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If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?

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20 answers to the question about the single most influential book are an indicator that the question might not be well-posed. I am therefore voting to close. –  Alex B. Feb 15 '11 at 6:07
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I don't understand Alex B.'s comment. It is clear that different people would give different answers of what the single most influential book would be to them, and that is why it is a big-list community wiki question. –  Jonas Meyer Dec 30 '11 at 4:05
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I'd love to contribute an answer here but before I do can you tell me how you're defining "beginning of your career"? Do you mean the beginning of your math college/graduate math education? Or do you mean the beginning of your first post-doc? –  Schmitty Feb 8 '12 at 16:43
    
For me this book is definitely Mac Lane's Categories for the working mathematician. It showed me that mathematics is a unity. But I actually read this when I was young, so probably this doesn't fit to the question (which has been misunderstood already a couple of times). –  Martin Brandenburg Dec 8 '13 at 0:36

29 Answers 29

A Mathematician's Apology by G H Hardy. I did in fact read this in high school, and it raised my view of mathematics from a thing of utility to a thing of beauty and wonder. It inspired me to go on to study mathematics at Cambridge myself.

It's a pity that the "introduction" by C P Snow is longer than the original and contains a rather depressing view of Hardy's later life. I would recommend readers to skip the introduction altogether and concentrate on Hardy's own words.

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I would recommend this, but ignore anything about it that turns you off from mathematics! In particular, not all mathematicians believe that you can't be good at math past your young adulthood and that teaching, writing or applying math are a waste! :) –  Jamie Banks Jul 24 '10 at 0:48
    
@Katie: Yes, good point, I agree. I guess I just didn't pay much attention to those aspects when I was young, since it all seemed a long way off! –  Neil Mayhew Jul 24 '10 at 4:20
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@Katie Hardy was a bitter, frustrated researcher who'd had been creatively blocked for years when he wrote this and it shows. There are so many counterexamples to the myth about math and age he created in this work. –  Mathemagician1234 Dec 6 '11 at 8:32
    
@Mathemagician1234, this is true! The connection between age and math creativity by its own would be far away from inspiration, the way the original question refered to. –  nullgeppetto Jan 22 at 15:11

Polya's "How To Solve It"

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Is there any particular reason why you're recommending this book? –  user489 Jan 23 '12 at 18:50

When I was in my fourth year of high school I got a copy of What is Mathematics? by Courant and Robbins. That book showed to me that Mathematics is far more than a "boring tool" to do Physics and opened up new worlds. I would recommend it to any bright high school kid with an interest in math and sciences.

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+1 I love that book! –  becko Jun 11 '12 at 0:00
    
An intelligent math major who got a PhD from Columbia took an entire year to get through just the first chapter with doing all the problems. A year! Long ago I tried and gave up. It is definitely not an innocent's book. –  George Frank Jun 13 at 18:42

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

alt text

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i downvoted, because i don't like that book. its popularity is/was a bit of a fad, if you ask me, and there's nothing very mathematical about it. –  ixtmixilix Jul 14 '11 at 17:07
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@ixtmixilix: I loved the book, but I think it's worth reading for the clever puns and funny self-referential dialogues. It also has some much-needed (but obvious) rejoinders to philosophers like Lucas and Penrose. It also has an extended, and fairly good, treatment of the proof of Gödel's theorem; I wouldn't say "there's nothing very mathematical about it". But I agree that it's not a "book every mathematician should read". –  ShreevatsaR Sep 22 '11 at 16:31
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Is there any particular reason why you're recommending this book? –  user489 Jan 23 '12 at 18:50
    
I liked the book (and thus upvoted). It's been years since I read it, but I remember working out some calculations with a recursive function $G$ that I thought were very cool back then. –  Olivier Bégassat Dec 8 '13 at 0:47
    
I absolutely loved this book for the word play, puns, dialogs, bibliography, and typography. If you want a nice, easy-to-follow proof of Goedel's Incompleteness theorem, I would go with Newman & Nagel's Goedel's Proof. –  Ron Gordon Jan 22 at 15:09

William Dunham's "Journey through Genius."

Well, rather that is the book I read that made me want to be a mathematician.

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This is the best book for non-mathematicians, to show them real, beautiful mathematics. A lot about the history of mathematics, but it actually has real proofs inside, not only history. I liked maths before this book, but this took my love to a whole new level. –  Edan Maor Jul 21 '10 at 7:05
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The Mathematical Universe by the same author is also very good. –  Neil Mayhew Jul 21 '10 at 17:10
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Dunham's "Euler: The Master of Us All" had that effect on me when I was in high school. –  Kevin H. Lin Jul 28 '10 at 19:48

Proofs from the Book

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Solomon Lefschetz is purported to have said, "Don't come to me with your pretty proofs. We don't bother with that baby stuff around here." I don't generally agree with him, but I do a bit when I read Aigner's book: organizing a text around not theorems but pretty proofs results in a certain preciousness. With a few notable exceptions (the proof of Two Squares via Thue's Lemma has become inspirational to me, although not immediately when I read it there) the number theory section was rather disappointing. –  Pete L. Clark Dec 6 '11 at 9:49
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Is there any particular reason why you're recommending this book? –  user489 Jan 23 '12 at 18:51

I am not a mathematician but Flatland: A Romance of Many Dimensions blew my mind. I read it when I was a college student in a class on Special Relativity and wish I had read it way earlier.

