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>Mathematicians are famous for embracing the fact that they have no intuition. That's just plainly false. Mathematics is all about intuition. That's how we al
by romwell 8y ago
>Mathematicians are famous for embracing the fact that they have no intuition.
That's just plainly false. Mathematics is all about intuition.
That's how we all know that Riemann hypothesis is true, regardless of whether there's a proof of it. Or that Fermat's Last Theorem holds.
That's how we still do math, even though we never really had a solid system of axioms for it -- and Godel showed that, in some sense, it's not even possible.
That's how even famous theorems were proven after many attempts, with people accepting proofs with errors in them.
That's how Calculus was invented before the notions of the limit, derivative, and integral -- the most fundamental ones! -- were solidly written down. Newton and Leibniz built up the math upon heresy, and everyone took it, because it felt right.
What you are writing about is surprise that mathematics still gives to practicing mathematicians.
And your first quote is a trick on how to extend one's intuition, not abandon it!
- retsibsi 8y ago> And your first quote is a trick on how to extend one's intuition, not abandon it! Can you elaborate on this? Do you mean that the point is 'to free up your intuition, abandon your hope of truly visualising this stuff, and make do with a hybrid of simplified visualisation and abstract thought'?
- romwell 8y agoIn a way, yes. But also: it is surprising how well building up the intuition in 3 dimensions works for N dimensions. You can build up a very working, visual intuition for Linear Algebra just by doing everything in 3D, for example. There's a textbook that takes this approach: "Practical Linear Algebra". But, of course, this comes with a caveat: it works until it doesn't. The volume of a unit sphere increases for dimensions 1-5, then starts to go down[1]. There exists an exotic R^4: a manifold that is homeomorphic, but not diffeomorphic to R^4[2]. And so on. But for a lot of (intro) differential geometry and topology, (e.g. concepts like covering spaces), I haven't found many examples where intuition based on 3 dimensions breaks. A friend of mine[3] once told me that geometry is the art of getting correct results from incorrect diagrams. I think that the quote we're discussing is in a similar spirit. [1]https://en.wikipedia.org/wiki/Volume_of_an_n-ball https://en.wikipedia.org/wiki/Volume_of_an_n-ball [2]https://en.wikipedia.org/wiki/Exotic_R4 https://en.wikipedia.org/wiki/Exotic_R4 [3]Dmytro Karabash
- nimonian 8y agoWe know that FLT is true because Andrew Wiles has proven it. We don't know that RH is true. We have strong reasons to believe it is true since we have discovered directly a large number of roots on Re(z) = 1/2 and none anywhere else. I do believe intuition exists in the process of doing mathematics, but I don't think either of these are good examples.
- trukterious 8y agoHow about in the process of reading a proof? Each single step is validated by the intuition of the reader.
- hodgesrm 8y agoI have to disagree. The lesson I have drawn from studying mathematics is that intuition cannot be trusted and that even obvious things must be proven. Or you must admit they cannot and convert those things to axioms. Russell's Paradox [0] is a good example of intuition going wrong in the field of set theory. As I recall Zermelo-Frankel (ZFC) set theory eliminated the known paradoxes of naive set theory but it took decades to complete. There are formally undecidable propositions in ZFC but that's different (to me at least) from incorporating outright contradictions.