Timeline for What is Realistic Mathematics?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Jan 27 at 11:50 | comment | added | Andrea Marino | I support this point of view. What I find nice of mathematics is that developing something beyond the (physically useful) boundaries actually yields something within the boundaries. See for example the Galois proof of non-existence for a général radical-based formula for the degree 5 equations. That needed complex numbers, which aren't that real. Some other times, as with the axiom of choice (see accepted answer), allowing for exotic objects actually make mathematics easier; the circle, for example, does not exist in nature, since it encompasses a transcendental number... | |
| Dec 4, 2017 at 11:27 | comment | added | Thomas Benjamin | (cont.) you will find information helpful to your program. | |
| Dec 4, 2017 at 11:26 | comment | added | Thomas Benjamin | (cont.) plus a compactness principle: every infinite subtree of the full binary tree has an infinite path ." The phrase, "$WKL_{0}$ is equivalent over $RCA_{0}$ to" the Hahn-Banach theorem for separable Banach spaces means that the formalisation of "every infinite subtree of the full binary tree has an infinite path" in the language of second-order arithmetic is equivalent to the separable Hahn-Banach theorem over the subsystem $RCA_{0}$. I would encourage you to read Simpson's article (my copy is the November 24, 2009 draft, but I presume it has been published somewhere) because in it | |
| Dec 4, 2017 at 11:06 | comment | added | Thomas Benjamin | @AndreasThom: Actually, Hahn-Banach for separable Banach spaces is equivalent over $RCA_{0}$ to $WKL_{0}$, where $RCA_{0}$ (Recursive Comprehension Axiom) is the susbsystem of second-order arithmetic which forms (in the words of Spephen Simpson from his preprint, "The Godel Hierarchy and Reverse Mathematics", which can be found under title on the Web) "a kind of formalized or computable mathematics...the $\omega$-models of $RCA_{0}$ are precisely the nonempty subsets of $P$($\mathbb N$) which are closed under Turing reducibility", and $WKL_{0}$ (Weak Konig's lemma) "consists of $RCA_{0}$ | |
| Nov 20, 2010 at 19:51 | vote | accept | Andreas Thom | ||
| Nov 20, 2010 at 19:51 | |||||
| Aug 8, 2010 at 17:43 | comment | added | Andreas Thom | @gowers: But I see what you mean. I think the Axiom of Choice or the use of ultrafilters have their merits since they allow in many situations for short proofs of statements which hold in the realistic model if and only if they hold in ZFC (by some model theoretic argument). Hence, it is useful but not realistic. You are right, on a secondary more sophisticated level, one has finds unrealistic assumptions useful since they lead not too far abroad if one includes a careful model theoretic analysis. | |
| Aug 8, 2010 at 17:36 | comment | added | Andreas Thom | @gowers: Well, if there would be a reasonable theory of numbers less than 10^100 and it would save some other trouble, then I would consider it. "Taking a grain of sand from a sandheap still gives a sandheap. Hence, there are infinitely many grains of sand in a sandleap." I want to say: Some aspects of the theory of natural numbers have their obvious limitations when applied to reality; they are producing counterintuitive statements. Being able to avoid those would be good but I do not see any reasonable way how this could be done. I think that the natural numbers are pretty realistic. | |
| Aug 8, 2010 at 16:40 | comment | added | Steve Huntsman | @gowers: I think Ed Nelson is suggesting the plausibility of something not entirely different than what you are saying. docs.google.com/… | |
| Aug 8, 2010 at 13:26 | comment | added | gowers | I agree with you that some parts of mathematics are more "realistic" than others. What I was questioning was your apparent use of the "is helpful for studying" relation to capture this intuition. My example may not be perfect, but I think that this relation will let in mathematics that is not realistic. Also, if you say that only separable Banach spaces are relevant, why don't you also say that only rational numbers with numerators and denominators less than 10^100 are relevant? Or do you? | |
| Aug 8, 2010 at 5:27 | comment | added | Andreas Thom | Maybe I would even go a bit further and say that the consequences of the Hahn-Banach theorem are not part of the percieved reality. Nobody has ever encountered an invariant mean on ${\mathbb Z}$! On the other side, nobody ever found it necessary to consider non-continuous linear operators from one Hilbert space (defined everywhere) to another. Doesn't it seem more realistic to strive for continuity of all such operators and not for the existence of the mean. Of course, it is difficult to come with a definition. | |
| Aug 8, 2010 at 5:12 | comment | added | Andreas Thom | For most purposes, only separable Banach spaces are relevant. Cannot Hahn-Banach can be proved with Countable Choice of Dependent Choice in this case? Of course, non-separable Banach spaces such as $\ell^{\infty} {\mathbb N}$ are relevant. For example, I see that for the existence of Banach limits or means on amenable groups (both are elements in the dual of $\ell^{\infty} {\mathbb N}$), Krein-Milman (which uses Tychonoff) is practical and seems right. However, the non-constructive nature of the mean and the unintuitive consequences give me the feeling that these objects are not realistic. | |
| Aug 8, 2010 at 4:19 | comment | added | Harry Gindi | +1 for the fact that you typed out that number. =)! | |
| Aug 7, 2010 at 21:33 | history | answered | gowers | CC BY-SA 2.5 |