Physicists like me often think that mathematicians are unduly obsessed with rigor. We’re used to hearing that things that we use all the time (like the path integral in QFT) are not well-founded mathematically. Then we think that the mathematicians must be really dense for not accepting the path integral, when it’s obviously right (we have the experiments to prove it!). What’s the point in searching for rigor if you already know something is right?
But from what I’ve been reading recently, it seems like I’ve been too hard on the mathematicians. They aren’t interested in rigor for the sake of rigor. A couple of days ago I read Michael Harris, who says “that the basic unit of mathematics is the concept rather than the theorem, that the purpose of a proof is to illuminate a concept rather than merely confirm a theorem, and that the replacement of deductive proofs by probabilistic or mechanical proofs should be compared, not to the introduction of a new technology for producing shoes, say, but rather to the attempt to replace shoes by the sales receipts.” Less interestingly, here’s Terrence Tao talking about rigor and intuition in math. And Peter Woit is always arguing that the way to make progress in physics is by a better understanding of the mathematical structures behind the Standard Model and other physical theories. I don’t totally understand this argument (how will a deeper understanding give us new predictions?), but it’s very interesting.