Maths and Physics #4: The Long Divorce

At the end of World War II came the “long divorce” of physics and maths. Physicists only worked with well-established maths; and mathematicians had no interest in physics. Neither side looked to advances in the other field for new ideas or seeds that might be relevant to their own fields. Why had this happened?

 

Part of the reason was that mathematicians feared that their field was becoming a “ragtag of unconnected ideas and results”. Kurt Godel’s theorems had struck a dagger at the very heart of maths – maths seemed to be in tatters. Best for mathematicians to decide how their field could proceed, they felt. Another reason was that physicists found they were able to make progress with existing maths. And lastly, applied physics was in vogue, esp. solid state physics. The engineering mindset – approximations were acceptable as long as they worked – was becoming the norm. This, of course, made theoreticians in both physics and maths wary, uncomfortable and even contemptuous of such physics.

 

Even in pure theory, methods like Feynman’s “mystified his (physics) colleagues as much as it horrified mathematicians” – the way it handled infinities seemed absurd, but it worked. Since it was easy to use and yielded perfect answers, it spread quickly. As Freeman Dyson said:

“Mathematical rigour was the last thing that Feynman was ever concerned about.”

Mathematicians were (not surprisingly) not OK with brushing infinites under the carpet without proper reasons or rigour.

 

Then, the two fields began to un-divorce. An instance here, another there.

 

Yang and Mills worked out a purely mathematical field theory inspired by the symmetries in Maxwell’s equations – it came to be called the Yang-Mills gauge symmetry. But some of its predictions on the behavior of massless particles made no sense, at least not when it was proposed. Physicists were understandably dismissive of such a theory. It would take over 2 decades for experiments to show the theory was right after all, but until then, this seemed like yet another all-maths-no-physics “theory”.

 

Roger Penrose made numerous inventive contributions to both physics and maths. Like twistors. The concept was too complex for most physicists, and it didn’t serve any particular need either. It was therefore ignored. But Penrose persevered anyway – his enthusiasm and personality helped. Not only twistors, he argued, there were other areas where physics and maths needed to learn from each other.

 

Freeman Dyson advocated greater interactions between physicists and mathematicians. Group theory from maths might be relevant to physics, he said. As with Penrose, there weren’t too many takers, but eventually Dyson was proved right on the use of group theory in physics.

 

The Weinberg-Salam theory predicted the existence of three exceptional particles that mediate the electroweak force. Not only had the particles not been found (when the theory was proposed), the maths was plagued with infinities. Gerard ‘t Hooft found a way to systematically remove the infinities and with that, physicists were willing to “hunt” for the particles predicted by theory. And soon those particles began to be detected.

 

Identical ideas began to crop up in theoretical physics and pure maths. Was there a “pre-established harmony” between the two fields after all?

 

The long divorce was finally coming to an end.