Talking to students in different areas and taking different classes in math, physics, electrical engineering I have been struck by the differing amounts of rigor in use.
I know little about economics but I'm told that they use measure theory and prove rigorous theorems. Yet in physics, perhaps the most mathematical of the sciences, they solve differential equations and whatever else they do without having taken real analysis (usually, as far as I know). Is this for purely historical reasons or do economists really have more need for existence theorems for example?
Another question I have is about probability. It is taught in different ways to undergrads (with talk of different formulae for discrete and random variables) and graduate students (measure theoretically). Is there some great reason for this? I guess to do Brownian motion and similar you need measure theory (is this true?) but does measure theory give you better theorems or proofs than the undergrad approach where both are possible? In information theory it seems they are happy with undergrad level stochastic processes in their textbooks. Would they be improved by using measure theory?
So, as mathematicians (or perhaps physicists, economists, information theorists etc if that is who you are) do you think the different fields have it right?
Best Answer
I think the shortest answer is that if these other fields don't have enough rigor, the mathematicians will make up for it. In fact, a large number of important mathematical problems are just that: mathematicians working to fill in the gaps left by physicists in their theories.
On the other hand, if an economist tried to publish some grand result that used flawed mathematics, it certainly wouldn't pass through the economics community unnoticed. That being said I have read some (applied) computer science papers which spin a result to sound much grander than it is specifically by appealing to a lot of semi-relevant mathematical abstraction.
As they say in the comments, a random PhD theoretical physicist might not know measure theory, but there are certainly many mathematicians without mastery of physics working on physics equations. Similarly, an economist is unlikely to know group theory while a (quantum) physicist must. The point is that as a community we can achieve greater results.
As to the reason measure theory isn't taught to undergraduates: it's hard! Many undergraduates struggle with real analysis, and even the basic proofs underlying rigorous measure theory require mastery of a first course in real analysis, which is a stretch at a lot of universities, especially for non-mathematics majors. (Of course, at some prestigious schools undergraduate calculus is taught with Banach spaces, so I'm talking about the general undergraduate populace)