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MathematicsClassical texts that should not be missing from any shelf
[+36] [28] precarious
[2012-01-13 23:03:03]
[ reference-request soft-question big-list ]
[ https://math.stackexchange.com/questions/98885/classical-texts-that-should-not-be-missing-from-any-shelf ] [DELETED]

It seems to me as if many modern texts are rather streamlined. They are designed not to expect too much from the reader but they often miss the depth of respective classical literature.

The purpose of this record is to collect highly recommended classical texts. Quality and depth of the subject matter should serve as a benchmark. Suitability for beginners should be irrelevant.

For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art.

The final list will contain each recommended text with $10$ or more votes.

In order to make this work, please restrict yourself to one proposal per answer/comment.

Thank you for your contribution!

  1. Rudin - Principles of Mathematical Analysis, 3rd Edition, 1976
  2. Graham, Knuth, Patashnik - Concrete Mathematics, 2nd Edition, 1994
  3. Munkres - Topology, 2nd Edition, 2000
(2) Baby Rudin .... - Adam
(2) What level of books are you looking at? Euclid's Elements pops into mind but I have the feeling that's not what you're aiming for... - Scaramouche
@Scaramouche When I asked the question I thought about books from the 20th century. Clearly, this idea was biased. In the end, the list should contain what the community considers to meet the criteria stated in the original question. - precarious
@Brandon Thank you for the link. In this list, the focus should be on classical texts. - precarious
Could you clarify what classical means here? What is it about Hoffman and Kunze that isn't modern? - Dylan Moreland
@Dylan You are right that Hoffman and Kunze is modern in the sense that it represents the state of the art. However, it was published more than $40$ years ago and turned out a success. - precarious
(8) This question seems overly broad. Off the top of my head, I can easily think of about 15 titles that fit the criteria -- which, by the current design of the question, would entail 15 answers. I understand that this is a big-list question, but a question this broad could conceivably yield something like 70 answers, which I feel is too many. - Jesse Madnick
(8) I also disagree with the opinion that modern texts "often miss the depth of respective classical literature." - Jesse Madnick
(1) @Jesse Of course there are recent texts of outstanding quality and depth. I did not mean to offend anyone. I apologize if the words I chose conveyed that impression. -- I share your expectation of many answers. However, I think that the final list will be quite managable. Currently, there does not exist a single text with 10 or more votes. - precarious
(7) I am not sure how to reconcile the adjective "classical" with the requirement that the book should somehow be "state of the art". For instance, Mac Lane's book fails the second condition, as does Katznelson which I would otherwise suggest. Do you really mean something like "could still be used as a standard text today"? - user16299
(4) The books should really contain a line why they are good. If I'm coming here because I want to read a good introductory book, for example, then I'd like to know why that one is the right choice. - Nikolaj-K
I'm just curious, why "$10$ or more votes"? Is there any statistical reason? Also, the question does not have $10$ upvotes. - Karatug Ozan Bircan
(7) I agree with @YemonChoi; the requirements that a text be "classical" and yet reflect the "state of the art" are often mutually exclusive. The only exceptions I can think of are highly specific and introductory books (e.g. the first 7 chapters of Rudin's Principles reflect, IMVHO, the state of the art in "beginning real analysis"). For broader or more "advanced" material (e.g. "functional analysis", "number theory") all "classical" books will fail to reflect current perspectives. Many people below are answering the question as if it is "list books that you consider essential." - leslie townes
(1) @Karatug Creating $10$ fake accounts is not worth it. :-) - precarious
@leslie Indeed, Rudin's book is the first which made it. - precarious
(1) It seems like "classical" is supposed to mean "successful by some measure". I think it's going to become (has it already?) another "here are some good books" list. Rudin's book is great but it seems pretty "streamlined" to me. - Dylan Moreland
(1) May I repeat: what does "state of the art" actually mean in this context. All three of the Rudin books everyone mentions, and arguably his Fourier Analysis on (LCA) Groups book which few seem to know of, are classics, but they don't have the "cutting edge" feel that the phrase "state of the art" implies to me. - user16299
(3) @DylanMoreland are you implying that people on MSE might sometimes answer the question they think is there rather than the question actually worded? I am Shocked, Sir, Shocked I Tell You. - user16299
(3) To express what others have already said, I think this question is too vague to admit meaningful answers and has become yet another "what are some good books" thread. I am quite tempted to vote to close, but I would prefer not to act unilaterally. - Zev Chonoles
Can this list include shorter papers in general? Even papers such as Leibniz's on calculus? - Doug Spoonwood
(1) @Zev I understand the stated objections. Unfortunately, my definition of a certain category of texts does not match the precision of mathematical language. In spite of that, there is a great number of people who made reasonable proposals, who voted, or who contributed in some other way. Nevertheless, I think that the list should be closed at some point. I propose to leave it open for another day, update the list, and shut down. -- My thanks go to all contributors. Embarrassed, I have to admit that I do not own any of the books in or nearly in the list so far. :-( - precarious
(2) WHAT DOES "STATE OF THE ART" MEAN? - user16299
For instance: why not suggest: Hardy & Wright; Hardy's Divergent Series; Zariski-Samuel; Naimark's Normed Rings; Zygmund; Cartan-Eilenberg (classic but very non-state-of-the-art); Banach's original book... - user16299
Hilton-Stammbach; Dixmier's von Neumann algebras book; Simmons's Introduction to Topology and Modern Analysis; Sutherland's Introduction to Metric and Topological Spaces, Khinchin's Notes on Continued Fractions, Loomis's Abstract Harmonic Analysis, Duren's Theory of $H^p$ Spaces... - user16299
These are all fine books, but with the possible exception of Zariski-Samuel which I don't really know, they do not give an idea of where their respective subjects are now at, and so fail to "expose the present state of the art". I feel that the same could be said for several of the "usual suspects" that have been left as answers to this question. - user16299
[+16] [2012-01-14 21:11:35] Fredrik Meyer