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Free Google eBook –  The Chaz 2.0 Jan 21 '12 at 2:07
    
@TheChaz Thanks! –  becko Jun 11 '12 at 0:06

Anybody who wants to be a serious mathematician better read W. Rudin's "Principles of mathematical Analysis". It gives a rigorous foundation to the basic notions analysis and introduces the reader to the world of rigor, after the sloppy days of calculus courses. One must learn the notion of rigor properly if one wants to be a mathematician. More than anything else, it is an exercise in the rectitude of thought. No other book is so universally used that would teach this notion, than Rudin.

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It hopefully will be replaced for this purpose eventually by the far superior book by Charles Chapman Pugh,Line. –  Mathemagician1234 Dec 6 '11 at 8:34
    
@Math: Whence my suspicions of a literary agenda... –  The Chaz 2.0 Jan 21 '12 at 2:08

Not a book, but an essay: "Politics and the English Language" by George Orwell.

What? What?

(I note that the original question doesn't say that the book has to help with mathematics. It also seems to conflate 'influential' with 'should be read'; as others have pointed out, there is no pressing reason for someone who wants to be a mathematician to read the influential books rather than the useful or the interesting ones.)

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T.W.Körner, The Pleasures of Counting. It shows how mathematics is alive.

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ALL of Korner's books show this,mau. Young math students can benefit from careful reading of ANY of the Cambridge master's texts. –  Mathemagician1234 Dec 6 '11 at 8:52

This is an extremely broad question, especially given the wide variety of mathy people here, but I'll bite.

HSM Coxeter's Introduction to Geometry is a book that was very important to the development of my interest in mathematics and inclination towards its geometric aspects.

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Although I've never read a Coxeter book from front to back, I've always enjoyed what of his books I have read. –  Ryan Budney Nov 1 '10 at 18:23

Men of Mathematics by E T Bell

picture of book cover

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Yes. Although it gets a lot of criticism these days for its historical inaccuracy and exaggerations, it makes for very interesting and inspiring reading. –  ShreevatsaR Aug 4 '10 at 17:15
    
I liked the challenges that E.T. Bell posed to the reader. That was a lot of fun back in high school. –  Zach Conn Aug 5 '10 at 2:15
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I look forward to the sequel, "Women of Mathematics". –  Greg Martin Dec 6 '11 at 9:27
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Chapter I: Emmy Noether Chapter II: Ada Lovelace ... –  Pratik Deoghare Dec 6 '11 at 9:41
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Is there any particular reason why you're recommending this book? –  user489 Jan 23 '12 at 18:52

Following Noah's lead I will mention;

"The Man Who Loved Only Numbers"

and

"How to Read and Do Proofs"

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Is there any particular reason why you're recommending these books? –  user489 Jan 23 '12 at 18:52

I've been rereading Littlewood's Miscellany recently. It's a very readable collection of the writings of J. E. Littlewood, carefully edited by Béla Bollobás. Any budding mathematician will draw much inspiration from it. I like A Mathematician's Apology, but if I was forced into choosing only one book, it would be Littlewood's Miscellany.

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I'll recommend two, which are similar in that they take fairly elementary mathematical problems and give very thorough and careful "talking out loud" illustrations of how a proper mathematician would go about thinking them through - what's really going on, what's a good example, what's a definitive counterexample, how to generalise, how to realise you've reached a dead end, and so on. "Proofs and refutations" by Imre Lakatos (just one, geometrical, problem, in glorious detail). "Mathematics and plausible reasoning Vol 1" by G. Polya (a little more advanced, and much more satisfying, than "How to solve it").

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Geometric Algebra by Emil Artin. Though not for the beginner, it can do wonders for an intermediate undergraduate in terms of expanding their horizons and helping them appreciate the beauty and interconnectedness of mathematics. It did for me and I think convinced me that I'm a geometer at heart.

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I haven't read this particular book, but I definitely think geometric algebra is worthwhile for pretty much all mathematicians. –  Zach Conn Aug 5 '10 at 2:16
    
+1 for a great classic that showed the deep connection between classical geometry and algebra. –  Mathemagician1234 Dec 6 '11 at 8:33

Probability Theory: the Logic of Science.

