You've given me not the slightest evidence that you have any idea how math works. And a lot of evidence to the contrary. — fishfry

Our discussion, throughout this thread has never been about "how math works". We have been discussing fundamental axioms, and not the application of mathematics at all. You are now changing the subject, and trying to claim that all you've been talking about is "how math works", but clearly what you've been talking about has been pure math, and pure abstraction, not application.

This is not true. — Metaphysician Undercover

Ok, so the conversation is shifting now to arguing about what we've been arguing about. Slow day at Chez fishfry, so I'll play. But full disclosure, my heart's not in it.

You've been making arguments about "pure math", and "pure abstractions". — Metaphysician Undercover

Arguing in the sense of describing to you how modern math sees certain things, such as sets and order relations.

So it is you who is making a division between the application of mathematics, — Metaphysician Undercover

Me personally? You give me too much credit. Those divisions were there long before I was born. I am just talking about them. But actually we've never been talking about pure versus applied math at all, I don't know where you're getting this. Applied math is the use of math is fields like physics, economics, biology, and so forth. We haven't been talking about that at all. We've only ever been talking about pure math. The meaning of 2 + 2, the nature of sets, the nature of order relationships, how mathematicians formalize things.

"how modern math works", and pure mathematics, and you've been arguing that pure mathematics deals with pure abstractions. — Metaphysician Undercover

LOL. I'd be glad to argue that any day. Pure math deals with pure abstractions? What's your counter proposition? That's like saying barbering deals with cutting hair. There is no sensible negation to the proposition.

You've argued philosophical speculation concerning the derivation of mathematical axioms through some claimed process of pure abstraction, totally removed from any real world concerns, rather than the need for mathematics to work. — Metaphysician Undercover

On the contrary, I have argued that the choice of mathematical axioms is pragmatic. Possibly not that much in our convos but in general. The axioms are chosen because they let you build up good theories above them. I've always argued that. But "real world" concerns are not involved, that I agree with.

So your chess game analogy is way off the mark, because what we've been discussing here, is the creation of the rules for the game, not the play of the game. And, in creating the rules we must rely on some criteria. — Metaphysician Undercover

My chess analogy is perfectly apt. By what criterion is the rule for how the knight moves chosen? Why is the lower right-hand square always white ("white on the right") and never black as is often erroneously portrayed by careless prop artists in movies?

Nowhere do I dispute the obvious, — Metaphysician Undercover

Now THAT is funny. You do nothing but, starting from "2 + 2 does not denote the same mathematical object as 4," several years ago, right up to the present moment. You constantly dispute the obvious.

that this is "how modern math works". That is not our discussion at all. — Metaphysician Undercover

For the past several weeks I've been explaining to you that mathematical sets have no inherent order and you've been arguing that this is somehow "wrong." So we have definitely been discussing "how modern math works."

What I dispute is the truth or validity of some fundamental principles (axioms) which mathematicians work with. — Metaphysician Undercover

How can you argue with the truth of things that are not claimed to be true? Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY says that. This is your own personal strawman. And it's a tremendous misunderstanding on your part that anyone is claiming the axioms of math are "true." The axioms are strings of formal symbols, true in some models and false in others. Your failure to comprehend this is a great failing of yours.

This is why the game analogy fails, because applying mathematics in the real world, is by that very description, a real world enterprise, it is not playing a game which is totally unrelated to the world. — Metaphysician Undercover

I have never had any interest in applying math to the real world. I wonder why you think I do, or should? I'm with the great British mathematician G. H. Hardy, who argued in his great essay, A Mathematician's Apology, that the beauty of an area of math is measured by how utterly useless it is; and that by this criterion his own field, that of number theory, is the most supremely beautiful area of math. How ironic, then, that number theory, which was supremely useless for 2000 years, has in only the past few decades become the core technology behind Internet security and cryptocurrencies. Hardy would spin in his grave. Hardy was played by Jeremy Irons in The Man Who Knew Infinity, highly recommended. A very rare math film that gets the math right and tells a great human story too, the tragic story of Ramanujan. A must-see for all readers of this site.

Pure math is not about the real world. Now you may not like that, and you math think it "should" be otherwise, but I am only telling you how it is. You can't argue with me about that. I don't know why you persist in trying.

So the same principle which makes playing the game something separate from a real world adventure, also makes it different from mathematics, therefore not analogous in that way — Metaphysician Undercover

It's entirely analogous. Chess is a formal game, there's no "reason" why the knight moves as it does other than the pragmatics of what's been proven by experience to make for an interesting game. And there are equally valid variations of the game in common use as well.

I have a proposal, a way to make your analogy more relevant. Let's assume that playing a game is a real world thing, *as it truly is something we do in the world, just like scientists, engineers and architects do real world things with mathematics. Then let's say that there are people who work on the rules of the game, creating the game and adjusting the rules whenever problems become evident, like too many stalemates or something like that. Do you agree that "pure mathematicians" are analogous to these people, fixing the rules? — Metaphysician Undercover

Perhaps you're thinking of engineers.

Clearly, the people fixing the rules are not in a bubble, completely isolated from the people involved in the real world play. Of course not, they are working on problems involved with the real world play, just like the "pure mathematicians" are working on problems involved with the application of math in science and engineering, etc.. — Metaphysician Undercover

My God man, pure mathematicians are not concerned with the problems of the world. And when they are, they are doing applied math, not pure.

Plato described this well, speaking about how tools are designed. A tool is actually a much better analogy for math than a game. The crafts people who use the tool must have input into the design of the tool because they know what is needed from the tool. — Metaphysician Undercover

Math is justified only by itself. That the physicists find it useful is good for them. It's not what drives math. You truly don't understand this. Now you are perfectly entitled to argue that things SHOULD be different. But you can't credibly argue that they ARE different, because they are not. Just ask a physicist about math, they'll tell you the mathematicians are off in the clouds totally untethered from the real world. As if that's a bad thing!

In conclusion, your claim that "pure mathematicians" are completely removed from the real world use of mathematics is not consistent with the game analogy nor the tool analogy. Those who create the rules of a game obviously have the real world play of the game in mind when creating the rules, so they have a purpose and those who design tools obviously have the real world use of the tool in mind when designing it, and the tool has a purpose. So if mathematics is analogous, then the pure mathematicians have the real world use of mathematics in mind when creating axioms, such that the axioms have a purpose. — Metaphysician Undercover

You didn't move me with such a weak and fallacious argument. Your argument that math is concerned with the real world was true a thousand years ago, but has not been true for a long time.