Introduction to Commutative Algebra by Atiyah and MacDonald.


Good book, but: "For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art." - user16299
(2) I don't understand the "but". - Fredrik Meyer
I presume that A and M, since it did not set out to be "state of the art" when it was written, is not going to be a good picture of "the present state of the art". I agree that it's a classic, but so are many other books which could be listed here, in which case this thread will rapidly bloat - user16299
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[+13] [2012-01-14 23:51:10] Ben Blum-Smith

Complex Analysis by Lars Ahlfors.


(2) WHY would anyone still use Ahlfors when there are now a half a dozen much more readable texts at the same level of comprehensiveness and sophistication?!?I'll tell you why-because Ahlfors taught at Harvard and as a result,his textbook's been canonized whether it deserves it or not. I'm sorry,this kind of iconic rubber-stamping of textbooks really irritates me. - Mathemagician1234
(14) @Mathemagician1234 : That's one of the silliest comments I've ever read. Why on earth do you think that people care that Ahlfors was at Harvard? Rudin was at Wisconsin, but people still regard his book as a classic. Ahlfors was perhaps the greatest complex analyst of the 20th century, and his book is regarded as a classic first because it was the first real modern textbook on the subject, but second because of its elegant writing, perfect balance between analytic and geometric perspectives, etc. - Adam Smith
(1) @Adam I don't agree-I think it's dry and unpleasant in a lot of places. I understand it's historical significance,but I think the canon of texts on the subject has passed it by. I WILL say that the 3rd edition is VASTLY improved over the 1953 original.But to be honest,for a graduate course,I'd rather use either Greene and Krantz or Narasimhan and Nievergelt combined with Jones and Singerman.(continued) - Mathemagician1234
@Adam (comtinued) And for the record without beginning an argument here-I SERIOUSLY didn't appreciate the demeaning-and patently FALSE-comments you made here at math.stackexchange.com/questions/98299/… - Mathemagician1234
(5) @Mathemagician1234 I think you need to take books a little less eriously. There's more to life than books you know (but not much more, not much more ...) - though the first statement is true, Stack should be a place to browse when you have free time - offering assistance as you please - not to discredit the work of say Ahlfors in a bizarre way (not to mention he's been dead for ages, I hardly think you're taking into consideration the way the texts today are produced: authors of perhaps 50 years old would invariably of used Ahlfors and drew much knowledge from it. - Adam
(2) For the record, I mentioned the book because I love working out of it. I really enjoy how it's written. It's an ambition of mine to work through the whole thing. - Ben Blum-Smith
For the record, the only mental note I have about "mathemagician" is from months ago, something to the effect of "this guy has some sort of literary agenda" - The Chaz 2.0
@The Chaz My mentor instilled in me a vast love and appreciation of the literature and I review textbooks and monographs for the MAA. I make no apologies for that. I DO read the current research literature whenever I can,of course. - Mathemagician1234
Thanks for that comment. I'll quit now! - The Chaz 2.0
"For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art." (Disclaimer: I have heard good things about Ahlfors when I was an undergraduate, tempered with caution; my point is that this question claims not to be just seeking the usual big list, but is getting just that.) - user16299
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[+12] [2012-01-15 02:47:00] John D. Cook