Or anything by Edwin T Jaynes.

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Is there any particular reason why you're recommending this book (and author)? –  user489 Jan 23 '12 at 18:53

Recommending one single book at the beginning of a young mathematician's career is a little like asking someone what particular vitamin they should make sure is in a child's diet. It's absurdly restrictive.

That being said-there are certainly 3 books I would recommend without reservation to any young student just getting interested in serious mathematics: Micheal Spivak's Calculus, Klaus Janich's remarkable Topology and Paul Halmos' I Want To Be A Mathematician.

The last one in particular inspired me to leave pre-med to begin the path to be a mathematician. The other 2 are remarkable works that will begin to open the edifice of modern mathematics to the novice.

I can recommend a hundred others,but those are the absolute must-reads for the beginner to me.

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Title: The Mathematical Experience

Authors: Davis and Hersh

Short Description: A really accessible and funny introduction to the philosophy of mathematics. I think the description of the "ideal mathematician" is particularly hilarious.

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Visual Complex Analysis by Tristam Needham.

I always like to see mathematical problems in pictures whenever I can, and this one pushes the 'keep it visual' approach to the limits.

Needham won an award for some of the work in there.

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Euclid's Elements

Newton's Principia Mathematica

Ideally in the original languages of Ancient Greek and Latin respectively! No, just kidding. But they are true classics that any accomplished mathematician should read at some point during their career. Not because they'll teach you something you don't already know, but they provide a unique insight into the mind of these giants.

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I think both are a waste of time for a working mathematician. –  BBischof Jul 21 '10 at 14:50
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That's right, because neither Euclid or Newton contributed anything to maths. –  Noldorin Jul 21 '10 at 14:52
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I think the point stands that studying Euclid or Newton is likely to be an inefficient way to learn the subjects. There are modern books that can give the presentation more efficiently and elegantly (e.g., Euclid didn't have algebraic notation). "Studying the classics" in mathematics is often a bad idea (not least because they didn't have LaTeX in the Old Days). –  Akhil Mathew Jul 21 '10 at 23:15
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Fair point, especially regarding 'inefficiency', but disagreed in general. Getting an insight into the minds of the masters is a valuable thing to me - providing you have the time! Then again, I'm also a (very amateur) historian, and thus value it in that respect too. –  Noldorin Jul 22 '10 at 7:28
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Also re Euclid, I agree that it would be pointless for almost any modern person to slog through 100% of the Elements. For example, Euclid does number theory in geometrical notation, and that is not something that anyone but a historian of ancient mathematics will care to read in detail. But any mathematcian who hasn't read at least the first 47 propositions of the Elements is like a playwright who has never read Shakespeare. –  Ben Crowell Feb 8 '12 at 17:30

Nicolas Bourbaki's Éléments de mathématique (specifically Topologie Générale and Algèbre).

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Is there any particular reason why you're recommending this book? –  user489 Jan 23 '12 at 18:51

the man who loved only numbers, innumeracy, a beautiful mind. these three books have shaped my thinking and love of mathematics...books on math..not exactly a lot of math in them however.

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I didn't read "a beautiful mind" book butI don't like the "A Beautiful Mind" movie. I don't recommend anyone to waste his time on that movie –  Zeta.Investigator Sep 3 '12 at 13:52
    
Your third link doesn't work. –  Julian Kuelshammer Nov 11 '12 at 9:43

I would read a book about Perelman's proof of the Poincaré conjecture (or even the papers themselves). Oh, you mean the book had to be written when I was starting?

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dummit and foote's abstract algebra

it taught me, more than anything, how to be precise.

disclaimer: i'm computer science not math

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This question does not have a unique answer. I will concur with Jonathan in that Jayne's "Probability Theory: the logic of science" is a great book.

This book changed my life as a scientist, converting me into a fervent Bayesian. For me it was a truly irreversible experience when I, for the first time, understood and comprehended that probability (as applied to understanding the real physical world) essentially stems from our lack of knowledge, our incomplete information, of reality. Fantastic book, although I admit that Jayne's style might not suit everyone's taste.

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god created the integers by Stephen Hawking is the best book...... for mathematics.

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There are so many, and I've already seen three that I would mention. Two more of interest to lay readers:

The Man Who Knew Infinity by Robert Kanigel. Excellently written, ultimately a tragedy, but a real source of inspiration.

Goedel's Proof by Nagel & Newman. Really, a beautiful and short exposition of the nature of proof, non-euclidean geometry, and the thinking that led Goedel to his magnificent proof.

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I do recommend: Hugo Steinhaus - one hundred problems in elementary mathematics Hamilton - Perelman's proof of the Poincare conjecture and the geometrization conjecture

Also there are many great books written by polish authors (especially Kuratowski, Banach, Mostowski, Steinhaus, Leja) but I am not sure is it available in english language.

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