The problem is that you have been "reporting" falsely. — Metaphysician Undercover

By your ignorant measure. As measured by reality, I've been reporting accurately.

You consistently claimed, over and over again, that "pure mathematicians" work in a realm of pure abstraction, completely separated, and removed from the real world application of mathematics, and real world problems. — Metaphysician Undercover

Yes.

That is the substance of our disagreement in this thread. My observations of things like the Hilbert-Frege discussion show me very clearly that this is a real world problem, a problem of application, not abstraction, which Hilbert was working on. And, the fact that Hilbert's principles were accepted and are now applied, demonstrates further evidence that Hilbert delivered a resolution to a problem of application, not a principle of pure abstraction. — Metaphysician Undercover

Hilbert's famous quote is that ""One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs"" I suggest you meditate on this point.

Of course Hilbert did some applied math too. He famously offered to help Einstein finish general relativity, only for Einstein to discover, almost too late, that Hilbert was trying to finish first and gain credit. Fortunately Hilbert had an error in his calculations and in the end, graciously conceded priority to Einstein. Otherwise we'd say to this day, "He's no Hilbert!" instead of "He's no Einstein."

I can't say that I see what I'm supposed to comment on. The geometry used is the one developed to suit the application, it's produced for a purpose. With the conflation of time and space, into the concept of an active changing space-time, Euclidean geometry which give principles for a static unchanging space, is inadequate. Hence the need for non-Euclidean geometry in modern physics. — Metaphysician Undercover

Yet both theories are internally consistent. So math alone can't determine truth. I believe you've conceded my point.

Sorry fishfry, but this is evidence for my side of the argument. "The unreasonable effectiveness of math" is clear evidence that the mathematicians who dream up the axioms really do take notice, and have respect for real world problems. That's obviously why math is so effective. If the mathematicians were working in some realm of pure abstraction, with total disregard for any real world issues, then it would be unreasonable to think that they would produce principles which are extremely effective in the real world. Which do you think is the case, that mathematics just happens to be extremely effective in the real world, or that the mathematicians who have created the axioms have been trying to make it extremely effective? — Metaphysician Undercover

Of course math is inspired by the world. It's just not bound by it. A point I've made to you a dozen times by now.

Our discussion, throughout this thread has never been about "how math works". — Metaphysician Undercover

News to me.

We have been discussing fundamental axioms, and not the application of mathematics at all. — Metaphysician Undercover

I agree with that. I've never had the slightest interest in applied math. I'm with Hardy. Math is worthwhile to the extent that it's useless. Of course he was being a bit facetious, I suppose. And in the end even his belovedly useless number theory came to be indispensable to the world.

You are now changing the subject, and trying to claim that all you've been talking about is "how math works", but clearly what you've been talking about has been pure math, and pure abstraction, not application. — Metaphysician Undercover

LOL. Now that's funny. As if I've ever been talking about anything else.

@Meta surely this convo has run its course, don't you agree?

The geometry used [Non-Euclidean] is the one developed to suit the application, it's produced for a purpose. With the conflation of time and space, into the concept of an active changing space-time, Euclidean geometry which give principles for a static unchanging space, is inadequate. Hence the need for non-Euclidean geometry in modern physics — Metaphysician Undercover

Wow! I am really impressed to realize Omar Khayyam (1048-1131) had the perspicacity to realize his efforts at Non-Euclidean geometry involved notions of space-time. Thanks, MU. I would not have guessed. :chin:

Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true." The axioms are strings of formal symbols, true in some models and false in others. — fishfry

Many mathematicians and philosophers of mathematics regard certain axioms and theorems to be true not just relative to models. It might even be the dominant view.

Many mathematicians and philosophers of mathematics regard certain axioms and theorems to be true not just relative to models. It might even be the dominant view. — TonesInDeepFreeze

My sense of the matter is as follows. The overwhelming majority of working mathematicians are not set theorists or involved in foundations. They pay no attention to set theory and would be hard-pressed to even name the axioms. It's not like your average anabelian geometer ever gives explicit though to the truth or falsity of the axiom of replacement. The question doesn't come up.

Among those who study foundations, it must be abundantly clear that the axioms are arbitrary and not literally true, since it's consistent to accept or deny Foundation, Powerset, and other axioms that are never questioned in standard math. Powerset negation is its own cottage industry these days, even though it's an extremely niche interest from a mainstream point of view.

It's hard for me to believe that anyone thinks the axioms of set theory are literally true about the world; or even about the abstract world of mathematics. There may be a few.

So at best I would say that "a few" mathematicians claim the axioms are literally true in some sense.

The foremost philosopher of set theory, Penelope Maddy, argues persuasively that the axioms are chosen pragmatically, on a variety of practical grounds. See https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf for example.

Among philosophers, who could seriously argue that the axioms of set theory are "true" in any meaningful sense; or even meaningful in any meaningful sense! I am less familiar philosophy than math, but my sense is that just as with mathematicians, most philosophers aren't concerned with the axioms of set theory at all, let alone their truth.

That's my sense of the matter. I'd go further and say that it's perfectly clear that some of the axioms, such as the axiom of infinity, are literally false. That strengthens my point to @Meta, which is that the axioms are chosen pragmatically for their utility in developing math, and not for any real-world reasons.

I would add to all that the growing importance of neo-intuitionist, constructivist, and category-theoretic approaches, in which set theory is not only false, but irrelevant. (Category theorists pay lip service to set theory by defining "small" or "locally small" categories where certain collections are required to be sets, and some of the category theorists do worry about such things, but in the mainstream of category theory, they'd don't worry too much about set theory).

But of course you said, "certain axioms," and I suppose if you want to say that high school notions like unions and intersections are true or instantiable in the real world, you'd have a point. At least for finite sets. But finite sets are not of much interest to set theorists. Finite sets belong to combinatorics.

The overwhelming majority of working mathematicians are not set theorists or involved in foundations. They pay no attention to set theory and would be hard-pressed to even name the axioms. — fishfry

I would expect that there is a wide range of interest in foundational axioms among mathematicians - from no interest to intense interest. But even among mathematicians with only little interest in foundations, there are those who regard certain axioms as true without having to quality by saying "only relative to models". For example, there are mathematicians who regard the Peano axioms as true, without saying "but only relative to certain models".

Moreover, as a special case, the logical axioms are true in every model, so those are axioms that would be said to be true without qualification as to models.