Inequalities by G. H. Hardy, J. E. Littlewood and G. Pólya. (1934)

Gathers into one place techniques and results useful in many areas of mathematics.


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[+11] [2012-01-14 20:46:49] Brandon Carter

Algebraic Number Theory, by Cassels and Frohlich.


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[+11] [2012-01-14 23:39:51] Matt

Algebraic Geometry by Robin Hartshorne


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[+11] [2012-01-15 10:17:25] Karatug Ozan Bircan

Topology from the Differentiable Viewpoint [1], John Milnor.

[1] http://rads.stackoverflow.com/amzn/click/0691048339

(1) That's exactly the answer I wanted to post! - M Turgeon
(1) @MTurgeon One of the reviews on Amazon says that this book is "the best math book ever written." - Karatug Ozan Bircan
(5) As one of my professor once said: "anything written by Milnor is worth reading." - M Turgeon
@MTurgeon: Agree! I have read Milnor's Morse Theory, Topology from the Differentiable Viewpoint, and Characteristic Classes. I enjoyed reading each one of them. - Paul
I actually think Char. Classes is better than this one, but that is my dirty little secret. - user641
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[+9] [2012-01-14 23:53:21] Ben Blum-Smith

Algebra, by Michael Artin.


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[+8] [2012-01-14 20:09:58] Pacciu

Talking about classics:

Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press.

The first edition was published in 1922 and the second in 1944. Neverthless, after 90 years, Watson's Treatise is still the standard reference book on the theory of Bessel functions (e.g., it has 10334 citations in Google Scholar [1]). Its latest reprint was released in 1996.

[1] http://scholar.google.it/scholar?hl=it&q=watson%20a%20treatise%20on%20the%20theory%20of%20bessel%20functions&btnG=Cerca&lr=&as_ylo=&as_vis=0

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[+8] [2012-01-15 07:50:54] Pierre-Yves Gaillard

Elements of Mathematics, Nicolas Bourbaki.


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[+8] [2012-01-15 09:38:35] Pierre-Yves Gaillard

Emil Artin, Galois Theory, Lectures Delivered at the University of Notre Dame.

Freely and legally available at Project Euclid [1].

[1] http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197041

"For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art." - user16299
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[+7] [2012-01-13 23:20:42] Samuel Reid

I would say that the following are without a doubt considered some of the paradigmatic "classics",

  • Algebra, by Serge Lang (3rd Edition), 2002
  • Principles of Mathematical Analysis, by Walter Rudin (3rd Edition), 1976
  • Calculus, by Michael Spivak (4th Edition), 2008
  • Categories for the Working Mathematician, by Saunders Mac Lane (2nd Edition), 1998

