Some books that you might like to read on mathematics and more

Serge Lang was one of the great number theorists of the last century, who is perhaps best known for his influential conjectures and his many books. His conjectures include: the Mordell-Lang conjecture, the Bombieri-Lang conjecture, the Lang-Trotter conjecture, and many others. His books include all kinds: monographs, graduate textbooks, undergraduate textbooks, and more.

Today I want to talk about books. I love to read books of all genres; although my favorites are: mathematics, history of mathematics, and history in general. I did just read “Under the Dome” by Stephen King, so I am not limited to nonfiction—that is fiction—right?

Since it is the end of the year, I thought that some of us might have time to catch up on our reading. So I plan to share a few of my favorite books—I hope that you might let me know some of yours.

I knew Lang when I was faculty at Yale University in the mid-seventies. During our spring break one year, Dick Karp visited for a whole week. It was great having Karp visit the Computer Science department, and he and I had many fun discussions. On the last night of his visit, Karp, Lang, and I went out for dinner. I recall on the way to my car, Dick causally asked Serge what he had done during the spring break? Serge quickly replied:

I wrote a book.

Karp seemed at a loss for a reply. I had nothing to say, but I immediately thought, what did I do this last week? Besides enjoyable conversations and general stuff I had no answer—I certainly did not write a book. I did nothing. I do believe Lang, since he wrote many many books—as I said earlier he was extremely prolific. Oh well.

Let’s turn to reading books, not writing them.

My Book List

Here are some books that I have enjoyed, in no special order:

${\bullet}$ Math Talks for Undergraduates by Serge Lang. This book is a series of lectures for undergraduates. It is wonderful. He covers a wide range of topics, and demonstrates his great ability to explain things clearly. One example is the chapter on the famous ABC conjecture. This is due to Joseph Oesterlé and David Masser, and is:

Given ${\varepsilon>0}$, there exists a constant ${K}$ such that for all non-zero relatively prime integers ${a,b,c}$ with ${a+b = c}$, we have the inequality

$\displaystyle \max(|a|,|b|,|c|) \le K \cdot N_{0}(abc)^{1+\varepsilon}.$

Here ${N_{0}(x)}$ is defined as the product of the distinct prime factors of ${x}$:

$\displaystyle \prod_{p | x} p.$

Note, ${N_{0}(p^{100}) = p}$ where ${p}$ is a prime. Lang proves a version of the conjecture for polynomials, and explains why the conjecture is “the greatest conjecture of the century.”

${\bullet}$ The File by Serge Lang. This is one of the strangest books ever written, but it is hard to not find it compelling. It is a collection of letters that Lang wrote and were written to him. One letter after the other.

The letters all concern Lang’s campaign to keep a candidate out of the National Academy of Sciences. Lang was upset that the candidate, who worked in social science, was misusing mathematics to “prove” some point. Lang stated clearly in his letters against the candidate that this use of mathematics was not worthy of membership to the Academy. The defenders letters, of course, argued back the opposite point.

I was once given advice by a friend who is an attorney. He said:

Never throw out any letter you are sent, and never send anyone a letter.

This book is a tribute to this maxim. People wrote letters to Lang, and said things in those letters, that they never wanted to become public. But, they did become public.

${\bullet}$ Analytic Number Theory by Donald Newman. Analytic Number Theory is a deep and beautiful area. It is known for extremely technical proofs, which often require pages of complex formulas, careful estimates, and messy calculations. Yet, Newman takes you on a quick tour of some of the best results in 75 pages. He gives essentially full proofs of some of the major achievements of analytic number theory: Roth’s Theorem, the Waring Problem, and the Prime Number Theorem. This is not a book to read to become an expert, or even to become well versed in the area, but there is something magical about this book. All in 75 pages.

${\bullet}$ The Honors Class: Hilbert’s Problems and Their Solvers by Ben Yandell. This is a book on the famous list of 23 problems of David Hilbert. What I like so much about this book is the history behind the solutions to the problems. In some cases Hilbert problems were “solved” for decades, yet eventually it was discovered that the solutions were wrong or had gaps. Part of my “hidden” agenda is to remind us all that even the immortals make mistakes, have proofs with gaps, and are human.

${\bullet}$ The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O’Shea.

${\bullet}$ Poincaré’s Prize by George Szpiro.

