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H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics:

...I have tried to say to students of mathematics that they should read the classics and beware of secondary sources

In trying to learn more, I feel that I have accumulated a bookshelf full of secondary sources which I have left largely unread. So I would like to take heed of Edwards's advice and read some classics for the wonderment of it and for the respect I would gain for mathematicians of the past. I would like to exercise creativity over rigour, at least for a little while.

But this begs the question: what are the classics? And this is what I hoped to ask about.

Which primary sources do you feel are suitable for self-study? Naturally this is a broad question, but people must have their favourites, and I am hoping for some recommendations on the basis of these personal preferences.

Mikhail Katz
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    Stephen Hawking's "God Created The Integers" is a nice compilation of classical mathematics, with papers by Euclid, Newton, Fourier, Turing, and others. – Jair Taylor Aug 19 '14 at 06:32
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    What's suitable for one person's self-study is not suitable for another's. It depends a lot on your background and your interests. – Robert Israel Aug 19 '14 at 06:32
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    The recreational-mathematics tag is probably not what you want (but if it is, then the masters are Henry Ernest Dudeney, Martin Gardner, and Ian Stewart). – Gerry Myerson Aug 19 '14 at 06:54
  • Nice book by Edwards, even though it's a secundary source. :) – Raskolnikov Aug 19 '14 at 07:38
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    I haven't read it myself, but I've heard good things about Shannon's original paper on information theory, "A Mathematical Theory of Communication". Also, some volumes of Spivak's A Comprehensive Introduction to Differential Geometry walks readers through some foundational papers in differential geometry. – littleO Aug 19 '14 at 07:48
  • I support @InTransit 's answer and so does Laplace... In the words of Laplace: "Lisez Euler, lisez Euler, c'est notre maître à tous." or "Read Euler, read Euler, he is the master of us all". – KBusc Aug 19 '14 at 15:02
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    Mathematical literature to lose yourself in - I dread to ask, but who's your math tutor ? – Lucian Aug 20 '14 at 05:09
  • @Lucian: I never dreamt i'd get to the level that i'm at, this is whack ... What about math, how come I wasn't ever good at that? – Nikolaj-K Aug 25 '14 at 09:29
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    I would guess you would want to just look at the names that come up the most: Euler, Gauss, Lagrange, Fermat, Galois, Riemann, Poincare, Hilbert, etc, then find their most "ground-breaking" works. – Joe Moeller Jun 12 '15 at 06:06
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    @littleO I feel like if you're not emphasising rigour that rules out spivak, even if his proofs are fairly interesting and creative. He does do a good discussion of Gauss and Riemann though in volumes one and two. I haven't looked much further than that though. – snulty Jun 18 '15 at 19:50

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For introductory Number Theory, you could go with Gauss, Disquisitiones Arithmeticae. Don't worry, you don't have to read Latin, it is available in English and other living languages.

Gerry Myerson
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Three books: Euler's Introduction to the Analysis of the Infinite and Foundations of the Differential Calculus both translated by JD Blanton and published by Springer, also the very informative Analysis by its History by Hairer and Wanner. There are always the original papers by the biggies which are more often than not very interesting, illuminating and convey a sense of a firsthand encounter with the author(s).

InTransit
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Euclid's "The Elements". Greek. Old. It doesn't get much more classic than that. As well as his other writings. http://en.wikipedia.org/wiki/Euclid#Other_works

Also, the Principia Mathematica by Newton. As well as his other writings. http://en.wikipedia.org/wiki/Isaac_Newton#Mathematics

Archimedes probably deserves a mention as well. You know, pi and circles and all that. http://en.wikipedia.org/wiki/Archimedes#Writings

Also I've seen Gödel, Escher, Bach mentioned on this page several times. Perhaps not canonically 'classic'. But really an excellent book. Changed the way I've thought about life, science, philosophy, CS, and many other things. It's also probably a good gateway into Gödel's incompleteness theorem.

kingsfoil
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  • "Gödel, Escher, Bach" is outdated, especially when it comes to Artificial Intelligence. And for the rest, it's the predictable standard blather. – Han de Bruijn Jun 12 '15 at 11:21
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There are several Source Books that have made nice selections for you to pick from, e.g., Smith, Struik, Fauvel and Gray, Stedall. But for an extended read, you can do nothing wrong by immersing yourself in Gauss' Disquisitiones.

Per Erik Manne
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  • One of this year's Fields Medalists, Manjul Bhargava, did exactly that during his PhD at Princeton. I understand he went on to greatly improve what Gauss had done and others left untouched for two centuries. – user_of_math Aug 19 '14 at 11:12
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    A related quote by André Weil: "In 1947, in Chicago, I felt bored and depressed, and not knowing what to do, I started reading Gauss's two memoirs on biquadratic residues, which I had never read before (....) This led me in turn to conjectures about varieties over finite fields." – Per Erik Manne Aug 19 '14 at 11:46
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I'm not a mathematician and I'm advocating a secondary source here, but I'm very convinced that this book fully measures up to the greatness of the original classics: My personal favorite is Folland (1999).

I had been a total newbie to measure theory, topology, and functional analysis. It took me about 1.5 years to carefully study the first seven chapters (the book is extremely dense—it's like the Nutella of math textbooks), and it gave me a very deep working knowledge of real analysis. I owe more than 95% of everything I know about advanced mathematics to this fantastic textbook.

In addition, each chapter ends with a refined overview about the most important and distinguished classic articles and books the chapter builds on: It is worth having a look at these notes even if you are more interested in the original sources.

triple_sec
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  • "The Nutella of math books" is a phrase I will now attempt to integrate into my daily conversation every day. (+1) – clathratus Nov 16 '18 at 01:38
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I should say that as far as recreational maths is concerned and you have tagged this question as such, going through this book will be a very satisfying experience: Winning Ways for Your Mathematical Plays (all volumes). Games were not taken seriously by mathematicians as you will recall Euler himself dismissing the graph theory field as not serious maths even after having solved Seven Bridges of Königsberg. This book tries to go that extra mile.

user_1_1_1
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If you want to learn algebraic geometry, a classical paper is Jean-Pierre Serre's FAC. See here for the French original, and here for the Englisch translation. See here for some advertisement by Georges Elencwajg.

Community
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Martin Brandenburg
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Not exactly what you were looking for (because these are "secondary sources"), but maybe interesting nevertheless:

  1. Mathematics and Its History by Stillwell walks you through the history of mathematics showing original problems in modern notation with many good exercises at an undergraduate level and with lots of pointers to the original sources.

  2. Euler - The Master of Us All by Dunham provides a guided tour through the main ideas of many of Euler's original papers. (And fittingly starts with a Laplace quote: "Read Euler! Read Euler!")

Frunobulax
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Categories for the Working Mathematician by Saunders Mac Lane, is not technically primary, since it is not the "founding papers", but it is excellent, broad and exquisitely beautiful. And Saunders Mac Lane is one of the founders of category theory.

Nameless One
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Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter.

Although not strictly speaking a purely mathematical book, surely I would put it among the classics.

Tallmaris
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    I would have to put this down as a fine secondary source, but it isn't a classic in Edwards' sense of the word. – BMeph Aug 19 '14 at 18:55