Timeline for answer to Axiom of choice, Banach-Tarski and reality by user44143
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| when toggle format | what | by | license | comment | |
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| Jan 23, 2017 at 16:42 | comment | added | Pedro Lauridsen Ribeiro | @AbdelmalekAbdesselam "I find ACDC (I like this acronym)..." - I see what you did there... Anyway, concerning the (mathematical) physicist's stance on constructibility in mathematics, I had a similar discussion with an OP in the comments to my MO answer mathoverflow.net/questions/238153/… | |
| Jan 23, 2017 at 15:23 | comment | added | Abdelmalek Abdesselam | @Pedro: I find ACDC (I like this acronym) more intuitive than the full-fledged AC and it agrees with my standards of constructive math. If one gives up on it then there is not much analysis that survives, in particular Baire's Thm as you pointed out. I think it si OK to use AC for the first proof of some result like the on in general relativity you mentioned but then this should be considered a temporary fix awaiting a more satisfying constructive proof (eventually using ACDC). | |
| Jan 23, 2017 at 10:07 | comment | added | David Roberts♦ | @MattF. Sorry, 'get away with' is a very imprecise way of saying what I said in the second comment, second half of the first sentence. | |
| Jan 22, 2017 at 16:52 | comment | added | user44143 | @DavidRoberts, I agree with that comment, but it seems to mean your earlier comment, "one can get away with finitism", has little force. | |
| Jan 22, 2017 at 6:42 | comment | added | David Roberts♦ | @MattF. I would almost claim that one can't appeal to physics to justify infinitary mathematics, because physics doesn't need that logical strength. Recall that mathematics for years didn't itself use a foundation, and barring some corner cases, worked perfectly fine. | |
| Jan 22, 2017 at 3:16 | comment | added | user44143 | @DavidRoberts, if the physicists say "I don't like Ye's interpretation of my math" or "I don't need a mathematical foundation for my physics at all", how does an appeal to physics settle any of the mathematical questions at issue? Or more briefly: "necessary" by what standard? | |
| Jan 21, 2017 at 1:06 | comment | added | David Roberts♦ | One can get away with a strict form of finitism as one's mathematical foundation for physics, as shown by Ye (dx.doi.org/10.1007/978-94-007-1347-5, free draft copy here: phil.pku.edu.cn/cllc/people/fengye/…). From a review [1]: "most of the mathematics necessary for modern theoretical physics can be developed within a strict finitist framework. In Chapter 8, Ye outlines semi-Riemannian geometry sufficient for proving a version of Hawking’s singularity theorem." ([1] journals.uvic.ca/index.php/pir/article/view/13181/4184) | |
| Jan 21, 2017 at 1:02 | comment | added | Pedro Lauridsen Ribeiro | This also begs the question of whether separable Hilbert spaces suffice for applications, specially regarding (mathematical) physics: physics.stackexchange.com/questions/90004/separability-axiom-really-necessary/ | |
| Jan 21, 2017 at 1:02 | comment | added | Pedro Lauridsen Ribeiro | @AbdelmalekAbdesselam Agreed, also because Gelfand duality is actually equivalent to the Boolean prime ideal theorem, which is a bit weaker than AC. Nonetheless, the axiom of (countable) dependent choices (normally acronymed DC) is still non-constructive. A large part of analysis can indeed be done just with DC (Baire's theorem, for instance, is equivalent to it), but DeWitt's criticism still stands - he begs for an actual construction of the choice function, even if it is a countably dependent one. Thanks for the physics.SE link, I'll have a look. | |
| Jan 20, 2017 at 22:48 | comment | added | user13113 | Whenever constructibility is brought up, I always feel compelled to point out that the axiom of choice is the theorem of choice if you limit yourself to such. More precisely, $ZF+(V=L)\vdash C$ | |
| Jan 20, 2017 at 20:38 | comment | added | user21349 | @JamesWood: Physicists have been using infinitesimals since Newton. If you take any argument that uses infinitesimals, you can translate it into the language of either non-standard analysis or smooth infinitesimal analysis. NSA uses aristotelian logic, while SIA uses nonaristotelian logic. I think this is a good example of how utterly irrelevant such foundational issues are in physics. Physicists have a body of techniques that have been shown to be valid by comparison with experiment, not by showing their (self-)consistency within some foundational framework that is in style. | |
