I think the key, in part, is to resist that mathematicians’ tendency to abstract away individual problems into general solutions or categories of solutions or entire subfields, and spend some time with the specific problems that mathematicians are or have been interested in. But it also helps a lot if, in that specific problem, you get that mathematical move of discarding whatever doesn’t matter to the structure of the problem. After all, that’s a big part of what you’re trying to teach: how to think like a mathematician. You just to have to unlearn what a mathematician already assumes first.
Physicists have long looked to higher math for insights into the workings of the universe. “If a figure is so beautiful and intricate and clear, you figure it must not exist for itself alone,” John Baez, a professor of mathematics at the University of California at Riverside, said. “It must correspond to something in the physical world.” This instinct—the assumption that beauty will stand in for truth—has become a habit. Some physicists now worry that string theory’s mathematics have grown permanently unmoored from the real world—an exercise in its own complexity. And so modern theoretical physics has become, in part, an argument about aesthetics.