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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
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closed as not a real question by Asaf Karagila, Zev Chonoles Jan 17 '12 at 1:44

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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There is a fair amount of overlap with math.stackexchange.com/questions/21344/… –  Brandon Carter Jan 13 '12 at 23:27
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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 Jan 14 '12 at 16:09
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I also disagree with the opinion that modern texts "often miss the depth of respective classical literature." –  Jesse Madnick Jan 14 '12 at 16:10
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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 Jan 14 '12 at 23:48
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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 Jan 15 '12 at 1:11

28 Answers 28

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
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Could you please split up your list in order to enable individual voting? –  precarious Jan 13 '12 at 23:24
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Algebra, by Serge Lang (3rd Edition), 2002 –  precarious Jan 14 '12 at 0:15
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Principles of Mathematical Analysis, by Walter Rudin (3rd Edition), 1976 –  precarious Jan 14 '12 at 0:15
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Calculus, by Michael Spivak (4th Edition), 2008 –  precarious Jan 14 '12 at 0:15
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Categories for the Working Mathematician, by Saunders Mac Lane (2nd Edition), 1998 –  precarious Jan 14 '12 at 0:16

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.

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In order to enable individual voting: –  precarious Jan 14 '12 at 0:17
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Topology, by James Munkres –  precarious Jan 14 '12 at 0:17
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Algebra, by Saunders Mac Lane and Garrett Birkhoff –  precarious Jan 14 '12 at 0:18
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Introduction to Topological Manifolds, by John Lee –  precarious Jan 14 '12 at 0:18

Sorry for my definition of "classic" category.

  • Understanding probability, by Henk Tijms
  • Concrete Mathematics, by Graham, Knuth, Patashnik
  • Deterministic Operations Research, by Rader
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Thank you for your contribution. –  precarious Jan 14 '12 at 15:21
    
For individual voting, please refer to the comments below. –  precarious Jan 14 '12 at 15:21
    
Understanding probability, by Henk Tijms –  precarious Jan 14 '12 at 15:22
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Concrete Mathematics, by Graham, Knuth, Patashnik –  precarious Jan 14 '12 at 15:22
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Deterministic Operations Research, by Rader –  precarious Jan 14 '12 at 15:22

Personally, I was enlightened by

  • Hoffman, Kunze: Linear Algebra (2nd Edition), 1971
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"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 Jan 15 '12 at 22:27

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). Its latest reprint was released in 1996.

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Algebraic Number Theory, by Cassels and Frohlich.

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Introduction to Commutative Algebra by Atiyah and MacDonald.

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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 Jan 16 '12 at 5:07
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I don't understand the "but". –  Fredrik Meyer Jan 16 '12 at 9:00
    
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 Jan 16 '12 at 9:47

Algebraic Geometry by Robin Hartshorne

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

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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 Jan 16 '12 at 5:09
    
(so if you agree with my comment, you can downvote my answer) –  user16299 Jan 16 '12 at 5:09

Complex Analysis by Lars Ahlfors.

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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 Jan 15 '12 at 3:33
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@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 Jan 15 '12 at 4:40
    
@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 Jan 15 '12 at 4:56
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@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 Jan 15 '12 at 14:09
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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 Jan 15 '12 at 15:12

Algebra, by Michael Artin.

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Linear Algebra and Its Applications, Peter Lax.

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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 Jan 15 '12 at 0:01
    
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 Jan 15 '12 at 15:18

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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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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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.

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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 ) –  lentic catachresis Jan 15 '12 at 4:21
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@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 Jan 15 '12 at 5:03
    
And naturally,someone takes a point away. Terrific. –  Mathemagician1234 Jan 15 '12 at 5:04
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-1 For having (many) more than one text in the post. –  Austin Mohr Jan 15 '12 at 7:34
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@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 Jan 15 '12 at 7:50

Morse Theory by John Milnor.

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Disquisitiones Arithmeticae, Carl Friedrich Gauss.

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

$\bullet$ Internet Archive,

$\bullet$ Google Books.

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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 Jan 15 '12 at 8:00
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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 Jan 15 '12 at 8:59
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@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 Jan 15 '12 at 11:58
    
"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 Jan 15 '12 at 22:25

A Course in Arithmetic, Jean Pierre Serre.

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

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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 Jan 15 '12 at 22:26
    
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 Jan 16 '12 at 1:14
    
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 Jan 16 '12 at 5:01

Elements of Mathematics, Nicolas Bourbaki.

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Differential Geometry of Curves and Surfaces by Manfredo Do Carmo.

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

Freely and legally available at Project Euclid.

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"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 Jan 15 '12 at 22:25

Topology from the Differentiable Viewpoint, John Milnor.

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That's exactly the answer I wanted to post! –  M Turgeon Jan 15 '12 at 14:28
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@MTurgeon One of the reviews on Amazon says that this book is "the best math book ever written." –  the symplectic camel Jan 15 '12 at 14:37
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As one of my professor once said: "anything written by Milnor is worth reading." –  M Turgeon Jan 15 '12 at 17:52
    
@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 Jan 17 '12 at 4:47
    
I actually think Char. Classes is better than this one, but that is my dirty little secret. –  user641 Feb 5 '12 at 22:17

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.
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"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 Jan 15 '12 at 22:24
    
@YemonChoi Which text do you think I've referenced here doesn't fit? –  Doug Spoonwood Jan 16 '12 at 3:37
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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 Jan 16 '12 at 4:58

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.

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Do you feel that this matches the OP's original description "state of the art"? –  user16299 Jan 15 '12 at 22:23

Introduction to the Theory of Computation, Michael Sipser.

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