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 What is the single most influential book every mathematician should read?

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?

 (soft-question) (big-list) (reference-request)

edited Jul 21 '10 at 6:22 community wiki2 revs, 2 users 100%c4il

18  

–

20 answers to the question about the most influential book are an indicator that the question migh t

not be well-posed. I am ther efore votin g to close.single

 Al ex B. Feb 15 '11 at 6:07

15  

–

I don't understand Alex B.'s comment. It is clear that different people would giv e different a nswers of what

the most influential book would be to them , and that is why it is a big-list commu nity wiki question.single

Jonas Meyer Dec 30 '11 at 4:05

3  

–

I'd love to contribut e an an swer here but before I do can y ou tell me h ow y ou're defining "beginning of y our

career"? Do you mean the beginning of your math college/graduate math education? Or do you mean the

 begi nn in g of y our fir st post-doc? Schmitty  Feb 8 '12 at 16 :43

 

–

For me this book is definitely Mac Lane's . It showed me that

mat hema tics is a . But I actu ally read this when I was y oung, so probably this doesn't fit to the

question (which has been misunderstood already a couple of times).

Categories for the working mathematician

unity

Martin Brandenburg Dec 8 '13 at

0:36

28 Answers

 by G H Hardy. I did in fact read this in high school, and it raised y view of mathematics from a thing of utility to a thing of beauty  and wonder. It inspirede to go on to study mathematics at Cambridge myself.

 Mathematician's Apology 

t's a pity that the "introduction" by C P Snow is longer than the original and contains a ratherepressing v iew of Hardy's later life. I would recommend readers to skip the introductionltogether and concentrate on Hardy's own w ords.

ans wered Jul 21 '10 at 17 :07 community wikiNeil Mayhew 

7  

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I would recommend this, but ignore anything about it that turns you off from mathematics! In

particular, not all math ematicians believe that y ou can't be good at ma th past your y oung adulthood and

that teaching, wr iting or applying m ath are a w aste! :) Jamie Banks Jul 24 '10 at 0:48

 

–

@Katie: Yes, good point, I agree. I guess I just didn't pay mu ch a ttention to th ose aspects when I was

 y oun g, sinc e it all seem ed a l ong wa y off! Neil Mayhew Jul 24 '10 at 4:20

 

–

@Katie Hardy was a bitter, frustrated researcher who'd had been creatively blocked for years when he

 wr ote th is an d it show s. Th ere a re so m an y coun ter exa mp les to t he m y th abou t m at h a nd a ge h e cr eat edin this work. Mathemagician1234 Dec 6 '11 at 8:32

 

–

@Mathemagician123 4, th is is true! The connection between age and m ath creativ ity by its own would be

far away from inspiration, the way the original question refered to. nullgeppetto Jan 22 at 15:11

olya's "How To Solve It"

answered Jul 21 '10 at 5:57 community wikiIsaac

  –Is there any particular reason why you're recommending this book? user489 Jan 23 '12 at 1 8:50

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

answered Au g 4 '10 at 1 7 :06 community wikiChristian

0  

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i downv oted, because i don't like th at book. its popularit y is/was a bit of a fad, if y ou ask me, a nd there's

nothing very m athematical about it. ixtmixilix Jul 14 '11 at 17:07

 

–

@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. Italso has an ext ended, and fairly good, tr eatment 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

  –Is there any particular reason why you're recommending this book? user489 Jan 23 '12 at 1 8:50

 

–

I liked the book (and thu s upv oted). It's been year s since I read it, bu t I remember w orking out some

calculations with a recursive function that I thought were very cool back then.G       Olivier Bégassat Dec8 '13 at 0:47

 

–

I absolutely lov ed this book for the word play, pun s, dialogs, bibliography , and ty pography . If y ou 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 1 5:09

hen I was in my fourth year of high school I got a copy of by ourant and Robbins. That book showed to me that Mathematics is far more than a "boring

ool" to do Physics and opened up new worlds. I would recommend it to any bright high schoolid with an interest in math and sciences.

What is Mathematics? 

answered Au g 4 '10 at 22 :08 community wiki Andre a Mori

  –+1 I love that book!  beck o Jun 11 '12 at 0:00

illiam Dunham's "Journey through Genius."ell, rather that is the book I read that made me want to be a mathematician.

answered Jul 21 '10 at 5:47 community wikiNoah Snyder

 

–

This is the book for non-math ematici ans, to show them real, beautiful mathem atics. A lot about thehistory of math ematics, but it actua lly has real proofs inside, not only history . I liked math s before this

 book, bu t t his took m y lov e to a wh ole new lev el.

best 

Edan Maor Jul 21 '10 at 7 :05

  –The Mathematical Universe by the same author is also very good. Neil Mayhew Jul 21 '10 at 17 :10

  –Dunha m's "Euler: The Master of Us All" had th at effect on me wh en I was in hig h school. Kevin H. Lin Ju l

28 '10 at 19:48

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roofs from the Book 

edited Dec 6 '11 at 16:10 community wiki2 revs, 2 users 67%hmonroe

2  

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Solomon Lefschetz is purported to hav e said, "Don't come to m e with y our pretty proofs. We don't bother

 wi th th at baby stu ff ar oun d her e." I don't gen era lly ag ree w ith hi m, but I do a bit wh en I r ead A ig ner 's

 book: org an izin g a tex t a rou nd n ot th eorem s bu t pr ett y proofs resu lts i n a cer ta in p rec iousn ess. With a few 

notable exceptions (the proof of Two Squares v ia Th ue's Lemm a ha s become inspirat ional to me, alt houghnot immediately when I read it there) the number theory section was rather disappointing.

