Aki Anandam

In mathematical rigor, it’s important to have a continuous thread of proof. You start with the axioms, and then every subsequent result can be proved, using a continuous chain of precise logical reasoning. We should strive for the same thing when it comes to building intuition for subjects such as those in physics.

I claim it’s not enough to just have a vague understanding why something should be true. There should be a clear, definitive progression from starting principles to the end result which keeps the intuition intact and directly demonstrates a result. For example, there is a beautiful equation relating the Riemann curvature tensor to the commutator between covariant derivatives. One often gets a rough intuition why this should be true: the Riemann curvature tensor is determined by parallel transporting a vector along two dimensions. Parallel transport is governed by the covariant derivative, so the commutator of covariant derivatives in two directions is a sensible result. However, it is not obvious why this specifically must be the result. Why can’t it be anything else, or something that slightly resembles it? Some may place standard mathematical rigor above intuition, relegating intuition to an afterthought for which it’s not necessary to have such a clear causal link between first principles and the final result. After all, if we have the rigorous proof, we already know it’s true so why do we need a second, less rigorous (but more intuitive) explanation?

But I disagree with this line of reasoning. The purpose of physics isn’t just to know whether something is true. Especially theoretical physics. We can do experiments to determine validity. I posit that the purpose of theoretical physics (especially modern theoretical physics in the era of superhuman ai) is to understand why something is true.

I see physics transforming in the coming decades. Several physicists I have spoken to such as Cumrun Vafa have likened the future of physics to chess, where computers surpass humans in terms of capabilities, but humans still study it for their own enjoyment. If this is the case, then it’s clear to me that intuition and deep understanding (instead of just knowing whether something is true) is going to become more important. You could put two chess engines against each other, and they’ll play a near perfect game (almost definetely ending in a draw) but that’s unlikely to be a chess game people are so interested in. People love studying Tal’s queen sacrifices or Capablanca’s positional squeezes. In appreciating chess games, people don’t just care whether a move was the “correct” move (in the chess engine sense). They care about why it works, why Fischer took the bishop instead of the rook in the 1963 game against Byrne. It’s not enough just to see that the engine evaluates the move higher. Humans want to understand, in their own intuitive way, why something is true, and I think this will become a more prominent feature of theoretical physics in the future.

This is also an area I still see humans having a place in. Just as different contemporary chess players have their own style, different leading physicsts may have their own flair for how to understand things (in fact, they already do, but this isn’t as emphasized - people presently focus on accuracy and originality of publication). Future ai may be far superior at divining new equations and ensuring mathematical rigor, but after receiving such equations, humans may look toward other humans whom they admire to interpret, adding their own style and perspective. All the modern chess players have opening lines from the books, but Carlsen and Nakamura will not play the Ruy Lopez the same, because they have fundamentally different approaches, which people admire separately.

As a result, when it comes to human work in physics, I see the possibility of intuitive exposition usurping mathematical rigor. To be clear, I am in no way suggesting that mathematical rigor is not important. If I had to choose one, I’d take mathematical rigor, as it is more reliable at ensuring the validity of a result (especially those which cannot be easily tested experimentally). My point is that ai will likely be so good at such rigor that it is no longer necessary to have a human checking results. In this future, I think humans will take on the role of interpretation, in which case it will become necessary to have a complete intuition. By complete, I mean one with no holes, in which there is an continuous progression from the starting principles to the end result. This is what I have always valued, and it is what I strive for in my articles.