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.

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 upfront. 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 nonrigorous 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 nonrigorous 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. :) 


I think out of all the fields of mathematics, analysis has the MOST application. We are talking about the subject Newton created to be able to even talk about physics here! 


While I agree that the paragraph is largely a sales pitch, I think it does hit on something else. It says that real analysis doesn't involve applications to other science. I take this to mean that when you are doing (or studying in a first course) real analysis you don't look at applications to science. This is in contrast to calculas, whereas many of the problems in calculus books are focused on all kinds of problems from classical mechanics and other areas. Just a final note. I thing that Pugh's book is amazing, the best undergrad analysis text out there. Mainly because of the HUGE number of very good problems. 


Reading the whole paragraph as reported above, it is clear that it is quite different from the title of this question. Saying that "Xtheory involves no application in engineering" just means that an Xtheorist, in her job, doesn't employ engineer's tools or language, and as an Xtheorist she may even forget about engineering. It definitely does not mean that (i) problems of Xtheory were not originated from concrete problems of engineering, nor that (ii) the results of Xtheory have no applications in engineering. Actually, at the origin of even the most abstract mathematic theory there are concrete problems of applied science (maybe after several successive steps of abstractions), and also, the final applications are again back in concrete problems. Abstraction (from abstractus: p.p. of abs+trahere: to take (something) away from) is the usual process by whom we take the essence of a problem in order to focus on it, not to distract ourselves (distrahere, to take here and there) by other unessential feature of it, and with the advantage to solve once for all several essentially similar problems. 


The comment on the back of the book seems to be saying "This is a math book." It says nothing about analysis in particular. To the extent that the book is a pure math, it is not a book about physics, horticulture, or sociology. But so what? You could say the same about any area of math. This seems like a particularly odd thing to say about analysis since it can be applied so directly to other areas. 


I have a good undergraduate analysis book, "Real Analysis with Real Applications," by Kenneth R. Davidson and Allan P. Donsig. The book is divided into two parts. Part A deals with "Abstract Analysis" which includes theory, proofs, examples, and problems found in most undergraduate analysis books. Part B deals with "Applications" which include polynomial approximations, discrete dynamical systems, diff. eqns, fourier series and physics, wavelets, and optimization. 


@ Pete,Victor and Greg: First,I agree with your interpretation of the paragraph Greg's referring to. Secondly,despite loving Pugh's bookI call it "Rudin Done Right"I was also very disappointed at the very terse preface.You'd think someone with Pugh's teaching experience would have a LOT to say on the subject having taught so many years to some of the best students in the world. ThirdI seriously doubt one of the world's experts in differential equations thinks real analysis is devoid of real world applications.But that being saidwhat compels people teaching this course to strip it down to Bourbakian purity? Tradition? Or something darker and deeper? I'm waiting for a balanced text at this level that unifies physical applications with a comprehensive introduction to real analysis. If it never arrives,I may have to write it myself. 


Ok. That's rubbish. Real analysis has a lot of application in numerical algorithms. Let's say you want to calculate an integral using some numerical method. Before doing that you might want to determine whether the integral converges. How do you figure out whether your integral converges/diverges? Study real analysis and you'll be on your way to the answer. 

