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.

4 comments:

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

David Heider said...

In principle, I can agree with what Alexander expressed. Speaking in a purely logical sense, i.e., really a formal logical system, one would find that math proofs are in general not as logic as they seem. I know that there was a stream in mathematics in the first half of the 20th century around Whitehead and Russell that wanted to build mathematics entirely up by means of logic. This is an approach, which is not very useful, especially when wants wants to follow the more main-stream, applicable and still-for-the-ordinary-human-being-graspable math trends.

In principle, as I was told by a friend of mine, who studies mathematics, when it came to doing some proofs (I am a physicist, besides), was: "Don't worry about the rigor, one could make stuff rigors afterwards, but usually, you won't do it in some exrecises, If you don't have to. Proof: Laziness ;)"

I guess that it right. A good math proof is usually clearly written, and well-tracable. Since elegant proofs are short, and reduced to the essential, such proofs cannot be as rigoros as they are usually claimed to be. This is, however, more from a math perspective.

In physics, as tells me my experience, the attitude towards mathematics is that the resulst are important, and proofs that make use of some fundamental math-concept or revbeal some elegant calculatory technique. However, the definitions, results, examples and applications of results are in fact more important. One remarks in principle the difference between mathematics for physicists and engineers and applied mathematicians and the pure math. books. People claiming they have read Schwarz' Algebraic Topology (not the Topology for Physicists book) are highly doubtable. I have read the first 10 pages, and quitted after that. Too abstarct, too theoretical, too rigorous. In practice, even a practically working mathematician, as I have been told, doesn't need this rigor.

As for QFT, I can say that the math of QFT is primarily build by theoretical physicists, perhaps also a bit by mathematical physicists. There are only few mathematicians involved in the progress of QFT, although a clear matematical framework would certainly be desirable.

I think that math and physics are two separate sciences. In physics one needs a lot of math, and does a lot of math, but still is physicist, whereas mathematicians, wouldn't care about the argumen: "Yeah, but it's physically reasonable." - I have in fact read math books for physicists, where the authors attacked precsely that sort of argumentation. But someone who is ever calculated some cross-section or stuff, or even easier examples, such as magnetic fields for solenoids or stuff, knows that physicists will use precisely that sort of argumentation.

I guess with math and physics one must decide, whether one is more physicist or mathematician. Accordingly, one can choos the attitude. Personally, I am more the physicist, and yeah, I commute limes and integral and differntial operators in general. And my functions are well-behaved. And no, in general, I don't check the assumptions of the theorems I use for calculatory purposes.

Non-linearities can in general be treated perturbatively, and that's what one often does in QFT - but in the intuitive way. And this is, due to all the math formalism involved in calculations, very helpful.

Regards,
Dave

Alexander R Pruss said...

Indeed.

The puzzle is why all this nonrigorous mathematics works. You don't check the assumptions of a theorem and yet it works. Why? Two options:
1. The assumptions are nonetheless satisfied.
2. The theorem's conclusion holds (at least approximately?) in this case, even though the assumptions aren't satisfied.

But how did you know that 1 or 2 would be the case? Were you just lucky? I don't think so: this sort of success happens way too often in physics (and I expect in engineering as well) for it to be just luck.

Experience obviously plays a role. If inexperienced undergrads tried to do what you're doing, they would have to be lucky to succeed.

But how exactly does experience help here? Is there such a thing as inductive reasoning in mathematics? Or do we, given experience, trace the logical pathways with rigor but subconsciously?