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My Logics professor said once to em that even though there are many proofs for a given theorem, there's generally only one that feels like the "natural" explan
by Fargren 13y ago
My Logics professor said once to em that even though there are many proofs for a given theorem, there's generally only one that feels like the "natural" explanation of why a thing is how it is, the one that really explains why. I think many mathematicians believe this is so, and it's good to reach these proofs and not just any proof when we are trying to understand something.
For example, let's look at Euclides proof of the pythagorean theorem. It's true that it show's that the theorem holds, and therefore it shows why it holds. But it just feels awfully convoluted and round-away. It's talking about triangles, but it's going through seemingly unrelated constructions to do so. The proof by similar triangles is, to me at least, much more intuitive, and after reading it I feel I understand and not just know why the theorem is true.
- lmm 13y agoThe four-colour theorem is a good example of a simple result which doesn't have a "natural" explanation - the only known proof is a computerized check of hundreds of cases. I remember Imre Leader saying some people think it's just an "accident of nature", that doesn't "mean" anything (insofar as any mathematics has meaning). Oddly enough that doesn't bother me; in fact, it seems natural that some parts of mathematics are just "like that" with no underlying reason. It makes the world of mathematics seem all the richer if not all things are simple and logical consequences of other things.
- ColinWright 13y agoExactly so. For the record, Imre is one of my PhD siblings, and it's perfectly reasonable that we came to this point of view as a by-product of working with our supervisor (and "grandparents", Erdős and Adams)
- scott_s 13y agoColin, on days like this, I'm very glad you have not given up on HN entirely.
- Fargren 13y agoFor problems like that, I can't help but wonder if some day we'll discover that we are just not searching for the solution in the proper "language". Maybe some day someone will find a problem isomorphic the the for-color problem in a different field, and that problem will have a simple and natural explanation. Probably in a field we don't know today.
- tikhonj 13y agoTo me, that feels more like a limit on human cognition than a fundamental quality of the proof or the program. After all, you can summarize--or, perhaps compress--the proof into a program which checks the cases. It's simply a level removed from how mathematicians normally work. On the other hand, if a proof could not be reduced at all--and, given my very limited understanding of information theory, this is possible--then I would certainly agree that it's inherently more complex. Put another way, I think that using a computer to check cases like this is morally similar to using induction. It still exposes and exploits a certain simplicity in the domain.
- lmm 13y agoThere was no induction; the proof was essentially: a) all planar graphs can be reduced to these several hundred cases (this part is ordinary mathematics) b) observe that it's possible to four-colour each of these several hundred cases (and a computer program was used to verify it). To my mind the proof is as complex as that set of cases; the fact that we can write a relatively simple program to check them all doesn't really reduce the complexity or tell us anything about why they're all 4-colourable, in the same way that using a computer to check the riemann hypothesis up to some limit tells us nothing about why it holds.
- tikhonj 13y agoI know it was not induction. My point is that a computer program to consider all the cases is just another mathematical tool morally similar to induction. The program does tell us something, namely that the problem can be reduced to a bunch of similar cases all of which are colorable. The difference is that the insight is perhaps a level removed from a normal proof. Essentially, I think that it's the program and not its output that plays the philosophical role of the proof in this case. Of course, my perception is significantly colored by the fact that I'm mainly interested in programming languages rather than traditional math :P.
- auvrw 13y agoi think that's a good notion as long as your prof. didn't demand that you feel the same way s/he did about everything. also, proofs of older facts using newer techniques can illuminate a new theory rather than the fact at hand. for instance, the topological proof that there are infinitely many primes is nice because it makes you think about what separability means.