Oct 21, 2024 06:24 PM
https://bigthink.com/hard-science/physics-new-math/
EXCERPT: . . . Why should physics—rooted in making sense of real things in the world like apples and electrons—provide such good leads for solving some of the toughest problems in mathematics, which deals with intangible stuff, like functions and equations?
“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.
The process of physics inspiring mathematics is, in fact, as old as science itself. The ancient Greek mathematician and inventor Archimedes described how the laws of mechanics had spurred some of his most important mathematical discoveries. Then there’s Isaac Newton, who (alongside his contemporary, the German polymath Gottfried Wilhelm Leibniz) famously developed an entirely new kind of math—calculus—while trying to understand the motion of falling objects.
But 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.
Yet a reconciliation was afoot, spearheaded by the late British-Lebanese geometer Michael Atiyah. With rare intuition, and a little luck, Atiyah, also a Fields medalist, often alighted on areas that would later be of interest to theoretical physicists. [...] “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....” (MORE - missing details)
EXCERPT: . . . Why should physics—rooted in making sense of real things in the world like apples and electrons—provide such good leads for solving some of the toughest problems in mathematics, which deals with intangible stuff, like functions and equations?
“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.
The process of physics inspiring mathematics is, in fact, as old as science itself. The ancient Greek mathematician and inventor Archimedes described how the laws of mechanics had spurred some of his most important mathematical discoveries. Then there’s Isaac Newton, who (alongside his contemporary, the German polymath Gottfried Wilhelm Leibniz) famously developed an entirely new kind of math—calculus—while trying to understand the motion of falling objects.
But 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.
Yet a reconciliation was afoot, spearheaded by the late British-Lebanese geometer Michael Atiyah. With rare intuition, and a little luck, Atiyah, also a Fields medalist, often alighted on areas that would later be of interest to theoretical physicists. [...] “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....” (MORE - missing details)