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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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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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    $\begingroup$ 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! :) $\endgroup$
    – Jamie Banks
    Jul 24, 2010 at 0:48
  • $\begingroup$ @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! $\endgroup$
    – Neil Mayhew
    Jul 24, 2010 at 4:20
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    $\begingroup$ @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. $\endgroup$
    – Mathemagician1234
    Dec 6, 2011 at 8:32
  • $\begingroup$ @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. $\endgroup$
    – nullgeppetto
    Jan 22, 2014 at 15:11
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Polya's "How To Solve It"

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    $\begingroup$ Is there any particular reason why you're recommending this book? $\endgroup$
    – user489
    Jan 23, 2012 at 18:50
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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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    $\begingroup$ 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. $\endgroup$
    – George Frank
    Jun 13, 2014 at 18:42
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Proofs from the Book

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    $\begingroup$ 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. $\endgroup$
    – Pete L. Clark
    Dec 6, 2011 at 9:49
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    $\begingroup$ Is there any particular reason why you're recommending this book? $\endgroup$
    – user489
    Jan 23, 2012 at 18:51
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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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  • $\begingroup$ 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. $\endgroup$
    – Edan Maor
    Jul 21, 2010 at 7:05
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    $\begingroup$ The Mathematical Universe by the same author is also very good. $\endgroup$
    – Neil Mayhew
    Jul 21, 2010 at 17:10
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    $\begingroup$ Dunham's "Euler: The Master of Us All" had that effect on me when I was in high school. $\endgroup$
    – Kevin H. Lin
    Jul 28, 2010 at 19:48
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Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter:

alt text

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    $\begingroup$ 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. $\endgroup$
    – ixtmixilix
    Jul 14, 2011 at 17:07
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    $\begingroup$ @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". $\endgroup$
    – ShreevatsaR
    Sep 22, 2011 at 16:31
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    $\begingroup$ Is there any particular reason why you're recommending this book? $\endgroup$
    – user489
    Jan 23, 2012 at 18:50
  • $\begingroup$ 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. $\endgroup$
    – Olivier Bégassat
    Dec 8, 2013 at 0:47
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    $\begingroup$ 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. $\endgroup$
    – Ron Gordon
    Jan 22, 2014 at 15:09
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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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    $\begingroup$ Free Google eBook $\endgroup$
    – The Chaz 2.0
    Jan 21, 2012 at 2:07
  • $\begingroup$ I should point out there's a movie; we watched it in a 7th grade Algebra I class before spring or summer break last year (can't remember which) - it was very entertaining and well done, I thought, though perhaps a bit little-kiddish. $\endgroup$
    – Auden Young
    Feb 5, 2017 at 23:25
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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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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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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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    $\begingroup$ Although I've never read a Coxeter book from front to back, I've always enjoyed what of his books I have read. $\endgroup$
    – Ryan Budney
    Nov 1, 2010 at 18:23
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T.W.Körner, The Pleasures of Counting. It shows how mathematics is alive.

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    $\begingroup$ ALL of Korner's books show this,mau. Young math students can benefit from careful reading of ANY of the Cambridge master's texts. $\endgroup$
    – Mathemagician1234
    Dec 6, 2011 at 8:52
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Men of Mathematics by E T Bell

picture of book cover

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    $\begingroup$ Yes. Although it gets a lot of criticism these days for its historical inaccuracy and exaggerations, it makes for very interesting and inspiring reading. $\endgroup$
    – ShreevatsaR
    Aug 4, 2010 at 17:15
  • $\begingroup$ I liked the challenges that E.T. Bell posed to the reader. That was a lot of fun back in high school. $\endgroup$
    – Zach Conn
    Aug 5, 2010 at 2:15
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    $\begingroup$ I look forward to the sequel, "Women of Mathematics". $\endgroup$
    – Greg Martin
    Dec 6, 2011 at 9:27
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    $\begingroup$ Chapter I: Emmy Noether Chapter II: Ada Lovelace ... $\endgroup$
    – Pratik Deoghare
    Dec 6, 2011 at 9:41
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    $\begingroup$ Is there any particular reason why you're recommending this book? $\endgroup$
    – user489
    Jan 23, 2012 at 18:52
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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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    $\begingroup$ I haven't read this particular book, but I definitely think geometric algebra is worthwhile for pretty much all mathematicians. $\endgroup$
    – Zach Conn
    Aug 5, 2010 at 2:16
  • $\begingroup$ +1 for a great classic that showed the deep connection between classical geometry and algebra. $\endgroup$
    – Mathemagician1234
    Dec 6, 2011 at 8:33
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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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Following Noah's lead I will mention;

"The Man Who Loved Only Numbers"

and

"How to Read and Do Proofs"

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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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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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    $\begingroup$ I think both are a waste of time for a working mathematician. $\endgroup$
    – BBischof
    Jul 21, 2010 at 14:50
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    $\begingroup$ That's right, because neither Euclid or Newton contributed anything to maths. $\endgroup$
    – Noldorin
    Jul 21, 2010 at 14:52
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    $\begingroup$ 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). $\endgroup$
    – Akhil Mathew
    Jul 21, 2010 at 23:15
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    $\begingroup$ 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. $\endgroup$
    – Noldorin
    Jul 22, 2010 at 7:28
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    $\begingroup$ 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. $\endgroup$
    – user13618
    Feb 8, 2012 at 17:30
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Probability Theory: the Logic of Science.

Or anything by Edwin T Jaynes.

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    $\begingroup$ Is there any particular reason why you're recommending this book (and author)? $\endgroup$
    – user489
    Jan 23, 2012 at 18:53
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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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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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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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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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Nicolas Bourbaki's Éléments de mathématique (specifically Topologie Générale and Algèbre).

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    $\begingroup$ Is there any particular reason why you're recommending this book? $\endgroup$
    – user489
    Jan 23, 2012 at 18:51
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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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    $\begingroup$ 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 $\endgroup$
    – Zeta.Investigator
    Sep 3, 2012 at 13:52
  • $\begingroup$ Your third link doesn't work. $\endgroup$
    – Julian Kuelshammer
    Nov 11, 2012 at 9:43
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Every undergrad should read in areas outside mathematics especially in areas that can be influenced by mathematics. Theoretical physics and computer science are prominent examples. Biology and chemistry are not far. The DNA and polymers can be understood using knot theory and feynman's path integrals . Feynman's path integrals facilitated the quantization of nonabelian gauge field theories ( Quantum chromodynamics ) and is used to study complex systems and stochastic processes.

So here are two books that I found very interesting :

The road to reality by Roger Penrose Kleinert's path integrals in physics , financial markets & Stochastic processes

There are lots of books that talk about the applications of math in physical science just search Amazon.

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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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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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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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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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