Project Tuva hopes to introduce physics to the masses through entertaining lectures à la Feynman. Thrilled with the first lecture, I encouraged my busy girlfriend to take time over the weekend to watch the second lecture with me. Judging purely by her reaction to “The Relation of Mathematics and Physics,” Bill Gate’s efforts are not efficacious. But it’s too soon to write him a reimbursement check, as she falls asleep anytime we curl up to watch something on the couch. I, on the other hand, was completely engaged with this lecture, as was the original crowd it seemed. My favorite joke was in there, probably in its original form. These days it’s told as, A physicist seeks out a brilliant mathematician and asks, ” how do you visualize four-dimensional space?” The mathematician responds, “Well, first I visualize n-dimensions, and then I set n equal to four.” If I tell it just right, that can actually get a laugh out of my girlfriend. Since she is a math PhD student, by the way, I thought it appropriate to assign gender when contrasting mathematicians and physicists in what follows.
Feynman stresses that the physicist needs to assign physical meaning to the equations, whereas the mathematician can just keep hammering away with a rigorous approach to everything. He in essence says that the two attack a given problem in completely different ways, albeit work with the same mathematical machinery. When researching something new, the physicist need not bother with organizing his thoughts in a coherent framework that can then methodically be applied to gradually get to the heart of the problem. Rather, he let’s his intuition do that, and relies on conservation laws and simple models to try to get a grip on the problem, letting it simmer in his head for weeks or months on end, until finally one day, insight gleams forth under some relaxing state of mind, and everything falls together. The mathematician, meanwhile, works hard to put everything in place before she ever begins. She likes her problems well-posed, for then she can reach into her bag organized case of tools and bring her training to bear on the problem in a straightforward manner, deriving the result, establishing convergence, proving uniqueness, etc. That process certainly requires ingenuity and sometimes strokes of genius, but it follows one after the other in a laid out procedure.
The physicist follows a different line of attack. His mind is quite cluttered. I would compare him to a curious craftsman trying to open up a black box, unaware of what tool he might need next. He doesn’t have the luxury of an organized workspace; he must keep all of his tools at his fingertips. Sure, he possesses a deep understanding of mathematics. Otherwise, it’s unlikely he could ever forge connections among similar concepts in physics, let alone among the vast fields of physics, that are crucial to making significant progress. But he doesn’t concern himself with the likes of real analysis past a certain stage of utility.
I sort of like my black box analogy because it captures, if only in a highly simplified and over-generalized way, the interplay of these two fields. The boxes are the problems from physics, and the tools to open them are mathematics. History has shown that after opening one black box, the physicist is presented with another inside, as if he had just opened a Russian nested doll. Excited, he gets to work on trying to open the next black box, while the mathematician often occupies herself with polishing up the one he just opened. Or she might be developing a tool which he needs to open the next box, yet neither has any clue that such is the case at that point. Occasionally the tools to open the box have been around for many years but the physicist has not learned to use them. And certainly many a mathematician take a stab at opening the boxes themselves, more often than not only to pry it open part way until one day a physicist comes along and shakes it up and down, pours out all of its contents, and leaves it swinging from its hinges.
It used to be that mathematicians took all of their problems from physicists for the most part. However, now that math is infiltrating other fields at an overwhelming pace, mathematicians are beginning to learn less and less about physics. Yet the reverse is not true, while the concepts of physics are also finding numerous applications in other areas. It remains to be seen what will come of all of this, but I can say that being in a relationship with a mathematician has only heightened my respect for them overall, as the skill set they develop truly is unique.
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