${\bullet}$ Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen. These three books are on the Poincaré conjecture and its resolution. The last one is the most recent, and it focuses mostly on Grigori Perelman as a person. The book discusses the question: Why did he turn down the Fields Medal? You will read that he had turned down awards before, but you will still be puzzled—I think—why he turned the Fields Medal down. The books approach the Poincaré problem and its history from different angles, and I enjoyed them all.

${\bullet}$ Prime Obsession by John Derbyshire.

${\bullet}$ The Riemann Hypothesis by Karl Sabbagh. These books are on the hunt for a solution to the Riemann Hypothesis. The second book highlights the work of Louis de Branges who solved the Bierberbach Conjecture of Ludwig Bierberbach. This was a major open question in analysis, and de Branges deserves a great deal of credit for solving this difficult problem.

Unfortunately, since his great success he has been claiming that he is “about” to solve the Riemann Hypothesis. Sabbagh spent many hours with de Branges and presents a case study in how a mathematical problem can almost take you over. The Riemann Hypothesis is definitely a mathematical disease for some.

${\bullet}$ The Mathematician’s Brain by David Ruelle. Ruelle is a first class mathematician and tells some wonderful stories about his views of everything—including mathematics.

${\bullet}$ Structure and Randomness by Terry Tao. Or anything else by Tao. This book is a collection of lectures from his blog posts of the year 2007. His ability to explain things is wonderful. For example, his section on ultrafilters is a classic. I thought I knew what they are, but I was wrong. Only after reading this did I really “get” what they are.

${\bullet}$ Kolmogorov’s Heritage in Mathematics edited by Eric Charpentier, Annick Lesne, and Nikolai Nikolski. This is a collection of articles about one of the great mathematicians of the twentieth century.

${\bullet}$ Lost Horizon by James Hilton. A classic novel—the cover claims that it was the first paperback ever published. I do not know if this is correct, and I decided not to check.

Open Problems

What are some of your favorites? I would love to hear.

December 21, 2009 9:12 pm

“The File” makes Serge Lang come off as an unfriendly, crazy person. You should also read his article here: http://www.ams.org/notices/199511/forum.pdf in which he also portrays himself as an unfriendly, crazy person (again, using his favorite technique of publishing correspondences that their authors thought were private messages to him). He was, of course, also an AIDS denialist.

December 21, 2009 11:46 pm

I think that Lang was a person that could go over the edge and do strange stuff. The File is definitely not a normal book.

2. December 22, 2009 3:14 am

Mathematics: A Very Short Introduction by Timothy Gowers. It talks about numbers, proofs, infinity, approximations, multidimensional spaces, etc. I learned what numbers really are from Feynman’s lectures, my high-school textbooks were pretty good about explaining infinity and limits, during college I `got’ approximations, etc. But, later, I found the essence of all these things gathered in one little book. I wish I had it when I was 15. (Unfortunately, it wasn’t written yet at the time.)

3. December 22, 2009 4:55 am

I very much enjoyed The Music Of The Primes by Marcus Du Sautoy. Not that technical and accessible to non-mathematicians I still enjoyed it immensely. Also, a golden oldie but Fermat’s Last Theorem by Simon Singh is my favourite maths book of all time!

4. December 22, 2009 5:50 am

Mathematical Logic: A Course With Exercises Part I and II by Cori and Lascar. This is for me one of the best book to discover logic. It goes from propositional calculus to model theory.

5. December 22, 2009 6:51 am

I was meant to purchase Lang’s graduate algebra text as an alternate for my first year graduate class. I did, and found it impenetrable whereas Hungerford was an easy read. Others in my class concurred. But I never read anything else by Lang.

December 22, 2009 7:52 am

I loved ” my brain is open: the mathematical journeys of paul erdös ” by Bruce Schechter.
He is my favourite mathematician of all time.
I’m still waiting for a popular science book about the history of the p vs np question and related topics. That would be a wonderfull book 🙂

December 22, 2009 8:05 am

Thx for another nice post.

I want to mention two I recently read:

-Logicomix: a graphic novel coauthored by Papadimitriou. Probably there is not much new to those in the know, but it’s an easy read and interesting. Particularly interesting was one of its themes: “people who do logic go crazy”.

