tag:blogger.com,1999:blog-8148573551417578681.post1547943939986629184..comments2020-11-30T07:07:08.701-08:00Comments on Dark Buzz: Mathematicians care about proofsRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-8148573551417578681.post-15116654585732224592012-10-29T10:17:11.236-07:002012-10-29T10:17:11.236-07:00The argument on that page about Fermat's Last ...The argument on that page about Fermat's Last Theorem is wrong. The name of the mathematician is "Wiles", not "Wile".Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-75919068174866540012012-10-28T20:48:55.896-07:002012-10-28T20:48:55.896-07:00Wile's Proof of Fermat's Last Theorem htt...Wile's Proof of Fermat's Last Theorem http://www.coolissues.com/mathematics/Wile'sproofofFLT.html<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-26983188910948645512012-09-19T18:18:45.502-07:002012-09-19T18:18:45.502-07:00The abc conjecture is a statement about the intege...The abc conjecture is a statement about the integers. What would convince mathematicians that it is true? I say only a ZFC proof. It would be interesting if someone proved abc as a consequence of some large cardinal axiom, but that would not be accepted as a proof of abc unless it could be formalized in ZFC.<br /><br />If I am wrong here, then there should be some example of a non-ZFC proof being accepted. I do not know of any such example.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-57157644912693628142012-09-19T17:57:24.961-07:002012-09-19T17:57:24.961-07:00I don't understand what you're asking for ...I don't understand what you're asking for an example of. Any result on large cardinals would be accepted as "true" by mathematicians. Such a result wouldn't be "ZFC-true" but it would still be true in the sense that it was a correct consequence of some clearly stated axioms. <br /><br />It seems like you want an example of a question about ZFC which was answered using non-ZFC assumptions. Obviously such an example doesn't exist. If we interpret the abc conjecture as a question about the validity of a certain statement in the ZFC framework, then a proof cannot use axioms that are outside of ZFC. The point I've been making is that there's much more to mathematics than just ZFC.Bob Jonesnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-7492116214005734842012-09-19T16:37:40.005-07:002012-09-19T16:37:40.005-07:00The question here is whether a non-ZFC argument wo...The question here is whether a non-ZFC argument would ever cause the abc conjecture to be accepted as being true. I say no, and I say that nothing like that has ever happened in the history of mathematics.<br /><br />The statements of the <a href="http://en.wikipedia.org/wiki/Millennium_Prize_Problems" rel="nofollow">Millennium Prize Problems</a> say nothing about the proofs being in ZFC, but it is understood. A non-ZFC proof would not be accepted.<br /><br />If I am wrong, then give me some example. The closest example I know is <a href="http://en.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem" rel="nofollow">Wiles's proof of Fermat's Last Theorem</a>, which was accepted in spite of its apparent dependence on Grothendieck universes. But the experts were satisfied that the proof could be formalized in ZFC. No one said that we need to accept a non-ZFC proof because truth required changing the rules of the game.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-76592826336290337802012-09-19T15:11:34.566-07:002012-09-19T15:11:34.566-07:00"but then your theorems will only be publisha..."but then your theorems will only be publishable of the form 'if CH then XYZ.'"<br /><br />The point is that all mathematical results are of this form. When you publish a result, you're always explaining the consequences of some set of axioms, either ZFC or something else. Since mathematical results are contingent on the assumptions they make, they are always of the form "if A then B". They never assert the truth of a proposition in some absolute, objective sense, independently of any axioms.<br /><br />"If it turns out that Mochizuki has to assume axioms outside ZFC, then he will not be considered to have proved the abc conjecture."<br /><br />If you define the abc conjecture to be a statement about the consequences of the ZFC axioms, then no, he would not have proved the abc conjecture. But even if he's using new foundations which are not equivalent to ZFC, his work still shows that this alternative foundation has interesting and highly nontrivial consequences. That's why mathematicians are taking him seriously.<br /><br />"If I am wrong, then give me an example of some theorem outside ZFC that mathematicians accept."<br /><br />If you're asking for a theorem outside of ZFC which is true in the sense that it follows from the axioms of ZFC, then obviously that's an impossible task. But I've already given you many examples of topics which are considered legitimate parts of mathematics even though they are not based on ZFC...Bob Jonesnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-59933780662333811532012-09-19T14:38:52.147-07:002012-09-19T14:38:52.147-07:00The point of restricting to ZFC is to get a publis...The point of restricting to ZFC is to get a publishable proof that is acceptable to mathematicians. Sure you can assume the Continuum Hypothesis or the Riemann Hypothesis, but then your theorems will only be publishable of the form "if CH then XYZ." If it turns out that Mochizuki has to assume axioms outside ZFC, then he will not be considered to have proved the abc conjecture.<br /><br />If I am wrong, then give me an example of some theorem outside ZFC that mathematicians accept. Sure category theory and Grothendieck universes are formally outside ZFC, but they are being used to prove things that can be proved in ZFC. Logicians experiment with large cardinals, but they do not change the rules of the game. They might say, "It is a theorem of ZFC that if there is an inaccessible cardinal them ..." Give me some example of mathematicians changing the rules of the game to expand what is true or provable.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-36454571486472206982012-09-19T13:40:37.157-07:002012-09-19T13:40:37.157-07:00"No mathematician would consider rejecting Mo..."No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth."<br /><br />There are parts of this statement that I accept and other parts that I reject. I agree that mathematicians are interested in truth, but I don't think that mathematics has an "objective" or "absolute" notion of truth. In mathematics, truth is always relative to the assumptions you start with. If you want to do mathematics in ZFC, then that's great. There's lots of wonderful wonderful mathematics in ZFC. If instead you want to do mathematics in ZF and assume the negation of the axiom of choice, that's another perfectly interesting and worthwhile way of doing mathematics. Similarly, you can assume the continuum hypothesis is true, or you can assume it's false. There is no "absolute" sense in which a statement like the continuum hypothesis is true or false.<br /><br />"Nobody changes the rules of the game."<br /><br />This statement is completely false and does not reflect what mathematicians actually do. Mathematicians have studied alternative foundations for mathematics based on category theory and Grothendieck universes, for example. Modern set theorists experiment with large cardinals all the time. The fact that Mochizuki is working with alternative foundations for mathematics is further evidence that mathematicians can and do change the rules of the game.<br /><br />"So in fact Mochizuki cares very much about proving his theorems in ZFC, even if he likes to use tools that go outside ZFC."<br /><br />In some parts of algebraic geometry, you want to formalize certain intuitions about space, and going outside of ZFC turns out to be quite convenient. Going outside of ZFC can also produce many rich and interesting results with applications. So what's the point of restricting ourselves to ZFC? I don't see anything wrong with exploring alternative foundations as long as we remember which foundation of mathematics we're using.Bob Jonesnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-23736263841504141362012-09-14T10:46:49.016-07:002012-09-14T10:46:49.016-07:00Yes, we accept expert opinion that FLT has been pr...Yes, we accept expert opinion that FLT has been proved. But if the experts said, "we could not prove FLT but we decided that it was true so we changed the rules of the game", then we would not accept FLT.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-22968336981116094212012-09-14T10:40:08.129-07:002012-09-14T10:40:08.129-07:00Mathematicians definitely care about truth and abo...Mathematicians definitely care about truth and about proof, but most of all they care about whether the mathematics community accepts a proof to be true.<br /><br />For instance, FLT is considered by the mathematics community to have been proven by Wiles, but most mathematicians haven't even read his proof nor are qualified to evaluate it.Craignoreply@blogger.com