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So as I have said before in a previous question, I am taking a first course in Mathematical Analysis, and I'm quite excited. I just found out though that unlike the other professors at my university, my professor is using Real Mathematical Analysis by Pugh. I thought it was rather strange because I have read from so many places that Rudin's text on the topic is "the bible" of mathematical analysis, and also he is the only professor who doesn't use it. So I was wondering what some of you experienced mathemeticians thought of choosing this book over Rudin? Is this book a little easier to use then Rudin's? I have heard the Rudin is quite rigorous.

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In my own experience I have seen that the "bible" doesn't exist in mathematics. One needs to know a lot of different books, because every one of them has its strengths and its weaknesses. Even Rudin's "Principles of mathematical analysis" has weaknesses, IMHO. –  Giuseppe Negro Dec 7 '12 at 0:47
    
@GiuseppeNegro What do you think some of the weaknesses of Rudin is? –  TheHopefulActuary Dec 7 '12 at 0:50
    
There are comments on your question on Amazon (read the user reviews) including some comparative statements to Rudin. See: amazon.com/Real-Mathematical-Analysis-Charles-Chapman/… –  Amzoti Dec 7 '12 at 0:52
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It is way too concise, to say one. This leads to a tendency to present only the final product, hiding all the reasoning that leads you there. This can be very useful if you are reviewing things you already know, but can be very frustrating if you are studying for the first time. For an example of what I mean try having a look at the first chapter, and precisely at the theorem asserting that every positive real number has a square root. I find that the proof of that theorem is unreadable unless you already know what to do. –  Giuseppe Negro Dec 7 '12 at 0:58

2 Answers 2

up vote 15 down vote accepted

Every great book on a particular topic will have things another great book omits. Having seen both Rudin and Pugh, I would say they are both excellent choices for a rigorous course in mathematical analysis. Pugh's book may be easier to understand as Rudin is very terse.

Often what we call the bible is just what has been used for many years by many people, but newer alternatives exist and can be equally awesome too.

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Seconding Giuseppe Negro's comment that there are no "bibles" in mathematics: what is true is that there are some sources that do accomplish a certain difficult goal, without too many bad side effects, and earn a place in the "pantheon" for this attribute. Sometimes, however, a place is earned for "being impressive" rather than for "being helpful". Or for "being tough" rather than "being clear". One person's "rigor" is another's "tediousness", etc.

Pugh is a genuine mathematician, so the choices he made in putting together his book are surely reasonable. I have the impression that he chose to emphasize intuitive/pictorial things, rather than Cauchy-Weierstrassian, as would be Rudin's wont.

Most days, I agree (with D'Alembert, I think it was?, who said) "After belief, proof will follow".

EDIT: in fact, as Michael Harris observed, it is "Allez en avant, et la foi vous viendra": "Go forward, and faith will follow"... conceivably even more radical? Or is it less so...? :)

Although years ago I scoffed at this potentially seemingly frivolous unrigorous remark, by now I understand it in a different way. E.g., if one has a "physical intuition" that a thing is true, this often suggests "proof". And, from the other side, if a given question is "purely formal", sometimes meaning "of no genuine interest to anyone", then the "formal" (in a derisive sense) approach we find ourselves taking is indicative of lack-of-sense.

My advice would be to look at many sources, encompassing a range of viewpoints. One can argue that some sources are too fussy over small things, and that others are negligent. In the end, I think a professional mathematician wants to have experienced a certain amount of fussing-over-details, but in as many cases as possible seeing, in hindsight, that many of those details were effectively fore-ordained to be ok, ... rather than conceding that "the universe is hostile, and things tend to be false rather than true..."

That is, while many naive notions are, of course, incorrect, I claim that the good news (about elementary analysis, as of many other things) is that things turn out pretty well. That is, although it is entirely reasonable, and perhaps intellectually responsible, to be worried about details, it turns out that things are not as bad as they might have been. (One may argue that if this were not so we wouldn't be doing this at all.)

One minor-but-important disclaimer is that essentially all "introductory analysis" sources do limit their technical outlook, so that some questions which can be asked in relatively elementary terms, but which admit no real coherent answer in the same terms, are ... nevertheless... answered in sometimes-ghastly terms. My own pet case is about differentiation in a parameter inside an integral... :)

Summary: multiple sources. Look around. (And... there are no rules.)

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Interesting quote. I like it, and also happen to agree. Have any source on it? A quick Google turns up zilch. –  Noldorin Apr 6 at 0:06
    
@Noldorin, hm, ... I'd wager that this is quoted in E.T. Bell's (not entirely reliable... not to mention bias-laden) "Men of Mathematics", but I don't have my copy here. I can check this coming Monday. –  paul garrett Apr 6 at 3:53
    
Ah right, interesting. Well, please do check if you wouldn't mind; I'd appreciate it. –  Noldorin Apr 7 at 0:18
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@Noldorin, ... (found this thing again... :) The (corrected) attribution does not seem to be in Bell's book. but is given on turnbull.dcs.st-and.ac.uk/~history/Day_files/Day1029.html –  paul garrett Apr 7 at 18:02
    
That's great, thank you. I'm not sure d'Alembert meant precisely the same thing you did in your above answer, but I do think both are very worthwhile pieces of advice! –  Noldorin Apr 7 at 18:29

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