Timeline for How much of the axiom of choice do you need in mathematics?
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29 events
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| Aug 14, 2023 at 8:59 | vote | accept | Someone211 | ||
| Aug 13, 2023 at 20:09 | comment | added | Andrej Bauer | I elaborated on my point of view, in case anyone is interested. This thread is getting too long. | |
| Aug 13, 2023 at 18:16 | comment | added | Sam Hopkins | I don't mean to place myself in any camp, but when I was referring to the RH possibly lacking a "definite truth value" I meant in the sense of, say, S. Feferman's paper "Is the Continuum Hypothesis a definite mathematical problem?" (Incidentally, I think CH is a good example showing how the kind of evidence I mentioned above - namely, the work of Gödel and Cohen - could convince someone that a proposition they previously have thought of as a perfectly "normal" mathematical statement might lack a truth value.) | |
| Aug 13, 2023 at 18:00 | comment | added | Andrej Bauer | I repudiate the formalist's petulant expostulations. In any case, what business would a formalist have talking about truth values, a semantic concept, in the first place? | |
| Aug 13, 2023 at 17:58 | comment | added | Timothy Chow | @AndrejBauer I don't think that your proposal is an accurate reflection of what a formalist means by "neither true nor false." The formalist means something closer to, "RH is a syntactic string to which no truth value can be assigned." That is, the formalist denies that RH is a meaningful statement. | |
| Aug 13, 2023 at 17:55 | comment | added | Andrej Bauer | P.S. Ill-formed expressions and other kinds of meaningless statements (for instance those involving invalid definite descriptions) play no role in a discussion about truth values. They are not in the domain of the semantic interpretation function that assigns truth values. | |
| Aug 13, 2023 at 17:53 | comment | added | Andrej Bauer | @TimothyChow: no, it is not the same at all! "Neither true nor false" means $\neg p \land \neg\neg p$, which is just false (classically and constructively). "Not having a definite truth value" is necessarily a meta-theoretic statement, because internally to any logic every statement obviously has a truth value simply by virtue of being meaningful. However, meta-theoretically we could observe a statement $p$ whose interpretation in some Boolean or Heyting algebra is something other than $\bot$ or $\top$. | |
| Aug 13, 2023 at 17:50 | comment | added | Andrej Bauer | By the way, much the same discussion can be had about excluded middle and constructive mathematics, which is not just frugal, it's downright ascetic. The old school proceeded from philosophical positions, but I would say the new generation, especially people close to computer science, are in the first place pragmatic (although still speak fanatically at times). Personally, I am just fascinated by how rich mathematics gets once we drop excluded middle – one can get all sorts of things that are classically unthinkable. | |
| Aug 13, 2023 at 17:50 | comment | added | Timothy Chow | @AndrejBauer I think I did literally mean what I wrote. Sam Hopkins was referring to people who might "doubt that RH has a definite truth value." Isn't that the same as "neither true nor false"? Syntactic expressions that are not well-formed, and therefore cannot be interpreted, are neither true nor false, aren't they? | |
| Aug 13, 2023 at 17:46 | comment | added | Andrej Bauer | @TimothyChow: I do hope nobody will ever convince themselves about any statement, including RH, that it is neither true nor false. They would immediately disappear in a puff of logical smoke of inconsistency. (That's just my way of saying that you probably did not literally mean what you wrote.) | |
| Aug 12, 2023 at 18:21 | comment | added | Timothy Chow | @SamHopkins In theory what you say is correct, but I don't think I've encountered anyone like that. People I've met either already doubt that RH has a definite truth value, or else unprovability doesn't faze them. For example, I don't know of anyone who says that the unprovability of "ZF is consistent" in ZF is what convinced them that ZF is neither consistent nor inconsistent. | |
| Aug 12, 2023 at 15:29 | comment | added | Sam Hopkins | @TimothyChow: "Even if we accumulated evidence that we are never going to know which it is, that wouldn't tempt us to declare that the Riemann hypothesis is neither true nor false, would it?" - Doesn't it depend on the kind of evidence that we accumulate? For instance, one kind of evidence we could obtain is a proof that the truth/falsity of RH is independent of the usual axiomatic systems that mathematicians work in. That would be a reason to doubt that RH has a definite truth value (at least for some). | |
| Aug 12, 2023 at 3:27 | comment | added | Nik Weaver | @TimothyChow you're right. I confused myself because I was thinking about the fact that I'm not certain the axioms I am skeptical of are false. But that's not the point; the point is that even they are definitely false as a description of what I understand to be the set-theoretic universe, as long as they are consistent we can legitimately reason with them. | |