| Jan 20, 2017 at 20:27 | comment | added | user21349 | @MattF.: given what de Witt describes, would you count Hawking as a physicist who thinks the axiom of choice is relevant to physics? The fact that Hawking and Ellis used the axiom of choice doesn't tell us anything about their philosophical stance, if any. They probably used it simply as a convenience. Pedro Lauridsen Ribeiro's comment tells us that the proof holds without the axiom of choice, but it sounds like it's just a lot more work that way. | |
| Jan 20, 2017 at 20:20 | comment | added | Abdelmalek Abdesselam | @Pedro: Thanks for pointing out the new AHP article. However, I disagree about the spectral theorem (even for finite collections of commuting unbounded self-adjoint operators) needing the AC. As long as the Hilbert space is separable, it can be done with ACDC only. See this discussion on physics.SE physics.stackexchange.com/questions/43853/… | |
| Jan 20, 2017 at 19:20 | comment | added | user44143 | @JamesWood, I agree with you that de Witt's argument could go against $LEM$. Taking it further: given a theorem $T$, do physicists ever need to prove $T$? Could they do physics equally well using only $ZFC \vdash T$, as a theorem of $PRA$ which does not require $LEM$ or believing anything about sets? | |
| Jan 20, 2017 at 19:11 | comment | added | user44143 | @BenCrowell, given what de Witt describes, would you count Hawking as a physicist who thinks the axiom of choice is relevant to physics? | |
| Jan 20, 2017 at 19:00 | comment | added | mudri | What would such a critic make of the law of the excluded middle? I don't know physics, but “an explicit algorithm or construction” wouldn't use LEM in my book. | |
| Jan 20, 2017 at 17:53 | comment | added | user21349 | Some physicists want to use the axiom of choice, but some physicists don't. Based on Pedro Lauridsen Ribeiro's comment, I doubt that this is the case. I doubt that any physicists, think the axiom of choice is relevant to physics, but there may be cases where it's a nice convenience, and they may not know how to or want to bother with eliminating choice from their arguments. | |
| Jan 20, 2017 at 16:05 | comment | added | user44143 | @PedroLauridsenRibeiro Yes, many theorem-proving physicists like the spectral theorem; other physicists do not consider any theorems fundamental; de Witt would probably be in the middle. Saying that this theorem is fundamental to quantum mechanics is making a judgment about physics and the use of math in it, of the same sort as our judgments about math and the use of choice in it. | |
| Jan 20, 2017 at 14:54 | comment | added | Pedro Lauridsen Ribeiro | It's interesting to note, though, that by loosening the notion of "space", as @AndrejBauer puts it in his answer, one can indeed make the spectral theorem for commutative C*-algebras (a.k.a. Gelfand duality) constructive. More precisely, one replaces "compact Hausdorff topological space" by "compact, completely regular locale in a Grothendieck topos" in the statement of Gelfand duality - this was shown by B. Banaschewski and C.J. Mulvey in A Globalisation of the Gelfand Duality Theorem, Ann. Pure Appl. Logic 137 (2006) 62-103. | |
| Jan 20, 2017 at 14:46 | comment | added | Pedro Lauridsen Ribeiro | As for the lack of need for the axiom of choice in the "paraphernalia of theoretical physics", as DeWitt puts it, one should keep in mind that this statement may hold true for the definitions of the mathematical objects he lists, but not necessarily for the ensuing results. A typical example is the spectral theorem - a cornerstone of both Hilbert space and C*-algebra theories, and fundamental to quantum mechanics -, which relies on the Stone-Weierstrass theorem (for the continuous functional calculus) and also on the Riesz representation theorem (for the $L^\infty$ functional calculus). | |
| Jan 20, 2017 at 14:21 | comment | added | Pedro Lauridsen Ribeiro | It's worthwhile to mention that DeWitt's criticism to the result of Choquet-Bruhat and Geroch no longer applies. Jan Sbierski recently proved the existence of a maximal Cauchy development without using Zorn's lemma: On the Existence of a Maximal Cauchy Development for the Einstein Equations - a Dezornification, in Annales Henri Poincaré 17 (2016) 301-329. | |
| Jan 20, 2017 at 14:08 | history | answered | user44143 | CC BY-SA 3.0 |