## Friday, October 2, 2009

### The surprising effectiveness of non-rigorous mathematics in physics

It is well-known how surprisingly effective mathematics is in science. But it is perhaps even more surprising, I think, how effective non-rigorous mathematics is. Physicists by and large do not do mathematics with the rigor with which mathematicians do it (not that mathematicians are that rigorous—basically, I think of the "proofs" published by mathematicians as informal arguments for the existence of a proof in the logician's sense). But, amazingly enough, it works. Neither Newton's nor Leibniz's calculus was rigorous. Yet physics based on calculus did just fine before the 19th century when calculus was made rigorous. Physicists often make approximations—for instance, taking the first term or two in some expansion—without proving any bounds on the approximation, but tend to get it right. Likewise, it is, I suspect, not uncommon for a scientist to write down a set of partial differential equations governing some system, and then say things like "Solutions must be like this..." without ever proving that the equations in fact have a solution. (It won't do, logically speaking, to say: "It must have a solution since it describes a physical system." For in practice none of the equations describe physical systems—they describe approximations to physical systems.)

One might think that a mathematical proof that is not logically valid is like tracing your ancestry to Charlemagne with only two gaps. But it's not like that at all in the sorts of mathematical arguments physicists use. They do tend to get it right, despite not doing things rigorously.

DTH Rocket said...

Hi, I'm new to your blog.

I can totally relate to this post, as I'm currently in calculus and physics. My calculus teacher is always giving the physics teacher a bad rap, and my physics teacher is always discrediting what the calculus teacher says, and I'm in between, lost in bewilderment, head spinning, wondering what to do somewhere in between calculus and physics. For instance, everyting I learned about notation for vectors and such mathematical phenomena from math teachers (they're very particular about that sort of thing) was blown away when I got into physics and we were very loose about how we reported our answers, which symbols we picked for componets, etc. I tend to try the rigorous approach, because being enrolled in both subjects, it wouldn't hurt to get in the habit of the rigorous way, because it wouldn't hurt anything.

Like the blog!

Nichoteh said...

Dear Alex,

That's quite right. In physics, we often just work with some formal power series ring, and linearizations (whereas a mathematician would bother about the hairy analytic and non-linear details). When we get to fairly complicated theories like QFT, the math is probably ten to twenty years behind these manipulations! N