(6) Could you please split up your list in order to enable individual voting? - precarious
(1) You want me to give four different answers? This isn't a polling site, I'm simply answering your question. - Samuel Reid
(1) I find this list quite peculiar - admittedly it is a problem with the original question as it is extremely vague, but when you put (easy) Spivak in a list with cats for a working mathmo, something is probably out of place. - Adam
(1) @SamuelReid This is a big-list question, so voting in this case is for polling. - Alex Becker
(1) @Samuel I am sorry if I misunderstood the concept of this site. The original idea was to create a list based on community recommendations. It should contain proposals supported by a certain amount of people. Utilizing the voting mechanism in order to measure support seems to be natural. - precarious
In order to enable individual voting: - precarious
(11) Algebra, by Serge Lang (3rd Edition), 2002 - precarious
(21) Principles of Mathematical Analysis, by Walter Rudin (3rd Edition), 1976 - precarious
(7) Calculus, by Michael Spivak (4th Edition), 2008 - precarious
(10) Categories for the Working Mathematician, by Saunders Mac Lane (2nd Edition), 1998 - precarious
Irrational numbers, by Ivan Niven - Yuval Filmus
(3) If it were possible to downvote portions of a textbook, the portions of Rudin's Principles devoted to multivariable analysis, differential forms, and Lebesgue integration would receive a downvote from me. In my experience, even most people who regard this book as "essential" either never read those chapters, or found much better treatments of those topics elsewhere. (I try to say this whenever I recommend the book to someone, fearing that otherwise they will think they will think this book is the last word on those topics.) - leslie townes
That is true leslie; he even says on the first page of Lebesgue integration he is 'glossing over' the theory and it's perhaps "common knowledge" between people who have gone through the book to leave out everything past chapter 9. I only have the 2nd edition of Lang in the library - is it riddled with typos? I always considered that aspect essential reading but the amount of mistakes were monumental considered this was perhaps in excess of it's 10th printing (of the 2nd edition!). - Adam
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[+7] [2012-01-15 07:43:15] Pierre-Yves Gaillard

Disquisitiones Arithmeticae, Carl Friedrich Gauss.

A French translation is available at Internet Archive and at Google Books:

$\bullet$ Internet Archive [1],

$\bullet$ Google Books [2].

[1] http://www.archive.org/details/recherchesarithm00gaus
[2] http://books.google.com/books?id=0LdUAAAAYAAJ&dq=intitle%3Arecherches%20inauthor%3Agauss&pg=PP9#v=onepage&q&f=false

(1) I actually have a paperback copy of this book. I agree it's a remarkably important book historically and on those grounds,it's worth having and reading,but I don't think a math student loses out not having it anymore then they'd miss out if they didn't have a copy of Newton's Principia. - Mathemagician1234
(2) Dear @Mathemagician1234: Thanks for your comment. I consult the DA on a regular basis. Many (most?) of my MSE answers are based on them. I've never even opened Newton's Principia, I don't have the slightest idea about what they contain, and I'm ashamed of that. I can only talk about my personal experience. So, I'd say that my understanding of mathematics would have been even poorer than it is if I hadn't had the DA on my shelf for many years. - Pierre-Yves Gaillard
(2) @Mathemagician1234 "The purpose of this record is to collect highly recommended classical texts. Quality and depth of the subject matter should serve as a benchmark." -- I have not read the DA, but it certainly seems like it would meed these requirements set forth by the OP. Regarding your claim that the student will not lose out by not having it, don't you think that reading such a classic (assuming you see it as a classic) is a rewarding pursuit on its own? // On a separate note, I do not appreciate your disparaging comments on more than one recommendations on this thread. - Srivatsan
(1) "For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art." - user16299
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[+7] [2012-01-15 07:45:13] Pierre-Yves Gaillard

A Course in Arithmetic, Jean Pierre Serre.


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[+7] [2012-01-15 10:23:56] Karatug Ozan Bircan

Algebraic Topology [1], Allen Hatcher.

[1] http://www.math.cornell.edu/~hatcher/AT/ATpage.html

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[+6] [2012-01-14 23:54:27] Ben Blum-Smith

Linear Algebra and Its Applications, Peter Lax.


