# Princeton Companion to Mathematics

I just recently got my hands on a copy of the new Princeton Companion to Mathematics, and I fear that this is likely to seriously impact my ability to get things done for a while, as I devote too much time to happily reading many of its more than 1000 pages.

The book is an amazing document (and physically, a beautiful, if weighty object), unlike anything else I know of. Its coverage of mathematics and mathematical culture is very wide and sometimes deep, but it makes no attempt to be comprehensive. Thus, the accurate title “Companion to” rather than “Encyclopedia of”. The most remarkable aspect of the book is the extremely high quality of the contributions from a large number of different authors. It includes many wonderful long expository articles, mostly at a level that a good undergraduate math student could hope to appreciate, with much of the book accessible to an even wider audience. The articles are often written by some of the best researchers and expositors around. For example, one can find Barry Mazur writing on Algebraic Numbers, Janos Kollar on Algebraic Geometry, Cliff Taubes on Differential Topology, Ingrid Daubechies on Wavelets, Persi Diaconis on Mathematical Statistics, and many, many others of similar quality. The table of contents is available here.

The book also includes extensive articles on historical topics in mathematics and short biographies of a large number of mathematicians, as well as coverage of applications and a section largely devoted to describing the art of problem-solving and how mathematics really gets created. This section includes a beautiful set of five essays called “Advice to a Young Mathematician”, which give five different equally fascinating perspectives from some of the best in the subject about how they achieved what they did, as well as what they have learned from years of helping students become researchers. The authors of these pieces are Michael Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff, and Peter Sarnak. Luckily for all young (and old) mathematicians, this chapter is freely available here.

The person most responsible for this is clearly the editor (and author of some of the pieces), Fields Medalist Timothy Gowers, who had help from many others, including fellow Fields Medalist Terry Tao. Gowers has a weblog, and he has written about the book in these entries (and there’s a podcast interviewing him on the book web-site at PUP). Terry Tao has a posting about the book here.

If you’re looking for a gift for someone with a serious interest in mathematics, no matter what their background, you won’t do any better than this.

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### 23 Responses to Princeton Companion to Mathematics

1. Serifo says:

Is there any similar book in physics ? I’m just a math’ s undergraduate student with some curiosity in mathematical foundation of physics ( Quantum mechanics in particular). By the way , may I ask your views on Hilbert’s sixth problem ? Sincerely

2. Peter Woit says:

Serifo,

I know of nothing like this book in physics.

To be honest, I never understood exactly what Hilbert was looking for in this problem (“axiomatization of physics”). There are certainly parts of physics that are mathematically well understood and could be axiomatized, others where we still don’t understand things well enough to sensibly do this.

3. Tim May says:

Thank you, Peter, for this news. I just placed an online order for a copy.

As a former physicist (is there really such a thing?) who found GR and black holes the most interesting things in the world, circa the early 70s, I now am finding the “quantum measurement” constellation even more interesting. I recently did a demo of the “quantum eraser experiment” from a Sci. Am. article, the one involving some polarizers and a needle between two oppositely-polarized filters. Wow. The essence of the EPR/weirdness reality demonstrated in a darkened living room.

Sorry that this isn’t highfalutin’ string theory vs. loop quantum gravity vs. whatever, but it’s the essence of physics for me.

And math. I eagerly await the Gowers book.

–Tim May

4. Serifo says:

Thank you for the reply, it’s a pty there isn’t such a book in physics. When I was a high school student sometime ago, my physics teacher said ” more maths you learn, more easier is to learn physics ” , well next time I see her, I will say ” more maths you learn, more you struggle to understand physicist’s thoughts ” ! Maybe the starting point should be to axiomatize first those mathematically well understood theories then try gradually to capture the rest. Anyway sorry for bringing this off topic subject, cheerse to Bourbaki! Sincerely

5. Yummy ! I’ll get a copy as soon as possible. Thank you for this.

6. Navneeth says:

That seems wonderful. Thanks, Peter. Added to my want-list.

7. Navneeth says:

Fabien Besnard said: “Yummy!… ”

8. Serifo says:

I really hope they will produce : Princeton Companion to fundamental physics ! 🙂

9. Navneeth says:

One for Physics will require revision much often than one for mathematics. Maybe a web-based service will suit it better.

10. Thomas R Love says:

A book which does similar work is

A Panorama of Pure Mathematics (Pure and Applied Mathematics (Academic Pr)) by Jean A. Dieudonne

While the articles are at a much higher level ( suitable for a graduate student or PhD) so is the price (about \$225).

I read it while in graduate school and it gave me a taste of other fields of mathematics.

Several of the articles from the Princeton book were once available on line and I read several of them. They were at too low a level to interest me.

