Timeline for Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones
Current License: CC BY-SA 4.0
17 events
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| Jul 29, 2021 at 13:23 | comment | added | David E Speyer | I'll point out that my counterargument also involves an example of the phenomenon in this question. I point out that a positive density of grammatical mathematical sentences are of the form "$1=1$ or S", yet mathematicians spend almost no time trying to prove such sentences. This fact is not very mysterious, though... | |
| Jul 27, 2021 at 19:48 | comment | added | Timothy Chow | @მამუკაჯიბლაძე I would say that "naturally occurring" means arising in a context without any hint (on the surface at least) of the foundations of mathematics, e.g., the Riemann hypothesis or the inverse Galois problem or the Navier-Stokes equation or the smooth 4-dimensional Poincare conjecture. Topos theory is suspiciously close to the foundations of mathematics, so it's not too surprising to find unprovability cropping up. | |
| Jul 27, 2021 at 19:12 | history | edited | Timothy Chow | CC BY-SA 4.0 |
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| Jul 27, 2021 at 18:49 | comment | added | მამუკა ჯიბლაძე | Would, say, considerations in topos theory count as something "naturally occurring"? Say, if I need to consider a topos possessing an internal topos with a natural numbers object, would this count? | |
| Jul 27, 2021 at 17:32 | comment | added | Timothy Chow | @AndrejBauer The main published paper I know of is Finite functions and the necessary use of large cardinals, published in the Annals of Mathematics in 1998. I said a few words about this paper in another MO answer. You're correct that Friedman, for whatever reason, hasn't published most of his results of this type. | |
| Jul 27, 2021 at 13:29 | comment | added | Timothy Chow | @მამუკაჯიბლაძე "Naturally occurring" is admittedly a vague statement. If a large cardinal axiom turned out to be needed for a statement that was not obviously set-theoretic on the face of it, that would count, but I don't think that has yet happened. | |
| Jul 27, 2021 at 13:25 | comment | added | Timothy Chow | @DavidESpeyer I wasn't aware of this issue...I'll try to take a look soon. | |
| Jul 27, 2021 at 11:20 | comment | added | David E Speyer | Regarding the Calude and Jurgensen paper: We've discussed this before and it seems wrong to me; see my post here mathoverflow.net/questions/4454/… . I've never been able to understand why I am wrong. @TimothyChow, you know a lot more logic than I do, can you help? | |
| Jul 27, 2021 at 7:16 | comment | added | Andrej Bauer | Is there a published (as in "journal") account of Friedman's work? Some of it looks very interesting, but when I google for it I always find just some FOM announcements and manuscripts typeset in fixed-width fonts. | |
| Jul 27, 2021 at 4:28 | comment | added | მამუკა ჯიბლაძე | Is not the study of large cardinals an example of systematic investigation of unprovable statements in set theory? | |
| Jul 27, 2021 at 2:45 | comment | added | Timothy Chow | @WillSawin I agree with your points. But note: ZF is much stronger than PA. A priori, we might expect that many theorems that can be stated in the first-order language of arithmetic and that can be proved in ZF would be unprovable in PA. But this seems not to be the case. "Natural" arithmetical theorems of ZF almost always turn out to be provable in PA (case study: Fermat's last theorem); it's not the case that we have lots of arithmetical theorems of ZF sitting around that we suspect are unprovable in PA but just can't prove that they're unprovable in PA. | |
| Jul 26, 2021 at 23:55 | comment | added | LSpice | Thank you for linking the comment so that I didn't need to go in and do any editing, as I do occasionally. 😁 | |
| Jul 26, 2021 at 23:35 | comment | added | Will Sawin | It's not yet clear if there are many natural unprovable statements - the only thing we know is that there aren't many natural statements that we know how to prove are unprovable. This statement is not as mysterious, as we don't have nearly as many techniques available for proving statements unprovable in number theory as we do in, say, set theory. Certainly it's still pretty mysterious, though. | |
| Jul 26, 2021 at 23:33 | comment | added | Will Sawin | Surely our notion of "naturality" of mathematical statements is shaped by our experience of mathematics, which tends to push it in the direction of statements that are not just provable, but can actually be proved by our existing techniques? Specifically, I think our mathematical aesthetic evolves to select for statements that are provable, but where a proof is very hard to find. | |
| Jul 26, 2021 at 23:18 | history | edited | Timothy Chow | CC BY-SA 4.0 |
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| S Jul 26, 2021 at 23:06 | history | answered | Timothy Chow | CC BY-SA 4.0 | |
| S Jul 26, 2021 at 23:06 | history | made wiki | Post Made Community Wiki by Timothy Chow |