(1) Isn't Lax's book quite recent? While it looks to have an interesting perspective, I think the jury is still out as to whether it will prove a classic and enduring source. - user16299
Fair. First edition was published in 1997 and OP asked for "at least a decade" so I figured it was fair game. I study at NYU so perhaps Lax's work takes on more of a patina of authority than it would at another university. I think the book is really beautiful though. - Ben Blum-Smith
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[+6] [2012-01-15 07:38:18] Paul

Morse Theory by John Milnor.


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[+6] [2012-01-15 07:58:05] Paul

Differential Geometry of Curves and Surfaces by Manfredo Do Carmo.


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[+5] [2012-01-14 23:47:00] user16299

Katznelson's Introduction to Harmonic Analysis - for when Zygmund will break your shelf


Good book, but: "For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art." - user16299
(so if you agree with my comment, you can downvote my answer) - user16299
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[+5] [2012-01-15 07:46:44] Pierre-Yves Gaillard

Elementary Theory of Analytic Functions of One or Several Complex Variables, Henri Cartan.


Good book, but: "For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art." - user16299
Dear @Yemon: I see that you made almost the same comment about many of my answers. Thanks. In all cases, I suppose you agree that the book in question has "proven its value for one decade at least". Or perhaps you don't? I think your point is that the books I suggest don't "expose the present state of the art". For Cartan's book, it looks to me very similar, on both counts, to Ahlfors's book (for which I voted), and I didn't see any comment of yours about Ahlfors's book. - Pierre-Yves Gaillard
Bien sur - I have a copy of Cartan on my own shelf. As you guessed, it is the "state of the art" part of the question which I find poorly thought-out (my guess is that the old editions of Krantz might get closer to state-of-the-art). Thank you for pointing out that the same applies to Ahlfors, a book I know by reputation only; I shall go and post my stock comment - user16299
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[+5] [2012-01-15 09:13:36] james

Topology and Geometry by Glen Bredon.

This is a fairly recent book [1993, I think] but it's a great book for a graduate algebraic topology course. It certainly isn't easy-going, but there are pretty nice exercises at the end of each section. Additionally, the point-set section uses some nice notions [nets, for example] that the student may have missed out on in undergrad topology.


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[+4] [2012-01-14 15:25:41] precarious

Personally, I was enlightened by

  • Hoffman, Kunze: Linear Algebra (2nd Edition), 1971

"For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art." - user16299
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[+3] [2012-01-15 00:49:15] chango

Here are some in Analysis...

  • All books by Rudin (Principles, Real & Complex, Functional Analysis)
  • Introductory Real Analysis (Kolmogorov / Fomin)
  • Real Analysis by Royden

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[+3] [2012-01-15 15:08:19] Zach Langley

Introduction to the Theory of Computation, Michael Sipser.


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[+2] [2012-01-13 23:39:03] ItsNotObvious

Books that I have learned a lot from that probably belong in the "classics" category are

  • Topology, by James Munkres
  • Algebra, by Saunders Mac Lane and Garrett Birkhoff
  • Introduction to Topological Manifolds, by John Lee

Ok, the last one isn't really a "classic", per-se, since it's only about 10 years old but it is a very good book.


In order to enable individual voting: - precarious
(11) Topology, by James Munkres - precarious
(5) Algebra, by Saunders Mac Lane and Garrett Birkhoff - precarious
(8) Introduction to Topological Manifolds, by John Lee - precarious
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[+2] [2012-01-14 14:54:05] com

Sorry for my definition of "classic" category.

  • Understanding probability, by Henk Tijms
  • Concrete Mathematics, by Graham, Knuth, Patashnik
  • Deterministic Operations Research, by Rader

Thank you for your contribution. - precarious
For individual voting, please refer to the comments below. - precarious
Understanding probability, by Henk Tijms - precarious
(13) Concrete Mathematics, by Graham, Knuth, Patashnik - precarious
(1) Deterministic Operations Research, by Rader - precarious
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[+2] [2012-01-15 13:03:44] Doug Spoonwood

I haven't read the first one here at all, but it seems a favorite:

  • Set Theory by Thomas Jech.
  • Introduction to Metamathematics by S. C. Kleene.
  • Introduction to Logic by Alfred Tarski.
  • Calculus by Tom Apostol.