Pete L. Clar k Dec 6 '11 a t 9:49

  –Is there any particular reason why you're recommending this book? user489 Jan 23 '12 at 1 8:51

 am not a mathematician but blew my mind. Iead it when I was a college student in a class on Special Relativity and wish I had read it way arlier.

Flatland: A Romance of Many Dimensions

answered Jul 28 '10 at 1 5:46 community wikipuri

  –Free Google eBook The Chaz 2.0 Jan 21 '12 at 2 :07

  –@TheChaz Thanks!  beck o Jun 11 '12 at 0:06

nybody who wants to be a serious mathematician better read W. Rudin's "Principles of athematical Analysis". It gives a rigorous foundation to the basic notions analysis and

ntroduces the reader to the world of rigor, after the sloppy days of calculus courses. One mustearn the notion of rigor properly if one wants to be a mathematician. More than anythinglse, it is an exercise in the rectitude of thought. No other book is so universally used thatould teach this notion, than Rudin.

answered Au g 5 '10 at 17 :13 community wiki

user218

 

–

It hopefully w ill be replaced for this purpose ev entuall y by the far superior book by Char les 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

ot a book, but an essay: "Politics and the English Language" by George Orwell.

hat? What?

I note that the original question doesn't say that the book has to help with mathematics. It

lso seems to conflate 'influential' with 'should be read'; as others have pointed out, there is noressing reason for someone who wants to be a mathematician to read the influential booksather than the useful or the interesting ones.)

answered Sep 22 '11 at 5:22 community wikiuser16299

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

answered Jul 21 '10 at 13:03 community wikimau

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 ALL of Korner 's books show t his,m au . Y oun g m at h st uden ts ca n be nefi t fr om c ar eful rea din g of AN Y of th e

Cambr idge master's texts. Mathemagician1234 Dec 6 '11 at 8:52

en of Mathematics by E T Bell

edited Dec 1 1 '11 at 14 :25 community wiki2 revs, 2 users 86%Pratik Deoghare

 

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 Yes. A lth oug h i t g ets a lot of cr iti cism th ese day s for it s histori ca l in ac cu ra cy an d exa gg era tion s, it m ake s

for very interesting and inspiring r eading. ShreevatsaR  Au g 4 '10 at 1 7 :1 5

  –I liked the challenges that E.T. Bell posed to the reader. Tha t wa s a lot of fun bac k in h igh school.Zach Conn Au g 5 '10 at 2:1 5

1   –I look forward to the sequel, "Women of Mathematics". Greg Martin Dec 6 '11 a t 9:27

  –Chapter I: Emmy Noether Chapter II: Ada Lovelace ... Pratik Deoghare Dec 6 '11 at 9:41

  –Is there any particular reason why you're recommending this book? user489 Jan 23 '12 at 1 8:52

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

SM Coxeter's is a book that was very important to theevelopment of my interest in mathematics and inclination towards its geometric aspects.

 Introduction to Geometry

answered Jul 21 '10 at 6:28 community wikiJamie Banks

 

–

 Al th oug h I'v e nev er r ead a Coxet er book from fron t t o bac k, I'v e al wa y s enjoy ed wh at of his books I hav e rea d.

Ryan Budney  Nov 1 '10 at 1 8:23

ollowing Noah's lead I will mention;

The Man Who Loved Only Numbers"

nd

How to Read and Do Proofs"

answered Jul 21 '10 at 5:57 community wikiBBischof 

  –Is there any particular reason why you're recommending these books? user489 Jan 23 '12 at 1 8:52

've been rereading recently. It's a very readable collection of theritings of J. E. Littlewood, carefully edited by Béla Bollobás. Any budding mathematician will

raw much inspiration from it. I like , but if I was forced into

 

 Littlewood's Miscellany

 A Mathematician's Apology

'

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  , .

answered Au g 5 '10 at 18:1 8 community wikiBaudrillard

'll recommend two, which are similar in that they take fairly elementary mathematical

roblems and give very thorough and careful "talking out loud" illustrations of how a properathematician would go about thinking them through - what's really going on, what's a good

xample, what's a definitive counterexample, how to generalise, how to realise you've reached

 dead end, and so on. "Proofs and refutations" by Imre Lakatos (just one, geometrical,roblem, in glorious detail). "Mathematics and plausible reasoning Vol 1" by G. Polya (a littleore advanced, and much more satisfying, than "How to solve it").

answered Au g 4 '10 at 20 :12 community wiki

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

eometric Algebra

answered Au g 4 '10 at 21 :20 community wiki Vladim ir Sotir ov 

 

–

I haven't read this particular book, but I definitely think geometric algebra is worthwhile for pretty much

all mathemat icians. Zach Conn Au g 5 '1 0 a t 2 :1 6

  –+1 for a gr eat classic that showed the deep connection between classical geometry and algebra.

