I am a student and I would love to become a research mathematician one day.

So I would like to ask you---experts in mathematics but also in education---what are some influential ($\star$) books that I (and everyone in my situation) should ($\star \star$) read.


$\bigstar$ By that, I mean: (1) books that you think may help to develop positively the way one thinks about mathematics (in general or some areas); (2) books that help developing intellectual and mathematical maturity; (3) books that are in any way inspirational; (4) books that are useful because they offer (generally) unknown way to look at things in maths.

Note: These books do not have to be strictly mathematical (that is, textbooks).

$\bigstar \bigstar$ I know that question is rather subjective and quite broad, but I do think that it may be helpful for this site and I honestly hope that you will share your experiences, which may be extremely precious for many people. I welcome any suggestion to edit, improve (possibly retag) the question.

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    In my opinion the most inspiring book is André Weil's "Souvenir d'apprentissage" [English translation: The Apprenticeship of a Mathematician]. But I feel I cannot rationalize this enough to give it as answer. – quid Oct 23 '14 at 22:47
  • @quid I've looked at it: it is a great suggestion. Thank you very much. – Dal Oct 24 '14 at 12:51
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    See also quora.com/…. – J W Oct 25 '14 at 8:10
  • @JW Ok thank you. – Dal Oct 25 '14 at 14:30
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    See the MSE question What is the single most influential book every mathematician should read? located at math.stackexchange.com/q/262. – Benjamin Dickman Nov 8 '14 at 5:11

11 Answers 11

up vote 9 down vote accepted
+50

Here goes nothing. I have many other suggestions that are not listed; there is always one more example of a great math book:

"The enjoyment of Mathematics" H. Rademacher, O. Toeplitz. You can't beat this classic collection of serious (and beautiful) results with elementary proofs.

"A mathematician's miscellany" J. Littlewood. This is much more advanced, but it is a wonderful collection of math, anecdotes, and insights into the life of a professional mathematician. Sometimes it has a very pedantic tone, but always, always profoundly human.

"Development of mathematics in the 19th century" F. Klein. You can browse and browse and always extract a bit more insight into what those mythical giants were thinking about when they created modern mathematics.

Anything by M. Gardner. Our enterprise is a serious one, but its recreational side is a major source of inspiration and ideas. The older books are better in the sense that they portray better the network of curious minds that kept in contact through Gardner in the primitive pre-internet ages.

"Goedel, Escher and Bach" D. Hofstadter plus anything by R. Smullyan. Someone said that Goedel's Theorem is the one result from 20th century Mathematics that every mathematician should know.

Anything by Milnor. I have said it elsewhere. If you want to learn to write better, absorb the style of the masters.

Specific topics:

"Three-dimensional geometry and topology vol. 1" W. Thurston. Just open at random and swim into an ocean of ideas. The original Thurston notes are already available in the web. They are very different from the book, so you can swim into two connected oceans.

"Complex analysis: an introduction" L. Ahlfors. Although the language feels now a little old, you can open almost any page and find jewels of exposition. I am always surprised by the incredible precision in the choice of proofs, exercises, topics, and topic order. It is an incredibly methodic book and very thorough.

"Differential equations, dynamical systems and linear algebra" M. Hirsch, S. Smale. I was told to study by hunting for the many, many, typos. The newest edition has been cleaned, updated, and expanded by Devaney, but it remains a great book.

"Ordinary differential equations" V. Arnold. Even if this is not your thing, you should read at least the first three chapters.

"Introduction to commutative algebra" M. Atiyah, I. MacDonald. A short book, and very readable (at least the first few chapters).

  • Thank you for the advice! – Dal Nov 2 '14 at 10:28

I recommend How to Solve It by George Polya. I merely glanced at it in high school and it helped me to work on math problems. More recently I bought it and now read the whole thing annually, and it now governs how I help my students to solve problems.

This book definitely meets your first three criteria. Although it tries to be broader than just mathematics, it inspires you to both find answers to problems and to develop proofs that are both sound and intuitive. It definitely helps in mathematical maturity and it inspires. The fourth point is debatable. The basic techniques on How to Solve It are no longer "unknown" since so many other books were inspired by it. But the depth of Polya's work, based on his own significant mathematics work, goes beyond the other books. The book has a great balance of accessibility for all and depth for the mathematician.

  • +1 I had already read it, but I agree that it is a great book. Thank you very much for your answer, Professor :) – Dal Oct 25 '14 at 14:28
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    @Dal, Rory: My understanding is that the successor to Polya's books is Schoenfeld's (1985) Mathematical Problem Solving. If you enjoy Polya's work, then I would suggest Schoenfeld's, as well. – Benjamin Dickman Oct 25 '14 at 23:25
  • @BenjaminDickman: Thanks for the suggestion. I'll look into it. – Rory Daulton Oct 25 '14 at 23:48
  • @BenjaminDickman: Wow! A new copy of that book at Amazon.com costs $692.96! Central Pennsylvania public libraries do not have it. Do you have any suggestions on how to get the book at a reasonable cost? – Rory Daulton Oct 26 '14 at 0:55
  • Yikes! I would suggest requesting your library purchase a copy, or checking local universities (if you have access) to see whether they have one. I see used copies on Amazon for less, but it is still about $200 (too much, though it is a great addition to a collection). More generally, you might look to Schoenfeld's various papers; there seems to be a somewhat nice report here: files.eric.ed.gov/fulltext/ED218124.pdf#1 – Benjamin Dickman Oct 26 '14 at 3:38

Two very good books, inspiring autobiographies:

1) "A mathematician grappling with his century", by Laurent Schwartz. Another great french mathematician. This book is even better, in my opinion than the one of André Weil, "The Apprenticeship of a Mathematician", which is already very interesting.