The question doesn't come up. — fishfry

The question whether mathematical truth is merely model-theoretic doesn't usually come up in studies outside foundations. But the question whether a given mathematical statement is true or false comes up all the time. Indeed, the fact that a great many mathematicians don't even care about foundations leads to them saying about certain axioms that they are true (while they don't qualify "but only in certain models"). The axioms adopted in the field of study are often regarded as true, even without bothering about model-theoretic truth. Indeed, for a good number of mathematicians, it is repugnant to view mathematics as just symbol study with the formulations not expressing mathematical truths unqualified by models.

Among those who study foundations, it must be abundantly clear that the axioms are arbitrary and not literally true — fishfry

What you say must be is just not the case. There are indeed mathematicians in foundations and philosophers who regard certain axioms as true without having to add "but only in certain models". This is not being ignorant of model-theoretic truth, but rather to view that there are other senses of truth - mathematical realism (Godel being the most famous), operational, et. al, or even just a naive sense in which mathematicians regard certain axioms and theorems to express truths in their fields of study, ranging from concrete finitary truths to greater abstractions.

or at best I would say that "a few" mathematicians claim the axioms are literally true in some sense. — fishfry

Then that is at lease some movement from 'NOBODY' [all caps original]. Of course for an accurate quantification, we would need an accurate poll. But from my readings in mathematics, from conversations with mathematicians, and from reading posts of mathematicians, I have found that there are many who regard certain axioms and theorems to be true without having to qualify to models. And my impression, though not derived by polling, is that that is the case for most mathematicians. Ask some mathematicians "Is the fundamental theorem of arithmetic true?" Then when the answer is 'yes', ask "Do you mean it's true only in certain models of the language of arithmetic, or do you mean it's just true"? I bet you dollars to donuts that most would answer 'just true' to the second question.

Also, adding "literally" narrows your first claim. There are notions of truth including, model-theoretic, literal, realism, operational, true-to-a-concept, et. al.

Among philosophers, who could seriously argue that the axioms of set theory are "true" in any meaningful sense; or even meaningful in any meaningful sense! — fishfry

You may think it not wise to think that they are true or meaningful in a sense other than model-theoretic, but that does not entail that no mathematicians or philosophers (or even only a few) have that view. Indeed, for example, mathematical realism, broadly put, is the view that there are mathematical objects and truth about them independent of consciousness about them. Ordinarily, mathematical realism would regard that there are true axioms that are true even without having to qualify "but only relative to certain models". For a good number of mathematicians and philosophers, they recognize that sentences that are not validities are not true in every model, but they still regard certain axioms to be true in a sense other merely model-theoretic..

I'd go further and say that it's perfectly clear that some of the axioms, such as the axiom of infinity, are literally false. — fishfry

I don't know what your definition of 'literal' is when it comes to mathematics, but you are of course entitled to your own view about certain axioms, but that does not entail that no (or even only few) mathematicians share your view.

I would add to all that the growing importance of neo-intuitionist, constructivist, and category-theoretic approaches, in which set theory is not only false, but irrelevant. — fishfry

Yet people do work in constructivist and intuitionist set theory. I don't know a lot about category theory, but it can be axiomatized by ZFC+Grothendieck-universe.

you said, "certain axioms," and I suppose if you want to say that high school notions like unions and intersections are true or instantiable in the real world, you'd have a point. — fishfry

I mean only that there may be axioms some mathematicians don't believe to be true. For example, some mathematicians might regard the axioms of Peano arithmetic to be true but not, the axiom of choice, or whatever. I don't mean to say that those axioms that mathematicians do believe to be true don't include set theory axioms such as power set, schema of replacement, infinity, choice, and even some who believe that the continuum hypothesis is true and some who believe the negation of the continuum hypothesis is true, or certain large cardinal axioms.

Your claim was that 'NOBODY' [all caps original] believes axioms are true without model-theoretic qualification. Now it is that at best only a few believe axioms are "literally" true. 'literal' is not defined yet in this conversation as to mathematical truth, and we should expect that different mathematicians would have different definitions if you forced them to respond to it, but it is not the case that only a few mathematicians and philosophers hold that there is no mathematical truth other than model theoretic, and, it is at least my impression that most mathematicians and a fair number of philosophers do hold that there is mathematical truth other than the model-theoretic sense.

I don't know a lot about category theory, but it can be axiomatized by ZFC+Grothendieck-universe. — TonesInDeepFreeze

Which requires the existence of an inaccessible cardinal, the existence of which is not even provable in ZFC.

We're arguing over what other people think, we can't ever get to the bottom of that. @jgill posted a while back about the tiny percentage of overall math papers that are devoted to set theory. Few working mathematicians give any of these matters the slightest thought. In what sense could the Peano axioms be true in the real world? There are only $10^{78}$ hydrogen atoms in the observable universe.

Which requires the existence of an inaccessible cardinal, the existence of which is not even provable in ZFC. — fishfry

Yes, which makes it even more curious what one would mean by saying the axioms of ZFC are false, while proposing a theory that is equivalent to ZFC PLUS another axiom.

tiny percentage of overall math papers that are devoted to set theory. — fishfry

That doesn't entail that a lot mathematicians aren't aware of axioms, including those not of set theory and those of set theory. And, again, probably most mathematicians don't get hung up on mathematical logic and its model theoretic sense of truth, yet mathematicians speak of the truth of mathematical statements.

And it's not even a given that only a few mathematicians who do understand models in mathematical logic hold that there are other senses of truth, including realism, instrumental, true-to-concept, et. al. Indeed, we know that there are mathematicians who well understand mathematical logic but still regard a sense of truth no restricted to that of "true in a model".

We're arguing over what other people think, we can't ever get to the bottom of that. — fishfry

You made a clam about it. We don't have a scientific polling, but we can see that there are many people who don't think that mathematical truth is confined only to the model-theoretic sense.

In what sense could the Peano axioms be true in the real world? — fishfry

So 'real world' is now added to the question.

Again, that you view certain notions about mathematics to be untenable doesn't entail that there are not plenty of people who don't share your view

Yes, which makes it even more curious what one would mean by saying the axioms of ZFC are false, while proposing a theory that is equivalent to ZFC PLUS another axiom. — TonesInDeepFreeze

The axioms aren't false, either, any more than the way the knight moves in chess may be said to be true or false. It's just a rule that's been found by experience to make the game interesting.

That doesn't entail that a lot mathematicians aren't aware of axioms, including those not of set theory and those of set theory. And, again, probably most mathematicians don't get hung up on mathematical logic and its model theoretic sense of truth, yet mathematicians speak of the truth of mathematical statements.

