The cultural divide between mathematics and AI

So Godel proved that there are true theorems that are unprovable. My hunch is that there is a fine grained version of this result <-- that there is a some distribution on the length of the proof for any given conjecture. If true that would mean that we better get used to dealing with long nasty proofs because they are a necessary part of mathematics...perhaps even, in some kind of Kolmogorov complexity-esque fashion, the almost-always bulk of it

  > the primary aim isn't really to find out whether a result is true but why it's true.

I'm honestly surprised that there are mathematicians that think differently (my background[0]). There are so many famous mathematicians stating this through the years. Some more subtle like Poincare stating that math is not the study of numbers but the relationship between them, while others far more explicit. This sounds more like what I hear from the common public who think mathematics is discovered and not invented (how does anyone think anything different after taking Abstract Algebra?).

But being over in the AI/ML world now, this is my NUMBER ONE gripe. Very few are trying to understand why things are working. I'd argue that the biggest reason machines are black boxes are because no one is bothering to look inside of them. You can't solve things like hallucinations and errors without understanding these machines (and there's a lot we already do understand). There's a strong pushback against mathematics and I really don't understand why. It has so many tools that can help us move forward, but yes, it takes a lot of work. It's bad enough I know people who have gotten PhDs from top CS schools (top 3!) and don't understand things like probability distributions.

Unfortunately doing great things takes great work and great effort. I really do want to see the birth of AI, I wouldn't be doing this if I didn't, but I think it'd be naive to believe that this grand challenge can entirely be solved by one field and something so simple as throwing more compute (data, hardware, parameters, or however you want to reframe the Bitter Lesson this year).

Maybe I'm biased because I come from physics where we only care about causal relationships. The "_why_" is the damn Chimichanga. And I should mention, we're very comfortable in physics working with non-deterministic systems and that doesn't mean you can't form causal relationships. That's what the last hundred and some odd years have been all about.[1]

[0] Undergrad in physics, moved to work as engineer, then went to grad school to do CS because I was interested in AI and specifically in the mathematics of it. Boy did I become disappointment years later...

[1] I think there is a bias in CS. I notice there is a lot of test driven development, despite that being well known to be full of pitfalls. You unfortunately can't test your way into a proof. Any mathematician or physicist can tell you. Just because your thing does well on some tests doesn't mean there is proof of anything. Evidence, yes, but that's far from proof. Don't make the mistake Dyson did: https://www.youtube.com/watch?v=hV41QEKiMlM

I'd not heard that Feynman quote before, so thanks for sharing; I love it.

I'd include writing, art-, and music-making in that category.

I think you're missing the core component. We care __WHY__ the theorem is true. To be honest, the __IF__ part matters a lot less.

The thing is that the underlying reasoning (the logic) is what provides real insights. This is how we recognize other problems that are similar or even identical. The steps in between are just as important, and often more important.

I'll give an example from physics. (If you're unsatisfied with this one, pick another physics fact and I'll do my best. _Any_ will do.) Here's a "fact"[0]: The atoms with even number of electrons are more stable than those with an odd number. We knew this in the 1910's, and this is a fact that directly led to the Pauli Exclusion Principle, which led us to better understand chemical bonds. Asking why Pauli Exclusion happens furthers our understanding and leading us to a better understanding of the atomic model. It goes on and on like this.

It has always been about the why. The why is what leads us to new information. The why is what leads to generalization. The why is what leads to causality and predictive models. THe why is what makes the fact useful in the first place.

[0] Quotes are because truth is very very hard to derive. https://hermiene.net/essays-trans/relativity_of_wrong.html

I do think the why that the Four Colour Theorem is true is captured my statement. The reason why it is true is because there exists some finite unavoidable and reducible set of configurations.

I'm fairly sure that people are only getting hung up on the size of this finite set, for no good reason. I suspect that if the size of this finite set were 2, instead of 633, and you could draw these unavoidable configuration on the chalk board, and easily illustrate that both of them are reducible, everyone would be saying "ah yes, the four colour theorem, such an elegant proof!"

Yet, whether the finite set were of size 2 or size 633, the fundamental insight would be identical: there exists some finite unavoidable and reducible set of configurations.