-Alfred Tarski: Life and Logic, a biography of Tarski. He must have been one of the weirdest logicians ever. It has some history of logic in it. Related to this, I also liked Kreiseliana, which I read long ago. It is a memoir of Georg Kreisel, another weird logician. This was more of a collection of personal accounts, I remember.

December 22, 2009 9:34 am

The Tarski book is fun. I agree about logicians.

December 22, 2009 11:21 am

Most of the books you recommend are very good. I would particularly agree with The Honors Class and Prime Obsession. Another favorite of mine is Fourier Analysis by Thomas Korner. It has a good mix of pure math, applications, and history.

9. December 22, 2009 11:49 am

The Tarski biography is great. Reading it made me want to run out and grab a copy of Tarski’s collected works, but apparently it’s been out of print since before I was born!

For books very much in print, there is John Horton Conway’s “The Symmetries of Things”. It’s a great subway/bus book, but it’ll take you a while to work through.

For a different kind of book, the Princeton Companion to Mathematics is a must-have for any kind of mathematician or mathematically inclined fellow.

December 22, 2009 12:13 pm

The Princeton Companion to Mathematics is wonderful. There are so many great articles in it. It did take me a day to figure out the order of the section on famous mathematicians.

• December 23, 2009 5:38 am

It’s explained in the Preface 🙂

December 22, 2009 12:47 pm

Thanks for this post. My Amazon wish list just got even fatter.
My favorite math book is perhaps Visual Complex Analysis by Tristan Needham. Until I saw that book, I thought complex analysis was just some ugly bastardization of calculus used by engineers. Needham’s book showed me how wrong I was.

December 22, 2009 12:51 pm

“… and never send anyone a letter”…just write a comment in a blog !

IMO the Political Scientist was not alone in the misuse of mathematics: still today many papers in the orbit of the social sciences (including economics, AGT etc…), seems to me as a mathematization “avant la théorie” and therefore misleading as social science, althought maybe great as mathematics.

In any case he was polemic with or without Lang and in his later theses pointed correctly that identitary problems (to wich avalaible social theories has not answer, AFAIK) beeing it biological (such as racism), social (classism), cultural (i.e. different languages in a same political unit, also called nationalism) or ideological are the real problems. Heck…i forgot this is a mathematics-TCS blog!

Speaking about great books, for understanding why it is difficult to mathematize the social sciences, i would recomend Martin Hollis´s “The Philosophy of social sciences”.

December 22, 2009 11:15 pm

In the history of Mathematics and biography genre, I recently read “The Man who knew Infinity” by Robert Kanigel. Biography of Srinivasa Ramanujan. A must read for anyone to understand the genius from South India. Being a South Indian myself, I could very much relate to the cultural expositions in the book. Also the character of G.H.Hardy is presented very well. A somewhat sketchy idea of the work of Ramanujan for the layman like me is also presented.

13. December 22, 2009 11:23 pm

Out of their minds.

It is about the life of 15 computer scientists: Dijkstra, Tarjan, etc.

14. December 23, 2009 3:16 pm

IAS Professor Johnathan Israel published a short book this past November: A Revolution of the Mind: Radical Enlightenment and the Intellectual Origins of Modern Democracy. Chapter 1 is available on-line (for free) at Princeton University Press.

IMHO, Prof. Israel might reasonably have been omitted the word “perhaps” from the final line. 🙂

Also, happy holidays to all, and my special appreciation and thanks are extended to Prof. Lipton for this great blog.

——–

The last three decades of the eighteenth century were an age of much turmoil, instability, and revolutionary violence. But they were also an age of promise.

The emancipation of man via forms of government promoting the “general good” and life in a free society that accords protection to all on an equal basis, argued d’Holbach in 1770, is not an impossible dream: “if error and ignorance have forged the chains which bind peoples in oppression, if it is prejudice which perpetuates those chains, science, reason and truth will one day be able to break them” (si l’erreur et l’ignorance ont forgé les chaines des peuples, si le préjuge´ les perpétue, la science, la raison, la vérité pourront un jour les briser).

A noble and beautiful thought, no doubt, but was he right? That perhaps, is the question of our time.