And it's not even a given that only a few mathematicians who do understand models in mathematical logic hold that there are other senses of truth, including realism, instrumental, true-to-concept, et. al. Indeed, we know that there are mathematicians who well understand mathematical logic but still regard a sense of truth no restricted to that of "true in a model". — TonesInDeepFreeze

I truly can't argue about what the majority or substantial plurality or "some" or "a few" or whatever mathematicians believe. I have no data or evidence, neither do you. But the subject matter that most mathematicians work on, as evidenced by the number of papers published, is so far removed from foundations that I can't imagine that many mathematicians spend five minutes thinking about the subject in a year or in a career.

You made a clam about it. We don't have a scientific polling, but we can see that there are many people who don't think that mathematical truth is confined only to the model-theoretic sense. — TonesInDeepFreeze

We can talk math, or we can talk philosophy of math, but arguing popular opinion is not fruitful. What outcome are you looking for? Would you like me to go from "a few" to "a whole bunch?" I'm not sure what outcome would satisfy you. This is not a meaningful conversation.

So 'real world' is now added to the question. — TonesInDeepFreeze

That's what true and false typically mean. The axiom of infinity is manifestly false about the real world. At the very least it's inconsistent with contemporary physics. But it's an essential axiom of standard mathematics. Perhaps you can put your concept of truth into context for me such that the axiom of infinity could be regarded as even having a meaningful truth value other than it being generally accepted as an axiom of modern set theory.

Again, that you view certain notions about mathematics to be untenable doesn't entail that there are not plenty of people who don't share your view — TonesInDeepFreeze

"Plenty." Ok I can live with that. If, given that there must be 100,000 or so math professors in the world, I concede that "plenty" of them believe whatever you say they believe, would that satisfy you? Respected mathematician Alexander Abian wanted to blow up the moon; and prolific author of high-level math texts Serge Lange was an AIDs denier. The Unabomber had a doctorate in math, as did the guy who swindled the CIA during the Iraq war, Ahmed Chalabi. Mathematicians are human, they believe all sorts of things.

I'll go with "plenty" if this will mollify your sense of right and wrong here.

A progression of views (not necessarily your own):

(1) "Hilbert said that mathematics is only a meaningless game of manipulating symbols."

False. Hilbert was very much concerned with the contentual aspect of mathematics.

(2) "Al mathematicians view mathematics as only a meaningless game of manipulating symbols."

Clearly false.

(3) Formalism in mathematics is the view that mathematics is only a meaningless game of manipulating symbols.

False.

(4) There is a form of extreme formalism that views mathematics as only a meaningless game of manipulating symbols.

True.

(5) All mathematicians and philosophers hold that truth in mathematics pertains only to truth per models in mathematical logic.

Clearly false.

(6) All mathematicians and philosophers who understand truth per models hold that there are no viable senses of mathematical truth other than that of models.

Clearly false.

(7) Most mathematicians and philosophers who understand truth per models hold that there are no viable senses of mathematical truth other than that of models.

Not known. My impression is that it is false, but would deserve a poll.

(8) Among mathematicians who know nothing, or very little, about models in mathematical logic, all (with possibly only few exceptions) regard axioms (incuding Peano, set theory) as true only with regard to models.

False, essentially a contradictory claim.

(9) Among mathematicians who know nothing, or very little, about models in mathematical logic, only a few are familiar with the set theory axioms.

Not clear. My guess is that it is false.

(10) Of those mathematicians who are familiar with the axioms of set theory, all (with possibly only a few exceptions) view the axioms as false.

Almost surely false.

(11) Of those mathematicians who are familiar with the axioms of set theory, all (with possibly only a few exceptions) view the axioms as meaningful only as syntactic objects for syntactically proving other syntactic objects.

Clearly false.

/

In any case,

Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true." — fishfry

is false.

(not necessarily your own): — TonesInDeepFreeze

Well then I don't feel bound to justify them. I'll let you have the last word on almost all of this. The one thing I'd like you to explain to me is that if you deny that the axiom of infinity is "manifestly false about the real world," which is a statement I actually DID make, in what sense to you regard it as physically true? Or if not physically true, how is it meaningful to say it's either true or false in some other sense, Platonic, formal, or otherwise? How is the axiom of infinity different than the way the knight moves?

Here is Maddy (linked above) quoting Hallet about infinity:

Dealing with natural numbers without having the set of all natural
numbers does not cause more inconvenience than, say, dealing with sets
without having the set of all sets. Also the arithmetic of the rational
numbers can be developed in this framework. However, if one is already
interested in analysis then infinite sets are indispensable since even the
notion of a real number cannot be developed by means of finite sets only.
Hence we have to add an existence axiom that guarantees the existence of
an infinite set.

This is a pragmatic argument. Ontologically we could do without the axiom of infinity. We adopt it on purely pragmatic grounds, in order to get a decent theory of the real numbers. That makes it neither true nor false, words that are not meaningful in this context; but rather useful, which is my position on the matter.

The axioms aren't false, either, any more than the way the knight moves in chess may be said to be true or false. It's just a rule that's been found by experience to make the game interesting. — fishfry

That's your view. My point is not nor has been to convince you otherwise. Rather, my point is that no matter that it may be your view, it is not true that nobody (or only a few) people disagree with it.

I truly can't argue about what the majority or substantial plurality or "some" or "a few" or whatever mathematicians believe. — fishfry

Fine. And so there's not basis to claim that nobody (or merely a few) views axioms as true in a sense other than relative to models.

I have no data or evidence, neither do you. — fishfry

I have evidence from writings, conversations, and posts. From those, it is manifestly clear that it is false that "Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true."" Then, as to what the majority of mathematicians believe, I've stated my impression based on what I have read and heard from mathematicians, while I've said that of course that impression is not scientific.

But the subject matter that most mathematicians work on, as evidenced by the number of papers published, is so far removed from foundations that I can't imagine that many mathematicians spend five minutes thinking about the subject in a year or in a career. — fishfry

Again, that is the wrong road of argument for your position. I don't doubt that the vast majority of mathematicians don't care about foundations, in particular the model-theoretic notion of truth. But that only adds to my argument, not yours. Clearly, commonly mathematicians speak of the truth of mathematical statements, and even many mathematicians not occupied with foundations understand axioms in their field of study and often enough even the set theory axioms. So when such mathematicians say things like "the fundamental theorem of arithmetic" is true, then they don't mean it as "the fundamental theorem of arithmetic is true only in the sense that it is derivable in a consistent formal theory so that it is true in some models".

arguing popular opinion is not fruitful — fishfry

It's become a point of contention only because I responded to your claim about it, and not just in popular opinion, but your claim of totality of opinion.

