I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications to other fields of science. None. It is pure mathematics." This seems like a false statement. My first thought was of probability theory. And isn't PDE's sometimes considered applied math? I was wondering what others thought about this statement.



Real analysis has no applications?
As it happens, I just finished teaching a quarter of undergraduate real analysis. I am inclined to rephrase Pugh's statement into a form that I would agree with. If you view analysis broadly as both the theorems of analysis and methods of calculation (calculus), then obviously it has a ton of applications. However, I much prefer to teach undergraduate real analysis as pure mathematics, more particularly as an introduction to rigorous mathematics and proofs. This is partly as a corrective (or at least a complement) to the mostly applied and algorithmic interpretation of calculus that most American students see first.


Some mathematicians think, and I've often been tempted to think, that it's a bad thing to do analysis twice, first as algorithmic and applied calculus and second as rigorous analysis. It can seem wrong not to have the rigor up-front. Now that I have seen what BC Calculus is like in a high school, I no longer think that it is a bad thing. Obviously I still think that the pure interpretation is important. On the other hand, both interpretations together is also fine by me. I notice that in France, calculus courses and analysis courses are both called "analyse mathématique". I think that they might separate rigorous and non-rigorous calculus a bit less than in the US, and it could be partly because of the name.


In fact, it took me a long time to realize how certain non-rigorous explanations guide good rigorous analysis. For instance, the easy way to derive the Jacobian factor in a multivariate integral is to "draw" an infinitesimal parallelepiped and find its volume. That's not rigorous by itself, but it is related to an important rigorous construction, the exterior algebra of differential forms.


Finally, I agree that Pugh's book is great. As the saying goes, you shouldn't judge it by its cover. :-)