"For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art." - user16299
@YemonChoi Which text do you think I've referenced here doesn't fit? - Doug Spoonwood
(1) Apostol. Haven't used it; have heard good things about it; in what sense is it "state of the art"? (To be fair, I think the original question is poorly phrased, but I wanted to highlight this part. Otherwise I could just write down my favourite books of yesteryear.) - user16299
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[+1] [2012-01-15 03:30:04] Mathemagician1234

That's easy-and I'll focus on the books not already mentioned here.In no particular order of importance:

Theory of Functions by E. Titchmarsh

Lectures on Elementary Topology And Geometry by I.M. Singer and James Thorpe

Differential Topology by Victor Guillemin and Alan Pollack

Notes On Differential Geometry by Noel J.Hicks

General Topology by John Kelley

Differential Equations with Applications and Historical Notes by George F.Simmons

Elements of Differential Geometry by Richard Millman and Thomas Parker

General Theory Of Functions And Integration by Angus Taylor

Foundations of Differentiable Manifolds And Lie Groups by Frank Warner

Analysis And Solution Of Partial Differential Equations by Robert L. Street

Algebraic Topology by C.F.Maunder

Algebra by Roger Godement

Anything by either Einar Hille or Albert Wilansky

And those are just the ones I can think of off the top of my head. I'm sure I can come up with more.


(7) Funny, in a comment to another answer you said: WHY would anyone still use Ahlfors when there are now a half a dozen much more readable texts at the same level of comprehensiveness and sophistication?!? I would say exactly this about Kelley's book (well, perhaps in a less excited way ;D ) - Bruno Stonek
(1) @Bruno Fair enough. The difference is that Kelly's really a PROBLEM COURSE. It's not really a textbook in the conventional sense in point set topology-it's intended to be worked through and not really read in the conventional sense. So is Hicks,to a lesser degree. Alfhors claims to be a book that can be read and learned from whether you do all the exercises or not. That makes a big difference. But like I said,that's a fair objection,Bruno. - Mathemagician1234
And naturally,someone takes a point away. Terrific. - Mathemagician1234
(9) -1 For having (many) more than one text in the post. - Austin Mohr
I agree with Austin. These are fine books, but isn't the idea of these list questions to have one book (or just a few) per answer, rather than a barrage? - user16299
(3) @Yemon and everyone else: I apologize for the list,I didn't see the requirement of one per suggestion until I had it posted. My bad.If I had to go with just one,I'd go with Taylor. The best analysis book I've ever seen bar none.Singer and Thorpe would be a very close second,but it loses out because of the major flaw of having no exercises. - Mathemagician1234
Understood, Andrew. - user16299
@Yemon cont. Actually,on further review,it'd be a dead heat between Taylor and Simmons. Anyone studying differential equations-especially those for the first time-are losing out without this marvelous book to guide them. - Mathemagician1234
I'd STRONGLY disagree with Warner (I found him tedious) and Kelley (same as an above commenter), but I highly recommend Guillemin and Pollack. Probably one of my favorite texts of all time. - james
@james Warner is an advanced textbook and is really best read after a thorough introduction to manifolds via a kinder textbook like Guillemin and Pollack. I've always loved G&P as one of the all-time great textbooks, although lately it's gotten a bad rap for embedding all it's manifolds in Euclidean space. I think that's a perfectly natural and ok way to introduce them without the complications of charts to beginners.It's largely a matter of taste-I also love the books by John Lee and Loring Tu as well,which use "general" manifolds from jump. - Mathemagician1234
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[+1] [2012-01-15 15:07:51] Pierre-Yves Gaillard

A Concise Introduction to the Theory of Numbers, Alan Baker.

It seems to me that this book is not very popular, and I've never understood why.


Do you feel that this matches the OP's original description "state of the art"? - user16299
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