Would you like me to go from "a few" to "a whole bunch?" I'm not sure what outcome would satisfy you. This is not a meaningful conversation. — fishfry

I don't care what you go to. I am making my own point that it is not the case that NOBODY (or even only a few) people regard axioms as true other than model-theoretically.

This is not a meaningful conversation. — fishfry

If it was meaningful for you to make the claim, then it is meaningful for me to reply to it, and to reply to your replies.

So 'real world' is now added to the question.
— TonesInDeepFreeze

That's what true and false typically mean. — fishfry

Views of mathematical truth don't have to be limited to what is typical otherwise. Whether or not departures from "typical" are justified, my main point was that it is not the case that all (or nearly all) mathematicians regard truth as merely model-theoretic.

Moreover, some mathematicians do regard certain mathematical statements in what is arguably a typical sense of finitary combinatory statements being concrete and true. And, as I mentioned, validities are true no matter what the models.

The axiom of infinity is manifestly false about the real world. — fishfry

That's your view. But it doesn't refute my point that it is not the case that all (or nearly all) mathematicians and philosophers regard axioms as true only as pertains to models.

Perhaps you can put your concept of truth into context for me. — fishfry

It doesn't matter toward my point. I have not claimed nor disagreed with any notion of truth. I don't have to just to point out that it is not the case that nobody regards axioms as true except relative to models. This reminds me of an article I read today. The writer claimed that nobody finds Colbert funny. I don't have to opine whether Colbert is funny to point out that it is false that nobody laughs at his jokes.

Respected mathematician Alexander Abian wanted to blow up the moon; and prolific author of high-level math texts Serge Lange was an AIDs denier. — fishfry

That opens another question. Whether one agrees with notions of mathematical truth other than model-theoretic, I'd be inclined not to claim that thinking philosophically or heuristically of mathematical truth as rather than model theoretic is among the ilk of proposing detonation of the moon or claiming that AIDS doesn't exist.

if you deny that the axiom of infinity is "manifestly false about the real world," — fishfry

I neither denied it nor affirmed it.

Two different things: (1) "P is the case" and (2) "Nobody claims that ~P is the case".

Today when I read "Nobody thinks Colbert is funny", my first thought was not "But Colbert is funny" nor "I agree that Colbert is not funny", but rather how ludicrous it is to start an opinion article about American society with such a manifestly false claim as "Nobody thinks Colbert is funny."

[the axiom of infinity] is just a rule that's been found by experience to make the game interesting. — fishfry

If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'.

That's your view. My point is not nor has been to convince you otherwise. Rather, my point is that no matter that it may be your view, it is not true that nobody (or only a few) people disagree with it. — TonesInDeepFreeze

I have conceded the word "plenty." I can't continue to argue with you about what (a few, some, many, a strong plurality, a majority, an overwhelming flood) of people think. I won't respond any more to that subject. If you want to talk about whether the axiom of infinity may be meaningfully said to be true or false, that's a good conversation. If you want to argue about what people think, I can't engage on that anymore. Having conceded the word "plenty" already, I would think you would be happy, and that's as far as I'll go.

Fine. And so there's not basis to claim that nobody (or merely a few) views axioms as true in a sense other than relative to models. — TonesInDeepFreeze

99% of professional mathematicians are not involved in foundations (more or less objective number, I didn't look it up but recall @jgill's post regarding the percentage of recently published papers) and therefore have no professional opinion on the subject at all.

I have evidence from writings, conversations, and posts. From those, it is manifestly clear that it is false that "Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true."" Then, as to what the majority of mathematicians believe, I've stated my impression based on what I have read and heard from mathematicians, while I've said that of course that impression is not scientific. — TonesInDeepFreeze

Enough. No more of this for me.

Again, that is the wrong road of argument for your position. I don't doubt that the vast majority of mathematicians don't care about foundations, in particular the model-theoretic notion of truth. But that only adds to my argument, not yours. Clearly, commonly mathematicians speak of the truth of mathematical statements, and even many mathematicians not occupied with foundations understand axioms in their field of study and often enough even the set theory axioms. So when such mathematicians say things like "the fundamental theorem of arithmetic" is true, then they don't mean it as "the fundamental theorem of arithmetic is true only in the sense that it is derivable in a consistent formal theory so that it is true in some models". — TonesInDeepFreeze

No más, por favor

It's become a point of contention only because I responded to your claim about it, and not just in popular opinion, but your claim of totality of opinion. — TonesInDeepFreeze

You made your point then got tedious and are now beyond that.

I don't care what you go to. I am making my own point that it is not the case that NOBODY (or even only a few) people regard axioms as true other than model-theoretically. — TonesInDeepFreeze

I think you've expressed yourself with sufficient conviction on the matter.

If it was meaningful for you to make the claim, then it is meaningful for me to reply to it, and to reply to your replies. — TonesInDeepFreeze

Perhaps you are taking things a bit too literally.

Views of mathematical truth don't have to be limited to what is typical otherwise. Whether or not departures from "typical" are justified, my main point was that it is not the case that all (or nearly all) mathematicians regard truth as merely model-theoretic. — TonesInDeepFreeze

As you've said.

That's your view. But it doesn't refute my point that it is not the case that all (or nearly all) mathematicians and philosophers regard axioms as true only as pertains to models. — TonesInDeepFreeze

Enough. Please.

It doesn't matter toward my point. I have not claimed nor disagreed with any notion of truth. I don't have to just to point out that it is not the case that nobody regards axioms as true except relative to models. This reminds me of an article I read today. The writer claimed that nobody finds Colbert funny. I don't have to opine whether Colbert is funny to point out that it is false that nobody laughs at his jokes. — TonesInDeepFreeze

You have pointed it out.

That opens another question. — TonesInDeepFreeze

Ok!! I'm glad to change the subject.

Whether one agrees with notions of mathematical truth other than model-theoretic, I'd be inclined not to claim that thinking philosophically or heuristically of mathematical truth as rather than model theoretic is among the ilk of proposing detonation of the moon or claiming that AIDS doesn't exist. — TonesInDeepFreeze

The question of foundations is as far from the practice of most mainstream mathematicians as blowing up the moon or AIDs denialism. If someone is classifying the finite simple groups, they are not thinking about the axiom of replacement.

— TonesInDeepFreeze

if you deny that the axiom of infinity is "manifestly false about the real world,"
— fishfry

I neither denied it nor affirmed it. — TonesInDeepFreeze

It would be fun if you did, then we could have a conversation.