11. Coin says:

I know of nothing like this book in physics.

From what you’ve said here it kind of sounds a bit like Penrose’s “Road to Reality”, though it sounds like it may be operating at a deeper level than Penrose’s book.

12. JJ says:

As mentioned by Coin, Penrose´s “Road to reality” is somewhat similar, giving some sort of overview of modern theoretical physics. Another interesting book, a little more demanding is Lawrie´s “A Unified Grand Tour of Theoretical Physics” (Amazon link: http://www.amazon.com/Unified-Grand-Tour-Theoretical-Physics/dp/0750306041 )

13. theoreticalminimum says:

While I agree there is no such similar book for Physics, may I however point out that there exists the comparatively bulkier 5-volume set “Encyclopedia of Mathematical Physics“, which covers to some appreciable extent topics in theoretical and mathematical physics. Springer will publish a “Modern Encyclopedia of Mathematical Physics” in 2010 apparently.

14. Thomas R Love says:

theoreticalminimum recommended the “Encyclopedia of Mathematical Physics“, and it looks good (judging by the description) but the price is a killer:

Price:
EUR 1,240
USD 1,495
GBP 855

That represents about one third of my monthly take-home pay.

15. theoreticalminimum says:

I am acutely aware of the huge price, to the extent that people who would actually like to read anything from Elsevier would have to dig deep in their savings (recall for instance the resignation of the editorial board of “Topology” as protest against the outrageous prices of the publisher – find more here, here and here). That’s why we have gigapedia (hint) to be thankful for ;-).

16. F Borg says:

Paul Davies edited a successful physics tome “New Physics” in 1989 which has got a follow-up edited by Gordon Fraser, “The new physics for the twenty-first century” (Cambridge UP 2006). While they do not quite parallel the Companion to Mathematics these books still give nice overviews of what’s going on in physics.

Eberhard Zeidler is by the way responsible for “Teubner-Taschenbuch der Mathematik” whose first part has also appeared in English as “Oxford User’s guide to mathematics” (Oxford UP 2004). The second part goes a bit deeper into the subjects including math physics. Lots of math for the money! Nirmala Prakash has written a quite “friendly” book on “Mathematical perspectives on theoretical physics” (Imperial College Press 2003). This is math phys seen from a stringy perspective. Quite different emphasis from *classical* math phys employed in electrodynamics, hydrodynamics etc. A modern exponent of the Jeffreys & Jeffreys tradition is Michael Vaughn’s “Introduction to mathematical physics” (Wiley 2007).

Finally, for someone picking up physics I would recommend John Walecka’s recent “Modern physics” (World Scientific 2008) before they, if hooked, may proceed to Lawrie’s grand tour mentioned in an earlier posting.

17. F Borg says:

I just got a posting from Springer. E Zeidler’s vol 2 of his gigantesque 6 vol set on math physics is announced (this is what the Germans call an “introduction” … 1000 pages plus per vol so far). One can download an interesting historical outline of physics from ch 1 here

Though Zeidler can be associated w/ the Bourbaki tradition Zeidler’s writing may have a stronger pedagogical intent judging from the above excerpt. I can see the advantage of a single author oeuvre in terms of coherence and style — but this one is a truly daunting task both for the author and potential readers!

Zeidler’s Einleitung:

Volume I: Basics in Mathematics and Physics
Volume II: Quantum Electrodynamics
Volume III: Gauge Theory
Volume IV: Quantum Mathematics
Volume V: The Physics of the Standard Model
Volume VI: Quantum Gravitation and String Theory

18. Steve Myers says:

There is an encyclopedia of physics which is pretty good but I think it’s over 10 years old. (I don’t have it to hand here so don’t recall publisher.) I wouldn’t recommend the Penrose book — I’m sorry I bought it.

19. Steve Myers says:

If anyone’s still checking here: the book is edited by Lerner & Trigg. It’s from 1990 (second edition). VCH publishers. I find it useful.

20. Serifo says:

Ive got Penroses “ The Road to Reality “ ! Although the book is quite well organized in respect to the topics, I found lack of clarity in a lot of the ideas ! Maybe because Im just a young outsider in physics ; but Im convinced the main reason is :
He tries to write the ideas as simple as possible ( maybe too simple and informal ), then the ideas get mix up and so confusing ! Actualy, I found similar problem in most of the physics textbooks, the textbooks are too informal to be understood ! Maybe via axiomatic approach , it will be easier to learn physics !
To Hilbert and his sixth problem 🙂

21. Kaloyan Todorov says:

Something of an equivalent in physics would be The Feynman Lectures on Physics by (you guessed it) Richard Feynman. A worthwhile read indeed!