Timeline for answer to The most outrageous (or ridiculous) conjectures in mathematics by Mikhail Katz
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19 events
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| Mar 15, 2017 at 14:44 | comment | added | Mikhail Katz | Thanks, @Neal. The link in my answer is actually to the mathscinet review of the article. This mathscinet entry also provides a link to the article itself. Incidentally, I seem to recall other articles by Bishop where the idea of mathematics as being calculations on strings of integers is discussed more explicitly. Have you seen anything related? | |
| Mar 15, 2017 at 14:42 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 15, 2017 at 14:40 | comment | added | Neal | @MikhailKatz The link you edited into your answer is to commentary on Bishop's talk; I believe this is the article in question by Bishop. | |
| Mar 15, 2017 at 12:26 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 14, 2017 at 8:41 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 13, 2017 at 17:53 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 13, 2017 at 17:51 | comment | added | Mikhail Katz | Thanks, Gil. This is in Bishop's famous "Crisis in contemporary mathematics" text and other texts. I will add a reference to my answer. | |
| Mar 13, 2017 at 17:49 | comment | added | Gil Kalai | Dear Misha, this is quite interesting. When was this conjecture made and are there some relevant references/links? | |
| Mar 13, 2017 at 13:03 | comment | added | Todd Trimble | Actually, Mikhail, I agree with your last point: that the (somewhat tautologous) finitary nature of proofs was not what Bishop had in mind when he speaks of what is "meaningful" in mathematics. (At the moment of writing, I was being rushed out of the house by my daughter who needed to get to school on time.) I'm happy at this point to let Gil decide if this answer meets his criteria -- I thought he wanted mathematical conjectures, not philosophical claims. | |
| Mar 13, 2017 at 12:43 | comment | added | Mikhail Katz | The computer-based formalisations you mention would presumably include symbols for such cardinals but Bishop would likely consider both the cardinals in question and symbols that stand for them as meaningless. Therefore I still think that if one could show that a proof of FLT depends on large cardinals this would come close to a refutation of Bishop's claim. | |
| Mar 13, 2017 at 12:41 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 13, 2017 at 12:40 | comment | added | Todd Trimble | As for Fermat, it's an interesting question, but the likelihood of his having a proof could in principle be estimated if, one day, we get a better grip on the inherent complexity of any such proof. I don't think "likelihood" enters for your claim. | |
| Mar 13, 2017 at 12:38 | comment | added | Todd Trimble | No, a proof that adopts large cardinal hypotheses as axioms is still enacted in a script that in principle could be converted to 0's and 1's (and this is not unrealistic when one considers computer-based formalizations of even hard theorems). Actually, if you say Bishop formulated this as an article of faith, it's less clear to me that he thought it was falsifiable (and I for one don't think it is). | |
| Mar 13, 2017 at 12:32 | comment | added | Mikhail Katz | Is the question whether "Fermat had a marvelous proof for Fermat's last theorem" falsifiable? | |
| Mar 13, 2017 at 12:18 | comment | added | Mikhail Katz | @ToddTrimble, apparently Bishop thought it was (incidentally Kalai did not impose a specific falsifiability clause) otherwise he wouldn't have presented it as a meaningful assertion. I would conjecture that a proof of Fermat's last theorem that relies essentially on large cardinal hypotheses would constitute a falsification of Bishop's claim (of course MacLarty has argued that it doesn't essentially depend on such). Also if NSA is used to prove the Riemann hypothesis, many will probably interpret this as a refutation of Bishop. | |
| Mar 13, 2017 at 11:49 | comment | added | Todd Trimble | Is this falsifiable? What form would a falsification take? | |
| Mar 13, 2017 at 11:20 | vote | accept | Gil Kalai | ||
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| S Mar 13, 2017 at 10:19 | history | answered | Mikhail Katz | CC BY-SA 3.0 | |
| S Mar 13, 2017 at 10:19 | history | made wiki | Post Made Community Wiki by Mikhail Katz |