Two different things: (1) "P is the case" and (2) "Nobody claims that ~P is the case". — TonesInDeepFreeze

The former being interesting, the latter tedious beyond belief.

Today when I read "Nobody thinks Colbert is funny", my first thought was not "But Colbert is funny" nor "I agree that Colbert is not funny", but rather how ludicrous it is to start an opinion article about American society with such a manifestly false claim as "Nobody thinks Colbert is funny." — TonesInDeepFreeze

So you didn't change the subject after all.

If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'. — TonesInDeepFreeze

This could never be true. Physics has not been axiomatized at all. They can't even reconcile quantum mechanics and relativity. And the idea that set theory could ever be a foundation for physics seems to me to be an unlikely stretch. But at least that is an interesting and substantive topic in the philosophy of math and science.

I would say that if someone asks, "Is it meaningful to ask if the axiom of infinity is true or false; and if so, which?" I would be willing to argue any side of that. That the axiom of infinity both is and isn't meaningfully true or false; or that if it is, it's true; or that if it is, it's false. I could whip up a good argument for each of those three propositions meaningless, meaningful/true, and meaningful false.

How can you argue with the truth of things that are not claimed to be true? Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY says that. — fishfry

I don't see why this is a problem for you. You hand me a proposition, and I refuse to accept it, claiming that it is false. You say, 'but I am not claiming that it is true'. So I move to demonstrate to you why I believe it to be false. You still insist that you are not claiming it to be true, and further, that the truth or falsity of it is irrelevant to you. Well, the truth or falsity of it is not irrelevant to me, and that's why I argue it's falsity hoping that you would reply with a demonstration of its truth to back up your support of it.. If the truth or falsity of it is really irrelevant to you, then why does it bother you that I argue its falsity? And why do you claim that I cannot argue the falsity of something which has not been claimed to be true? Whether or not you claim something to be true, in no way dictates whether or not I can argue its falsity.

It's entirely analogous. Chess is a formal game, there's no "reason" why the knight moves as it does other than the pragmatics of what's been proven by experience to make for an interesting game. And there are equally valid variations of the game in common use as well. — fishfry

Either you are not getting the point, or you are simply in denial. Playing chess, is a real world activity, as is any activity. Your effort to describe an activity, like the game of chess, or pure math, as independent from the real world, as if it exists in it's own separate bubble which is not part of the world, is simply a misrepresentation.

Now, you admit that there actually is a pragmatic reason why the knight moves as it does, and this is to make an interesting game. And of course playing a game is a real world activity. so there is a real world reason for that rule. Now if you could hold true to your analogy, and admit the same thing about pure mathematics, then we'd have a starting point, of common agreement. However, the reason for mathematical principles beings as they are, such as our example of the Pythagorean theorem, is not to make an interesting game. It is for the sake of some other real world activity. Do you agree?

Of course math is inspired by the world. It's just not bound by it. A point I've made to you a dozen times by now. — fishfry

You keep on insisting on such falsities, and I have to repeatedly point out to you that they are falsities. But you seem to have no respect for truth or falsity, as if truth and falsity doesn't matter to you. Mathematics has been created by human beings, with physical bodies, physical brains, living in the world. It has no means to escape the restrictions imposed upon it by the physical conditions of the physical body. Therefore it very truly is bound by the world. Your idea that mathematics can somehow escape the limitations imposed upon it by the world, to retreat into some imaginary world of infinite infinities, is not a case of actually escaping the bounds of the world at all, it's just imaginary. We all know that imagination cannot give us any real escape from the bounds of the world. Imagining that mathematics is not bound by the world does not make it so. Such a freedom from the bounds of the world is just an illusion. Mathematics is truly bound by the world. And when the imagination strays beyond these boundaries, it produces imaginary fictions, not mathematics. But you do not even recognize a difference between imaginary fictions, and mathematics.

News to me. — fishfry

The reason why I can truthfully say that our discussion has never been about how math works, is that you have never given me any indication as to how it works. You keep insisting that mathematical principles are the product of some sort of imaginary pure abstraction, completely separated from the real world, like eternal Platonic Forms, then you give no indication as to how such products of pure fiction become useful in the world, i.e., how math works.

Wow! I am really impressed to realize Omar Khayyam (1048-1131) had the perspicacity to realize his efforts at Non-Euclidean geometry involved notions of space-time. Thanks, MU. I would not have guessed. :chin: — jgill

As I said, one can produce any sort of geometry depending on the particular purpose. My reference to space-time was in reply to fishfry's talk of a specific incidence, the use of non-Euclidian geometry in modern physics

I'll go with "plenty" if this will mollify your sense of right and wrong here. — fishfry

I don't seek to be assuaged. You don't need to assent to 'plenty' on my account. Rather, one can assent to it merely on the grounds that it is obvious.

If you want to argue about what people think, I can't engage on that anymore. — fishfry

Yet you write:

99% of professional mathematicians are not involved in foundations (more or less objective number, I didn't look it up but recall jgill's post regarding the percentage of recently published papers) and therefore have no professional opinion on the subject at all. — fishfry

You continue to miss the point. That a vast number of mathematicians don't care about foundations doesn't imply that the vast number of mathematicians don't think axioms are true except model-theoretically, as indeed the fewer who care about foundations then reasonably we would expect the fewer who think truth is merely model-theoretic.

You made your point then got tedious and are now beyond that. — fishfry

It may have been tedious, but my entries have been responses to your continued posting on the subject. It seems odd to me that one would continue to reply, and repeat some points that are essentially the same, but then complain that responses on point to your own replies are thereby tediousness.

Whether one agrees with notions of mathematical truth other than model-theoretic, I'd be inclined not to claim that thinking philosophically or heuristically of mathematical truth as rather than model theoretic is among the ilk of proposing detonation of the moon or claiming that AIDS doesn't exist.
— TonesInDeepFreeze

The question of foundations is as far from the practice of most mainstream mathematicians as blowing up the moon or AIDs denialism. — fishfry

Whether that is the case, my point is that having a foundational view that there is mathematical truth other than model-theoretic is not remotely outlandish in the class of advocating that we destroy the moon or that AIDS does not exist.

Two different things: (1) "P is the case" and (2) "Nobody claims that ~P is the case".
— TonesInDeepFreeze

The former being interesting, the latter tedious beyond belief. — fishfry

You claimed the latter, so it is reasonable to reply to it whether you find that tedious or not.

So you didn't change the subject after all. — fishfry

I addressed the additional matter of outlandishness; I didn't thereby declare that I am forever changing any subject or not changing it, especially as you continued to post as if I had not already made clear that the question of what people think is distinct from whether they are wise to think it.

