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Comment by jackpirate | Hacker News Reader

Try getting a math professor to admit that he doesn't actually have an intuitive understanding for any of the stuff he teaches. I believe the vast majority probably don't have any such understanding but they are understandably scared to admit it.

Mathematicians are famous for embracing the fact that they have no intuition. For example, here's a famous quote from Geoff Hinton:

To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it.

And here's a famous quote from G.H.Hardy:

In mathematics, you never understand things; you just get used to them.

The idea behind both of these quips is that there is no intuitive way to visualize these weird mathematical objects. The main reason people struggle with math (in my experience) is that they assume there must be an intuition. There's not. Math is weird. And all mathematicians feel that way about math.

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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 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!

> 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'?

In 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

[2]https://en.wikipedia.org/wiki/Exotic_R4

[3]Dmytro Karabash

We 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.

How about in the process of reading a proof? Each single step is validated by the intuition of the reader.

I 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.

> To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it.

another version that I heard is "Start with visualizing a N-dimensional space, then let N be 14"

> In mathematics, you never understand things; you just get used to them.

That was von Neumann.

He was a notorious formalist, and a damn good one; but it's hardly the only style of mathematics. Compare with Groethendieck who also invested a whole new field of mathematics, but had a completely different approach: saturate oneself in the problem until one understands it, then write the solution down.

http://www.landsburg.com/grothendieck/mclarty1.pdf

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