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. jltsiren · 1789859057 · focus · HN ↗

            I think it's actually the opposite. Intuition is the only thing LLMs can do, which is why they are prone to hallucinating when they can't validate their intuition against reality.

            There is an idea that human intuition, expertise, and critical thinking are largely pattern recognition. When you encounter a situation, your brain gives you a plausible starting point, based on what it has experienced before. You then continue with explicit reasoning, which is slow and inefficient, and try to validate your ideas. The more relevant the patterns you have learned are to the situation, the more likely you reach a useful conclusion.

            LLMs are largely the same, except that they cannot learn from experience in normal usage. And except that they experience the world only through symbolic data, while the human brain has access to plenty of sensory data.

            1. pegasus · 1789996396 · focus · HN ↗

               Kind of. First of all, reasoning LLMs can also do (a form of) reasoning. Second, yes you could say LLMs form a sort of intuition, but its domain is the space of human-produced text. It only translates to real world intuition to the degree that those intuitions make their way into the corpus the model has been trained on. You could say that, when it comes to anything other than textual prediction, their intuition is secondary, a reflection of a reflection, so it will always lag behind that of humans.

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