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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.

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The whole paragraph: "Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is pure mathematics, and it is sure to appeal to the budding pure mathematician." This is pretty clearly a sales pitch aimed at bright undergraduates who found calculus underwhelming. I wouldn't read anything more into it than that. –  Pete L. Clark Jun 3 '10 at 12:48
I just checked the MathSciNet review (by Sherif T. El-Helaly) of Pugh's book. It is overwhelmingly positive, except: "The only shortcoming of the book is the Preface. It is too short (six lines) and does not include any guidance for the instructor." –  Pete L. Clark Jun 3 '10 at 17:09
I think one might find Pugh not guilty of saying real analysis lacks applications, since the paragraph could be taken to mean only that when you're doing real analysis, you're not doing applications. That doesn't mean it doesn't have any. –  Michael Hardy Jun 3 '10 at 17:14
More from the MathSciNet search: C.C. Pugh has written several papers in the area of applied analysis, most recently a 2005 paper Convex dynamics and applications. (From the review: "Some connections of this result with the problem of digital halftoning and with the chairman assignment problem are evoked.") With regard to the current indictment of Prof. Pugh on the charge of claiming that analysis has no applications: I move for dismissal. Note also that the answer to the question in the title is obviously "False." I don't think it needs discussion or defense here. –  Pete L. Clark Jun 3 '10 at 17:36
If real analysis has no applications, what about p-adic analysis? :-) –  Greg Kuperberg Jun 3 '10 at 17:55

7 Answers 7

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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. :-)

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Amazingly, in Euler's "Introductio in analysin infinitorum", first published in 1748, the same point about teaching analysis twice is made. In the first line of the preface, Euler writes that "most difficulties encountered by students of infinitesimal analysis are due to the fact that as soon as they have learned elementary algebra, they direct their thoughts to this higher art", and consequently, never master it. One of the ideas behind "Introductio" was to make students comfortable with functions and power series algorithmically before taking up more conceptual parts of analysis. –  Victor Protsak Jun 3 '10 at 17:05
Needless to say, some of the material presented goes beyond what modern mathematicians typically learn about the scorned "calculus" part in $\textit{all}$ their analysis courses (including differential equations and numerical methods). I learned about it from Varadarajan's article in the Bulletin a few years ago. –  Victor Protsak Jun 3 '10 at 17:12
In 1748. Well, while mathematics is in a monotonic trajectory, it sometimes seems that mathematics education has a pre-periodic orbit. –  Greg Kuperberg Jun 3 '10 at 17:31
I'd like to add that in Germany, we haven't calculus and real analysis not really seperated or at least seperated in another way. We start in the first semester with epsilon-delta-stuff and everything formally correct, but usually use only the Riemann integral in the first two semesters and introduce measure only in the third semester. I would guess, we do more calculus in school than in the usual US high school - at least, we did when we had still 13 years of school. –  Lennart Meier Jun 3 '10 at 18:24
@Greg: Your claim concerning monotonicity rests on some possibly optimistic assumptions concerning the future of literate civilization. –  S. Carnahan Jun 4 '10 at 1:56

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!

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Newton invented calculus. Bolzano, Cauchy, and Weierstrass (among others) invented analysis. –  Harry Gindi Jun 3 '10 at 15:56
How do you define calculus? Analysis? One has epsilons and the other doesn't? Personally, my calculus teacher in high school presented both the epsilon delta side of things and the softer "infinitesmal" based approaches. I think that having both sides around is incredibly important - and I think they are both analysis. –  Steven Gubkin Jun 3 '10 at 17:47
AMEN,Steven.What is wrong with you "purists"? I shudder to think the state of both topology and differential geometry if you hard-liners were in charge of what gets taught to math students. –  Andrew L Jun 3 '10 at 21:41
Unfortunately, most physicists don't know much analysis. (This statement is left ambiguous, so that everyone will agree.) –  Jon Jun 3 '10 at 22:07
Hmmm... I thought that differential geometry had been out of the regular curriculum for quite some time (differentiable manifolds do not count as DG). –  Victor Protsak Jun 4 '10 at 4:18

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.

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I agree with what you said, except I find it distressing that maybe this "not having applications" thing is meant to be part of the sales pitch, as if this makes the whole subject better. It still chills me a bit, since I never thought people would think this way, but reading A Mathematician's Apology and talking to mathematicians young and old in the past few years have shown me that they actually do. –  yanzhang Jun 4 '10 at 15:41
@yanzhang Then it's time to buck this trend with texts like Zorich and Estep. Why this kind of Borbakian purism is so prevalent is harder to say.I think it's mainly a kind of faddish academic snobbery:"Don't dirty my beautiful proofs with your physics!"Like it's cool to be ignorant of mathematics beyond the structures.This to me is tantamount to saying you want to study the structure of music passionately and write the definitive treatise on pentameter,but you're proud you're tone deaf and can't play a note! It's crazy to me. –  Andrew L Jun 4 '10 at 16:12

Reading the whole paragraph as reported above, it is clear that it is quite different from the title of this question. Saying that "X-theory involves no application in engineering" just means that an X-theorist, in her job, doesn't employ engineer's tools or language, and as an X-theorist she may even forget about engineering. It definitely does not mean that (i) problems of X-theory were not originated from concrete problems of engineering, nor that (ii) the results of X-theory 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 (dis-trahere, to take here and there) by other unessential feature of it, and with the advantage to solve once for all several essentially similar problems.

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Surely another key difference is the verb: the book claims that analysis "involves [...] no applications", not "has no applications". It has plenty of applications; it's just that the mathematical treatment (and presumably the book in question) is self-contained, and doesn't rely on or refer to them. –  Peter LeFanu Lumsdaine Jun 3 '10 at 16:30

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.

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@ Pete,Victor and Greg: First,I agree with your interpretation of the paragraph Greg's referring to. Secondly,despite loving Pugh's book-I 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. Third-I seriously doubt one of the world's experts in differential equations thinks real analysis is devoid of real world applications.But that being said-what 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.

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Please don't reply to answers with another answer. Better to use the commenting system. –  Scott Morrison Jun 3 '10 at 23:55
Andrew: In case you don't know it already, there is a rich Russian tradition of such texts, eg the courses of Fichtenholz and Smirnov spring to mind. –  Victor Protsak Jun 4 '10 at 4:27
@ Victor Absolutely-more recently,the 2 volume epic by Zorich is now available in English. Sadly,it's not getting the attention it deserves. –  Andrew L Jun 4 '10 at 6:02
@Scott: My response was too long for the commentary section. Next time I'll break it into pieces. –  Andrew L Jun 4 '10 at 6:02
@Scott SIGH.I give up. –  Andrew L Jun 4 '10 at 16:07

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.

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