Timeline for How does mathematics work?
Current License: CC BY-SA 4.0
9 events
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| Jul 22, 2019 at 17:06 | comment | added | Frank Hubeny | @James_pic I don't know either if FLT can be shown or not using these other logics since I am not familiar with the proof. I agree with you that there are alternatives which means that we don't have to use classical logic for proofs. When we don't the justification for restricting ourselves would be some suspicion that these mathematical statements do not represent reality but are dependent on a proof existing in some acceptable logic. | |
| Jul 22, 2019 at 16:51 | comment | added | James_pic | Not all mathematicians accept the principle of the excluded middle. The constructivist school doesn't, and have managed to produce proofs of many results nonetheless. I can't remember off the top of my head whether FLT is among the results that can be proved constructively, but plenty of other "non-existence in an infinte set" results can. | |
| Jul 22, 2019 at 15:12 | comment | added | Frank Hubeny | @Hurkyl To avoid these details, perhaps it would be clearer to note that the reason mathematics works is because we use classical logic to prove mathematical statements (rather than some other logic). Why do we do that? Because we believe that mathematical statements (unlike statements about the future) follow the principle of bivalence. They are either true or false. That belief in bivalence for these statements is a realist position toward mathematical statements. So, at bottom, what makes mathematics work is a belief in realism regarding mathematical statements. | |
| Jul 22, 2019 at 15:03 | comment | added | Frank Hubeny | @JohnColeman If realism about mathematical statements is true, that is, the belief that mathematical statements are either true or false is true, then it would be a realist position that recognizes that. Taking a different stance would be potentially anti-realist. Noting that it depends on this belief merely notes that we have a choice regarding these statements that we don't usually consider. Considering that choice is neither realist nor anti-realist. If mathematicians stopped being realists they may assess FLT differently. | |
| Jul 22, 2019 at 11:39 | comment | added | John Coleman | "What makes mathematical statements about infinite domains work is a belief in realism" -- does this mean that if mathematicians stopped being realists then Fermat's Last Theorem would stop working? This seems to suggest that the truth of mathematical statements depends on the subjective state of mathematicians -- which is a profoundly antirealist position. | |
| Jul 22, 2019 at 4:06 | comment | added | user6559 | Aside: excluded middle does not imply two-valued logic; all Boolean algebras obey the law of the excluded middle. Also, one can adopt the law of noncontradiction without insisting on the excluded middle: e.g. intuitionistic logic in the form of Heyting algebra. | |
| Jul 21, 2019 at 21:09 | comment | added | Frank Hubeny | @Rusi He was an anti-realist. I don't want to suggest that he wasn't. His presentation of realism as characterized by the principle of bivalence for mathematical statements may be considered an anti-realist perspective of realism. | |
| Jul 21, 2019 at 20:29 | comment | added | Rushi | You should add that Dummett was an anti-realist!!! ie there's more nontriviality (and subtlety!) in what you are saying. | |
| Jul 21, 2019 at 13:28 | history | answered | Frank Hubeny | CC BY-SA 4.0 |