Thread

1. ForgotMyUUID · 1789802300 · focus · HN ↗

   I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.

   I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.

   Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.

   And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.

   1. bananaflag · 1789810853 · focus · HN ↗

      As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.

      1. fidotron · 1789821275 · focus · HN ↗

         Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that, and rightly so, certainly not in the way Poincaré was on about.

         Maybe it's been done, but I'd like to see an LLM recreate Euclid from questioning without having seen it during training.

         1. bananaflag · 1789823453 · focus · HN ↗

            > Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that

            Yes, I believe that, it's part of what I was implying (I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs)

            1. fidotron · 1789824197 · focus · HN ↗

               Appreciate the clarification, even if I disagree!

               I think we differ on what "mathematical intuition" is then. I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned.

               The Euclid example also shows my bias towards spatial intuition of mathematical concepts (which is deeply unfashionable) but also exposes exactly where at least current LLMs break down; they do the symbol based pattern matching version, but they cannot leap outside of that, at least today.

               1. bananaflag · 1789825251 · focus · HN ↗

                  > I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned.

                  If you want to catch them, surely you can find proofs they aren't able to produce.

                  1. fidotron · 1789826846 · focus · HN ↗

                     That's easy: basically all the spatial ones.

                     I used to be a game dev, and one of the interview questions someone came up with consisted of working out the surface area of a variant of Menger sponge to some given level of depth. The bifurcation for people that could do this vs those that couldn't was incredible, and did not follow obvious trends for academic achievement. (The same interview also included the gem "How wide is a pointer?" which also catches a frightening number of people).

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