2) "I Want to Be a Mathematician: An Automathography", by Paul R. Halmos.

  • +1 I had already read the second one, which is very good. I will have a look at the first one :) – Dal Oct 25 '14 at 14:29
  • Don't forget to tell me what you think of it ;-) – popo Oct 25 '14 at 20:34

As a counterpoint to Polya: The psychology of invention in the mathematical field by Hadamard.

As much as we all love Polya, Hadamard's study indicates that mathematicians don't often think in language...with Polya as a notable exception.

I would also recommend the Princeton Companion to Mathematics.

You've asked for a book, but the most inspiring piece of writing about mathematics I've ever read is William Thurston's On proof and progress in mathematics. The title is roughly self-explanatory, but particularly helpful to aspiring teachers of mathematics are his insights on the way the subject is communicated.

"A History of School Mathematics" edited by George M. A. Stanic and Jeremy Kilpatrick is a must for any mathematics educator. I understand that you plan on becoming a research mathematician, but even if you do achieve your ultimate goal, at some point you will probably also be in front of the classroom as a teacher. The book is more a collection of essays on topics that are organized in a chronological and cohesive way, and really only focuses on North American schools, but it is still valuable.

Plus, you can buy it here for relatively cheap (about a nickel a page).

  • You are right, and I'll surely have a look at it. Thank you :). – Dal Nov 2 '14 at 10:28

Assuming that you have still not started college, here is a list that I had read in my school days and pre college days which influenced me a lot and for a large part were the reason I choose to become a mathematician:

  1. Men of Mathematics by E. T. Bell; the best biographies on mathematicians that I have ever read, this two volume book is still in print and is still a classic. You should definitely read this.

  2. What is Mathematics? by Corant and Robbins; written by two master mathematicians, this book contains everything that you have ever encountered in math and much more. Ideally suited for any young student eager to see the world beyond the school curriculum.

  3. Dr. Riemann's Zeroes by Karl Sabbagh; a book on a famous problem which discusses many connections with different branches of math and physics. It will definitely show you that math is not a disjoint union of different subjects but is woven in a thinly knit fabric which encompasses all.

  4. A Mathematician's Apology by G. H. Hardy; a long essay on what it is like to do math and be a mathematician. It makes for a very enjoyable reading and is recommended to see some less known sides of a mathematician.

I hope this helps.

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    What is Mathematics? is an amazing book, it was what first got me to seriously ponder math and numbers – celeriko Nov 4 '14 at 2:12
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    While I enjoyed reading a Mathematician's Apology (many years ago), this book has caused mathematics education a lot of harm because of Hardy's strange views of applied mathematics and applications of mathematics. This issue was picked up from Hardy (and probably others) and amplified by Halmos in his essay "Applied mathematics is bad mathematics." Many mathematicians still feel this way which is very sad. – Joseph Malkevitch Nov 12 '14 at 3:27

Another fine book which is, in a way, a modern counterpart to What is Mathematics?, is

Concepts of Modern Mathematics by Ian Stewart

who has also written many other great books (see e.g. the list at Amazon).

Aleksandrov, Aleksandr Danilovich, and Andreĭ Nikolaevich Kolmogorov, eds. Mathematics: its content, methods and meaning. Courier Dover Publications, 1999.


                  Cover
This 1,103-page book was originally published in 1956. It is translated from the Russian and now reprinted by Dover (so it is relatively inexpensive). I purchased it for A.D. Aleksandrov's contributions, but the list of chapter authors is a who's-who in Russian mathematics.

A NYTimes review said, "What makes these volumes so readable as compared with usual mathematics textbooks is the emphasis here upon basic concepts and results rather than upon the intricate and wearying proofs that make such demands in conventional textbooks and courses."

Here is one little result I learned from this book: "Why is the half-torus rigid?."

  • This book seems very nice. Thank you for mentioning it (and also thank you for the link to that question). – Dal Jan 4 '15 at 21:42

There are already some fantastic answers here. A few that I love, but haven't seen mentioned: Mathematician's Delight- W W Sawyer: Anything by Sawyer is worth a look, especially for aspiring teachers, but this one is the easiest to get, and to get into.

Everything and More- David Foster Wallace: One one the best writers of recent times tackles infinity, and it predictably makes for fascinating reading.

The IMO Compendium- Djukic et al.: Incredibly thick, expensive and difficult, but there are enough great problems here (with, thankfully, worked solutions) to last you forever.

The Road to Reality- Roger Penrose: A Physics book, but unlike Hawking, Penrose doesn't skimp on the Maths.

The Art and Craft of Problem Solving- Paul Zeitz: Does what it says on the tin.

Which Way did the Bicycle Go?- Stan Wagon et al.: My favourite book of problems, and he still does a weekly question, sign up here: archive here: http://mathforum.org/wagon/

I rather admire the book by Sherman Stein (who has written lots of fine books) called Mathematics The Man-Made Universe.

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