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. watwut · 1789821641 · focus · HN ↗

            It does not imply that. He is talking about how people do math. Intuition is what you use when deciding what to try and how to think about things.

            Proof is the rigorous outcome.

            LLM running probabilistic loop is different kind of process.

            1. fidotron · 1789821770 · focus · HN ↗

               The parent comment literally said "You cannot do proof without intuition".

               Therefore, according to that logic, an entity producing proofs must have intuition.

               Edit to add: the parent commenter has now confirmed my interpretation of their statement.

               1. bunderbunder · 1789823092 · focus · HN ↗

                  Your unstated major premise here is that their intent was to make a universal statement about how proofs work and not just talking to humans about how they teach humans.

                  That premise seems unlikely to be correct.

                  1. fidotron · 1789823283 · focus · HN ↗

                     Why? The entire subject of conversation is triggered by things which are not humans producing proofs.

                     If it's possible for a machine to produce a proof without intuition then clearly a human could also do it too. (And in fact I'd argue I've seen many people like that, simply very good at pattern matching over memorised items).

                     1. bunderbunder · 1789826341 · focus · HN ↗

                        Because regardless of the point TFA is making, that interpretation makes less sense for the specific comment. It doesn’t fit with the immediate context, which was a response to a thoughtful comment about how humans do math. And it requires assuming a math professor doesn’t understand a very basic and obvious thing about their area of expertise.

                        That doesn’t really read as good faith engagement in the discussion. At best, it reads as being so AI pilled that you can’t even fathom that others might want to have a little side discussion about something other than AI.

                        1. fidotron · 1789827389 · focus · HN ↗

                           You realize the commenter has now confirmed my interpretation was right?

                           What is up with this whole sub thread of obvious hole digging?

                           1. [deleted] · 1789996646 · focus · HN ↗

                              [deleted]

                     2. watwut · 1789851003 · focus · HN ↗

                        That math prof was talking about his stance in discussion between mathematicians long before AI.

                        Plus, I studied math, I am from that environment. His description matches how math is done by people.

                        People who are good at pattern matching and memorize are, frankly, shit mathematicians. They are find in fun culture around math, but rarely in actual math. They cant really do it as science.

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