If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'.
— TonesInDeepFreeze

This could never be true. Physics has not been axiomatized at all. — fishfry

I didn't say "axiomatization of physics". I said "axiomatization of the MATHEMATICS for the sciences" [all-caps added]. Of course, though there is some consensus that set theory does axiomatize the branches of mathematics needed for the sciences, one may question whether indeed all of the needed mathematics is captured. But even a negative answer would not refute my point that among the salient reasons for adopting the axiom of infinity, at least we may say those reasons include an intent to lend support to axiomatizing the mathematics for the sciences, which is far beyond merely adding it to make things interesting. Also, I don't know that physics has not been axiomatized "AT ALL" [all caps added].

Yet you write:
You continue to miss the point. That a vast number of mathematicians don't care about foundations doesn't imply that the vast number of mathematicians don't think axioms are true except model-theoretically, as indeed the fewer who care about foundations then reasonably we would expect the fewer who think truth is merely model-theoretic. — TonesInDeepFreeze

I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?

It may have been tedious, but my entries have been responses to your continued posting on the subject. It seems odd to me that one would continue to reply, and repeat some points that are essentially the same, but then complain that responses on point to your own replies are thereby tediousness. — TonesInDeepFreeze

I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?

I addressed the additional matter of outlandishness; I didn't thereby declare that I am forever changing any subject or not changing it. — TonesInDeepFreeze

I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?

I didn't say "axiomatization of physics". I said "axiomatization of the MATHEMATICS for the sciences" [all-caps added]. — TonesInDeepFreeze

I'm sure the standard axiomatization of math is an overkill for that.

Also, I don't know that physics has not been axiomatized "AT ALL" [all caps added]. — TonesInDeepFreeze

You don't know at all if it's been axiomatized ? That's something that can be looked up. Or you don't know if even small parts of it have been axiomatized? Your sentence was a little ambiguous. I'm sure there are axiomatizations of parts of science. Newtonian gravity has a nice axiomatization in Newton's three laws.

[,,,] That a vast number of mathematicians don't care about foundations doesn't imply that the vast number of mathematicians don't think axioms are true except model-theoretically, as indeed the fewer who care about foundations then reasonably we would expect the fewer who think truth is merely model-theoretic.
— TonesInDeepFreeze

I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. — fishfry

I don't know what hyperbole you have in mind. Maybe 'nobody'. Because you seemed adamant with all-caps, and, as I recall, three variations of 'no', I didn't know it was hyperbole. So I merely replied to it at face value. Of course I would not have begrudged you then declaring it was only hyperbole. But still, I don't think what was hyperbolized was correct, even if given non-hyperbolized restatement.

Anyway, your response again misses my point. My point that you just quoted is not to take issue with your hyperbole, but rather to point out how your more recent argument goes wrong.

It may have been tedious, but my entries have been responses to your continued posting on the subject. It seems odd to me that one would continue to reply, and repeat some points that are essentially the same, but then complain that responses on point to your own replies are thereby tediousness.
— TonesInDeepFreeze

I used a figure of speech called hyperbole. — fishfry

Okay, but my point quoted above was not about that.

I addressed the additional matter of outlandishness; I didn't thereby declare that I am forever changing any subject or not changing it.
— TonesInDeepFreeze

I used a figure of speech called hyperbole. — fishfry

Do you mean the hyperboles "blow up the moon" and "AIDS denier"? If so, that's fine that you say now it was hyperbole. But I did take your comments at least to be a claim that a view that mathematical truth is not confined to model-theoretical is on its face preposterous even outlandish. I said that a lot of mathematicians don't view truth as merely model-theoretic, and you replied to the effect that there are intellectually talented people who believe a number of crazy things. It is reasonable for me to say that believing that truth is not merely model theoretic is not that kind of crazy, if it is even crazy at all.

I'm sure the standard axiomatization of math is an overkill for that. — fishfry

Some set theorists have pointed how we can reduce some axiomatic assumptions and still get the mathematics for the sciences. And even if ZFC is too productive, that doesn't refute that a good part of the interest in the axiom of infinity is to axiomatize (even if too productively) the mathematics for the sciences.

You don't know at all if it's been axiomatized ? — fishfry

No, not that I don't at all know. Rather, I don't know that it hasn't been axiomatized at all (i.e. hasn't been axiomatized to any extent whatsoever or with no progress toward axiomatizing it). That was in response to what you wrote, "Physics has not been axiomatized at all."

I don't see why this is a problem for you. You hand me a proposition, and I refuse to accept it, claiming that it is false. You say, 'but I am not claiming that it is true'. So I move to demonstrate to you why I believe it to be false. You still insist that you are not claiming it to be true, and further, that the truth or falsity of it is irrelevant to you. Well, the truth or falsity of it is not irrelevant to me, and that's why I argue it's falsity hoping that you would reply with a demonstration of its truth to back up your support of it.. — Metaphysician Undercover

I am on record as holding that the axioms of set theory are neither true nor false, as they are syntactic entities whose truth or falsity can only be determined after an interpretation, or model, is provided. This is perfectly in keeping with standard practice in mathematical logic.

After all the "Abelian axiom" that xy = yx is true in the real numbers, and false in the set of 2x2 matrices whose entries are real numbers. "It is snowing" is true in Alaska in the winter, but never in San Diego. It's not possible for the axiom of replacement to be true OR false in isolation from an interpretation.

If the truth or falsity of it is really irrelevant to you, then why does it bother you that I argue its falsity? — Metaphysician Undercover

Because it shows that you misunderstand the distinction between syntax and semantics, between a formal axiomatic system and its models. So if you say an axiom is true you're wrong, and if you say it's false you're wrong! An axiom isn't true or false. Now if you would supply a model, I can tell you whether it's true in that model.

And why do you claim that I cannot argue the falsity of something which has not been claimed to be true? Whether or not you claim something to be true, in no way dictates whether or not I can argue its falsity. — Metaphysician Undercover

I couldn't talk you out of arguing that 2 + 2 and 4 represent distinct mathematical objects. I suppose I shouldn't bother with the axiom of replacement, which actually is a bit of a subtle and powerful axiom schema.

Either you are not getting the point, or you are simply in denial. — Metaphysician Undercover

Or, logically, you are in one of those two states.