Mathemagician1234 Dec 6 '11 at 8:33

robability Theory: the Logic of Science.

r anything by .Edwin T Jaynes

edited Dec 6 '11 at 8:16 community wiki2 revs, 2 users 67%Jonathan Fischoff 

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

18:53

itle: The Mathematical Experience

uthors: Davis and Hersh

hort Description: A really accessible and funny introduction to the philosophy of 

athematics. I think the description of the "ideal mathematician" is particularly hilarious.

edited Dec 1 1 '11 at 14 :26 community wiki2 revs, 2 users 84%user7104

uclid's Elements

ewton's Principia Mathematica

deally in the original languages of Ancient Greek and Latin respectively! No, just kidding. Buthey are classics that any accomplished mathematician should read at some point duringheir career. Not because they'll teach you something you don't already know, but they provide

 uni ue insi ht into the mind of these iants.

true

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edited Jul 28 '10 at 1 5:50 community wiki2 revsNoldorin

0   –I think both are a waste of time for a working mathematician. BBischof Jul 21 '10 at 14 :50

  –That's right, because neither Euclid or Newton contributed anything to maths. Noldorin Jul 21 '10 at

14:52

9  

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I think the point stands that study ing Euclid or Newton is likely to be an inefficient way to learn thesubjects. There are modern books that c an giv e the presentat ion more efficiently a nd elegantly (e.g.,

Euclid didn't hav e algebraic notation). "Study ing th e classics" in ma thema tics is often a bad idea (not least beca use t hey didn 't hav e LaT eX in th e Old Da y s).  Ak hi l Ma th ew Jul 21 '10 at 23:15

 

–

Fair point, especially r egarding 'inefficiency', but disagreed in general. Getting an in sight into the min ds

of the masters is a valuable thing to me - providing you have the time! Then again, I'm also a (very amateur) historian, and t hus v alue it in that r espect too. Noldorin Jul 22 '10 at 7:28

 

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 Yes, rea din g t he "cl assic s" in m at hem at ics i s mor e of histori ca l in ter est (how did t he m ast ers t hink abou t

the math ematics they did) than of mathematical interest (how worthwh ile will it be to a working

mathematician). ShreevatsaR Sep 22 '11 at 6:28

ecommending one single book at the beginning of a young mathematician's career is a littleike asking someone what particular v itamin they should make sure is in a child's diet. It's bsurdly restrictive.

hat being said-there are certainly 3 books I would recommend without reservation to any oung student just getting interested in serious mathematics: Micheal Spivak's , Klausanich's remarkable and Paul Halmos' .

Calculus

Topology I Want To Be A Mathematician

he last one in particular inspired me to leave pre-med to begin the path to be aathematician. The other 2 are remarkable works that will begin to open the edifice of 

odern mathematics to the novice.

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

answered Dec 6 '11 at 9:02 community wikiMathemagician1234

isual Complex Analysis by Tristam Needham.

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

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

answered Feb 8 '12 at 16:25 community wikiRobert

, , . these three books havehaped my thinking and love of mathematics...books on math..not exactly a lot of math inhem however.

he man who loved only numbers innumeracy a beautiful mind

edited Dec 6 '11 at 16:03 community wiki2 revs, 2 users 67%cody 

 –

I didn't read "a beautiful mind" book but I don't like the "A Beautifu l Mind" movie. I don't recomm end any one to wa ste h is ti m e on t ha t m ov ie Zeta.Investigator Sep 3 '12 at 1 3:52

  – Your th ir d lin k doesn't wor k. Julian Kuelshammer Nov 1 1 '12 at 9:43

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icolas Bourbaki's Éléments de mathématique (specifically Topologie Générale andlgèbre).

answered Jul 21 '10 at 6:32 community wikiuser126

  –Is there any particular reason why you're recommending this book? user489 Jan 23 '12 at 1 8:51

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

answered Jul 28 '10 at 1 5:28 community wiki y atima 297 5

ummit and foote's abstract algebra

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

isclaimer: i'm computer science not math

answered Au g 5 '12 at 2 1:5 3 community wikisam boosalis

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

Jonathan

his book changed my life as a scientist, converting me into a fervent Bayesian. For me it was truly irreversible experience when I, for the first time, understood and comprehended thatrobability (as applied to understanding the real physical world) essentially stems from our

ack of knowledge, our incomplete information, of reality. Fantastic book, although I admithat Jayne's style might not suit everyone's taste.

answered May 25 '13 at 9:42 community wikiCinco Sauces

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

answered Jan 22 at 15:05 community wikiuser114917

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

ay readers:

 by Robert Kanigel. Excellently written, ultimately a tragedy, but real source of inspiration.

The Man Who Knew Infinity

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

answered Jan 22 at 15:13 community wikiRon Gordon