| Aug 11, 2023 at 23:51 | comment | added | Timothy Chow | @NikWeaver I don't understand the motivation for being "formalists about those axioms whose truth status is unclear." Why not be platonists about such statements, but simply agnostic about their truth status? This is what we do all the time with something like the Riemann hypothesis. That is, the "standard" perspective is that the Riemann hypothesis is either true or false, but we just don't know which. Even if we accumulated evidence that we are never going to know which it is, that wouldn't tempt us to declare that the Riemann hypothesis is neither true nor false, would it? | |
| Aug 11, 2023 at 21:20 | comment | added | Nik Weaver | I agree with this! | |
| Aug 11, 2023 at 20:27 | comment | added | Joel David Hamkins | Right, that is sensible. But then why does one hear so often in the arguments about foundations that certain principles/axioms are not needed for ordinary applications? (For example, in this question.) That feature seems irrelevant to the question of truth, which is what we should really care about when choosing our foundations. | |
| Aug 11, 2023 at 20:23 | comment | added | Nik Weaver | Maybe so, but I would place skepticism about stronger principles first. If the axioms I think are clearly true turn out to be just those which are needed in mainstream mathematics, then maybe that is some sort of confirmation, but this is not why I think they are true. | |
| Aug 11, 2023 at 20:16 | comment | added | Joel David Hamkins | I think your perception about the positions I describe is not correct, since the strong foundationalists are typically arguing from truth---they have discovered these fundamental truths that we should incorporate into our foundations because it will help us discover more truths. The weak foundationalists, in contrast, seem often to argue from a view of efficiency---stronger principles are not needed for the math that we want, but underlying the view there often also seems to be some skepticism lurking about stronger principles. It seems you find a natural home with the weak foundationalists. | |
| Aug 11, 2023 at 20:00 | comment | added | Nik Weaver | Perhaps this meshes with your distaste for hybrid foundations. If one believes, as I do, that mathematical objects actually exist in some meaningful, well-defined sense, and some statements about them are true while others are not, then maybe it makes sense to be platonists about those axioms that we are confident are true, but formalists about those axioms whose truth status is unclear. From an operationalist perspective I suppose that wouldn't make sense. Anyway, I ordered your book and look forward to reading a more detailed account of your analysis. | |
| Aug 11, 2023 at 19:58 | comment | added | Nik Weaver | It seems as though neither of your characters (strong foundationalist/weak foundationalist) is primarily concerned with what is true. Both positions sound quite operationalist. | |
| Aug 11, 2023 at 18:16 | comment | added | Joel David Hamkins | What I call for in part is greater philosophical engagement specifically with the dispute between strong foundations and weak. Weak foundationalists seem to have the reasonable goal of frugality of commitment; strong foundationalists claim fundamental discoveries, which will support fundamental mathematical insights and unification. How are we to adjudicate this dispute? | |
| Aug 11, 2023 at 17:31 | comment | added | Joel David Hamkins | mitpress.mit.edu/9780262542234. That issue is treated (briefly) in chapter 8. | |
| Aug 11, 2023 at 17:28 | comment | added | Nik Weaver | Which book? I need to read this. | |
| Aug 11, 2023 at 17:22 | comment | added | Joel David Hamkins | In my personal view, I don't find those hybrid foundations to be attractive. The view I was trying to criticize in my final paragraph is a quite common one (not fringe), the idea that we should take as a goal to have the weakest possible foundational axioms that support "ordinary" or "natural" mathematics. I have highlighted this core dispute between strong foundations and weak foundations in my book and elsewhere, since I think disagreement on this points lies at the core of many other disputes. | |
| Aug 11, 2023 at 17:14 | comment | added | Nik Weaver | ... I was once accused by Harvey Friedman of wanting to "BAN" (his all caps) some mathematical fields. When I pressed him to indicate where I had ever said that, his response was something to the effect of "by saying that certain kinds of math lack a clear philosophical justification, and therefore we should be formalists about them, I was discouraging people from pursuing those directions, and, in effect, attempting to BAN them". So I have become sensitized to words like "limiting" which seem a little strawman-ish. | |
| Aug 11, 2023 at 17:10 | comment | added | Nik Weaver | Joel, regarding your controversial counterpoint, if I said, say, that we should be platonists about countable DC but formalists about stronger forms of choice --- would I be doing a fundamental disservice to mathematics? I ask because you talk about "limiting ourselves to the ideas used in `ordinary' mathematics", which sounds like someone wants to somehow forbid non-ordinary mathematics. Maybe there are such people, but it sounds like a pretty fringe position. | |
| Aug 11, 2023 at 2:30 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Aug 11, 2023 at 1:43 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Aug 11, 2023 at 1:34 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |