Long read: Why Physics Is Unreasonably Good at Creating New Math

When we were young, we were told Math is fundamental to learn everything else. And as we grew up we began to appreciate it – why without we couldn’t appreciate Physics, without which we couldn’t comprehend Chemistry without which we couldn’t explain Biology, to Psychology leading to social sciences such as economics all the way upto philosophy. However, this essay by Ananyo Bhattacharya, the chief science writer at the London Institute for Mathematical Sciences and author of The Man from the Future, a biography about mathematician John von Newmann, argues otherwise. In particular, how math is prospering from physics or observations in the real world. As the essay elaborates, Newton’s development of calculus was thanks to the apple falling and not the other way around. The author argues that this trend is accelerating now.

“Now insights and intuitions from physics are unexpectedly leading to breakthroughs in mathematics. After going their own way for much of the 20^thcentury, mathematicians are increasingly turning to the laws and patterns of the natural world for inspiration. Fields stuck for decades are being unstuck. And even philosophers have started to delve into the mystery of why physics is proving “unreasonably effective” in mathematics, as one has boldly declared. That question hinges on a largely unappreciated, perplexing, and profound link between the rules that govern the behavior of the cosmos and the most abstract musings of the human mind.

Why is this happening?

““Physicists are much less concerned than mathematicians about rigorous proofs,” says Timothy Gowers, a mathematician at the Collège de France and a Fields Medal winner. Sometimes, he says, that “allows physicists to explore mathematical terrain more quickly than mathematicians.” If mathematicians tend to survey—in great depth—small parcels of this landscape, physicists are more likely to skim rapidly over vast tracts of this largely uncharted territory. With this perspective, physicists can happen across new, powerful mathematical concepts and associations, to which mathematicians can return, to try and justify (or disprove) them.”

But it wasn’t always like this as seen with the Newton calculus experience. What changed?

“…in the middle of the 20th century, the flow of new math from physics all but dried up. Neither physicists nor mathematicians were much interested in what was happening on the other side of the fence. In mathematics, an influential set of young French mathematicians called the Bourbaki group sought to make mathematics as precise as possible, rebuilding whole fields from scratch and publishing their collaborative work in an effort—they hoped—to facilitate future discoveries. Physicists, meanwhile, were excitedly developing path-breaking ideas such as the Standard Model—still physicists’ best theory of the atomic and subatomic world. For many of them, math was just a handy tool, and they had no interest in the austere vision of mathematics championed by the Bourbakis.”

The current reversion is attributed to Michael Atiyah, a Fields medallist:

““In the mid 1970s, he became convinced that theoretical physics was by far the most promising source of new ideas,” Nigel Hitchin, a mathematician and emeritus professor at the University of Oxford who collaborated with Atiyah wrote  in 2020 of his former colleague. “From that point on, he became a facilitator of interactions between mathematicians and physicists, attacking mathematical challenges posed by physicists, using physical ideas to prove pure mathematical results, and feeding the physicist community with the parts of modern mathematics he regarded as important but were unfamiliar to them.””

Why should physics feed mathematics?

“There are an infinite number of patterns and structures that mathematicians could study, he says. “But the ones which come from reality are ones which we have an intuition about at some level.”

Hitchin agrees. “Mathematical research doesn’t operate in a vacuum,” he says. “You don’t sit down and invent a new theory for its own sake. You need to believe that there is something there to be investigated. New ideas have to condense around some notion of reality, or someone’s notion, maybe.””

Atiyah went on to explain this more scientifically through how the human brain evolved over time:

“…human brains evolved to solve physical problems and this required the brain to develop the right kind of mathematics.” To do so, the brain must also have adapted to recognizing and appreciating mathematical patterns in nature. Atiyah even co-authored a brain-imaging study in 2014 that concluded the experience of mathematical beauty excites the same parts of the brain as beautiful music, art, or poetry. That might explain why physics can be a lodestar for mathematicians: The sort of math that emerges from studying reality is the sort our brains tend to like.”

The author fascinatingly goes on to cite several philosophers’ explanation of this phenomenon. A thoroughly thought provoking read.

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