———

@book{*, Author = {Jonathan Israel}, Publisher = {Princeton University Press}, Title = {A Revolution of the Mind: Radical Enlightenment and the Intellectual Origins of Modern Democracy}, Year = {2009}}

15. December 23, 2009 3:28 pm

Oh yeah … Daina Taimina’s Crocheting Adventures with Hyperbolic Planes had a terrific impact on our QSE Group’s geometric understanding of ruled state-spaces in quantum system engineering.

Bill Thurston wrote the introduction (which is terrific), and Taimina also references a 1991 article by Barry Mazur Number theory as gadfly that is both prescient (regarding the Shimura-Taniyama-Weil Conjecture) *and* terrific.

Many pages (including Thurston’s introduction) are browsable on Google Books.

This book is fun all the way through.

——

@book{*, Address = {Wellesley, MA}, Author = {Taimi{\c{n}}a, Daina}, Publisher = {A K Peters Ltd.}, Title = {Crocheting {A}dventures with {H}yperbolic {P}lanes}, Year = {2009}}

@article{Mazur:1991uq, Author = {Barry Mazur}, Journal = {Amer. Math. Monthly}, Number = {7}, Pages = {593–610}, Title = {Number theory as gadfly}, Volume = {98}, Year = {1991}}

December 23, 2009 5:19 pm

There are several good books on the history of specific branches of mathematics (as opposed to math in general). Some of my favorites:

Felix Klein and Sophus Lie by I. M. Yaglom. (A history of group theory leading up to Klein’s Erlangen Program.) The first time I read it I only got through the first couple dozen pages and put it down. The second time I picked it up I couldn’t put it down — it’s worth given it a second chance.

Beyond Geometry: Classic Papers from Riemann to Einstein (Dover), Peter Pesic, ed. Shows very neatly how general relativity was the culmination of more than 50 years of ideas on geometry. Very nice in that it includes editorial comments and introductions to each article, and all the articles are translated into English. Particularly interesting was the close entanglement between geometry and physics in the 19th century (in particular, back then “geometer” and “physicist” were more or less synonymous).

Graph Theory: 1736-1936 by Norman Biggs, E. Keith Lloyd, and Robin J. Wilson. History of graph theory from some leading graph theorists, up through 1936 (with an appendix on developments thereafter). Again, introductions and editorial comments on excerpts of translations of the original papers. Interesting to see how topology (in particular algebraic topology) originally grew out of graph theory.

17. December 26, 2009 6:08 pm

Can add to Noname’s Dec. 22 comment above that Logicomix made TIME Magazine’s list of Top 10 Nonfiction Books of 2009:

• December 26, 2009 6:11 pm

Oops, must have mis-HTML’ed the link—here’s the text of the link too. Seeing Christos Papadimitriou among the listed authors is what caught my eye.

December 27, 2009 10:23 pm

My best for 2009 is The Unravelers Mathematical Snapshots, edited by Dars, Lesne and Papillault (AK Peters). It’s a series of photographs of mathematicians working at the Institut des Hautes Études Scientifiques, accompanied by personal essays containing their reflections on the nature of mathematics, on doing mathematics, their personal mathematical development, collaboration, and much more. If you can’t have coffee with leading mathematicians, this book is the next best thing. (French original: Les déchiffreurs, Éditions Belin).
Sokal and Bricmont, in their book Intellectual Impostures (Profile Books), systematically expose the misuse of mathematics and physics terminology as props for philosophical and sociological arguments. (Same goal as Lang’s The File, but the presentation and argument is considerably better.)
On the recreational side, Tomoko Fusè’s Unit Origami Multidimensional Transformations (Japan Publications) only needs time, plenty of paper, and patience, as the author notes. In a similar way, Fink and Mao’s The 85 Ways to Tie a Tie (Fourth Estate) is an excellent way to slow down the morning rush to work. (It includes a basic introduction to knot theory.)

December 30, 2009 7:53 pm

Thanks for all the wonderful suggestions. A good book is a wonderful thing.

I’d add “The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number” by Mario Livio. There has been a lot of nonsense both said and written about the golden ratio. Livio clears up a lot of the false notions but also makes room for all the wonderful things that are actually true.

I second the recommendation for “The 85 Ways to Tie a Tie “.

January 21, 2010 11:08 pm

Hi everyone,

I would like to add two books which I find very stimulating, both written by Saunders MacLane:

– Mathematics: Form and fuction, and
– A mathematical autobiography.