Playing chess, is a real world activity, as is any activity. Your effort to describe an activity, like the game of chess, or pure math, as independent from the real world, as if it exists in it's own separate bubble which is not part of the world, is simply a misrepresentation. — Metaphysician Undercover

Doing set theory is a real world activity too, done by set theorists and undergrads the world over. Even high school students get a watered-down version of it. So my analogy holds. You're trying to say chess is "real world" because you can sit at a board and move the pieces. But it's still a formal game. You're being very disingenuous here. Sure the pieces are made of atoms, but there is no fundamental physical reason why the knight moves that way. And sure, set theorists are made of atoms too, but there is no fundamental physical reason to adopt or reject the laws of set theory.

Now, you admit that there actually is a pragmatic reason why the knight moves as it does, and this is to make an interesting game. — Metaphysician Undercover

Yes yes yes yes. That is correct. We are in agreement. People play chess for years and find that some rules give more interesting versions of the game than others. Who invented en passant pawn captures, or castling? These are obviously historically contingent developments, introduced for purely pragmatic reasons.

As are, thinking ahead here, each and every one of the axioms of set theory.

And of course playing a game is a real world activity. so there is a real world reason for that rule.[/quote[

That's an equivocation of "real world." The planets move in elliptical orbits for fundamental reasons having to do with the laws of nature (stipulating for sake of argument that there are laws of nature and that Newton and Einstein are on to something real). Chess pieces are made of wood or plastic, but their movements are not subject to the laws of nature. That is, if you drop them near the earth, they fall. But their moves within the game are arbitrary conventions of humans. Surely you can do better than to argue by equivocating these two notions.
— Metaphysician Undercover

Now if you could hold true to your analogy, and admit the same thing about pure mathematics, then we'd have a starting point, of common agreement. — Metaphysician Undercover

But I do. If you drop a mathematician from a height, he or she will fall in accordance with gravitational acceleration. But the axioms of set theory are historically contingent, pragmatically derived, matters of agreement. Like traffic lights. Red and green wavelengths are laws of nature. Which means go and which means stop is a social agreement. One which, if you violated it, can be fatal; but a social agreement nonetheless.

However, the reason for mathematical principles beings as they are, such as our example of the Pythagorean theorem, is not to make an interesting game. It is for the sake of some other real world activity. Do you agree? — Metaphysician Undercover

I agree that math is different from chess in that math is inspired by the real world (ancient bookkeeping and surveying), and has vast applications in the real world. I certainly agree that math is subtly different and that generations of philosophers have tried hard to put their finger on exactly what that means. Ideas like indispensability and so forth. Of course I agree with this point.

You keep on insisting on such falsities, — Metaphysician Undercover

That math is inspired by the world and not bound by it? To me this is a banality, not a falsehood. It's true, but so trivial as to be beneath mention to anyone who's studied mathematics or mathematical philosophy.

and I have to repeatedly point out to you that they are falsities. — Metaphysician Undercover

Since you're wrong on this point, repetition doesn't help. If you were right, you'd only have to say it once. That echoes Einstein's remark on being told that a hundred physicists disagreed with him. "If I'm wrong, one would be enough."

You actually disagree with the statement that "Math is inspired by but not bound by the world?" I propose to drill down on this because it's a clear point that we could discuss and perhaps shed some light. You disagree that math is inspired by the world? Or that it's not bound by it? I suspect you disagree with the latter. In which case I whip out non-Euclidean geometry as the classic example in support of my point.

But you seem to have no respect for truth or falsity, — Metaphysician Undercover

Not in axiomatic systems, no. Absent a model there is no truth or falsity.

as if truth and falsity doesn't matter to you. — Metaphysician Undercover

If you give me an axiomatic system plus a model, or interpretation, then truth or falsity can be determined, and matters to me.

Mathematics has been created by human beings, with physical bodies, physical brains, living in the world. It has no means to escape the restrictions imposed upon it by the physical conditions of the physical body. — Metaphysician Undercover

That's an interesting point. Yet you can see the difference between representational art, which strives to be "true," and abstract art, which is inspired by but not bound by the real world. Or as they told us when I took a film class once, "Film frees us from the limitations of time and space." A movie is inspired by but not bound by reality. Star Wars isn't real, but the celluloid film stock (or whatever they use these days) is made of atoms. Right? Right.

Therefore it very truly is bound by the world. — Metaphysician Undercover

Science fiction, abstract art, novels. Moby Dick is based on a true story of the Essex, a whaling ship sunk by a whale. But it's not bound by the story of the Essex. The characters and events are wholly made up. The point holds even more strongly for abstract art, science fiction, surrealism, and all other creative works of people.

Your idea that mathematics can somehow escape the limitations imposed upon it by the world, to retreat into some imaginary world of infinite infinities, is not a case of actually escaping the bounds of the world at all, it's just imaginary. — Metaphysician Undercover

Because my pencil is made of atoms? Are you now taking the cranky anti-Cantorian position? Everything since 1870 is bullshit? Is this your stance? Are you like this at the art museum too?

We all know that imagination cannot give us any real escape from the bounds of the world. — Metaphysician Undercover

Maybe you just don't have enough imagination. You seem to be taking a radical realist position of some kind whereby science fiction and abstract art and Star Wars either don't exist or aren't real or are lies that should be banned. What exactly is your position here? How far will you take this stance that imagination has no place in the world?

Imagining that mathematics is not bound by the world does not make it so. Such a freedom from the bounds of the world is just an illusion. Mathematics is truly bound by the world. And when the imagination strays beyond these boundaries, it produces imaginary fictions, not mathematics. But you do not even recognize a difference between imaginary fictions, and mathematics. — Metaphysician Undercover

I recognize the difference between pure and applied mathematics. And you seem to reject fiction, science fiction, surrealist poetry, modern art, and unicorns. Me I like unicorns. They are inspired by the world but not bound by it. I like infinitary mathematics, for exactly the same reason. Perhaps you should read my recent essay here on the transfinite ordinals. It will give you much fuel for righteous rage. But I didn't invent any of it, Cantor did, and mathematicians have been pursuing the theory ever since then right up to the present moment. Perhaps you could take it up with them.

The reason why I can truthfully say that our discussion has never been about how math works, is that you have never given me any indication as to how it works. You keep insisting that mathematical principles are the product of some sort of imaginary pure abstraction, completely separated from the real world, like eternal Platonic Forms, then you give no indication as to how such products of pure fiction become useful in the world, i.e., how math works. — Metaphysician Undercover

Well that's Wigner's point in the Unreasonable Effectiveness paper. I don't claim to have the words or the philosophical background to give a good account of how math, which is perfectly obviously a massive fiction, can be so darn useful in the world. A lot of people have taken a shot at the question. Surely you know this.