Math: the book

Today I continue a three-entry streak of praising things that are good.  While visiting IAS to give a talk, I noticed on several of my friends’ desks heavily-bookmarked copies of the Princeton Companion to Mathematics: a 1000-page volume that’s sort of an encyclopedia of math, history of math, biographical dictionary of math, beginners’ guide to math, experts’ desk reference of math, philosophical treatise on math, cultural account of math, and defense of math rolled into one, written by about 130 topic specialists and edited by the Fields medalist, blogger, and master expositor Timothy Gowers.

The best way I can explain what the PCM is trying to do is this.  Suppose that—like in the Hitchhiker’s Guide to the Galaxy—aliens are threatening to obliterate the earth along with all its life to make room for an interstellar highway.  But while the aliens are unresponsive to pleas for mercy, an exemption might be granted if the humans can show that, over the last four millennia, such mathematical insights as they’ve managed to attain are a credit rather than an embarrassment to their species.  To help decide the case, the aliens ask that humans send them an overview of all their most interesting mathematics, comprising no more than 1,000 of the humans’ pages.  Crucially, this overview will not be read by the aliens’ great mathematicians—who have no time for such menial jobs—but by a regional highway administrator who did passably well in math class at Zorgamak Elementary School.  So the more engaging and accessible the better.

I don’t know what our chances would be in such a situation, but I know that the PCM (suitably translated into the aliens’ language) is the book I’d want beamed into space to justify the continued existence of our species.

So what makes it good?  Two things, mainly:

  1. For some strange reason I still don’t understand, it’s written as if you were supposed to read it.  Picture a stack of yellow books (), and imagine cornering the authors one by one and demanding they tell you what’s really going on, and the result might look something like this.  Admittedly, there are plenty of topics I still didn’t understand after reading about them here—Calabi-Yau manifolds, K-theory, modular forms—but even there, I gained the useful information that these things are apparently hard for me even when someone’s trying to make them easy.
  2. The book is cheerfully unapologetic about throwing in wavelets, error-correcting codes, the simplex algorithm, and the Ising model alongside the greatest hits of algebra, geometry, analysis, and topology—as if no one would think to do otherwise, as if the former were part of the mathematical canon all along (as indeed they could’ve been, but for historical accident).  Nor does it dismay me that the book gives such a large role to theoretical computer science—with a 30-page chapter on complexity by Avi Wigderson and Oded Goldreich, as well as chapters on cryptography, numerical analysis, computability, and quantum computing (my tiny role was to help with the last).  There are also essays on computer-assisted proofs, “experimental mathematics,” innumeracy, math and art, and the goals of mathematical research; a guide to mathematical software packages; “advice to a young mathematician”; and a timeline of mathematical events, from the first known use of a bone for counting through Shor’s factoring algorithm and the proofs of Wiles and Perelman.

But enough!  I must now descend from Platonic heaven, reenter the illusory world of shadows, and finish my grant proposal … alright, maybe one more puff …

30 Responses to “Math: the book”

  1. Raoul Ohio Says:

    Math shares a lot with Physics and Computer Science, but consider one glaring difference: If you have studied CS or Physics, say to the MS degree level, you can understand a lot of what is going on in most subfields. But this is not at all the case in Math. Primary reasons for this include: math covers a lot more ground than other subjects, and abstraction is important.

    Another factor is the scarcity of well written, fun to read, advanced math books. Some applied and computational math books are excellent; Watson’s “Bessel Functions” is quite the page turner for those who understand calculus, suitable for reading in bed while drinking coffee. Books by Knuth, Henrici, Hille, and Wilkerson come to mind. I look forward to reading the PCtM.

    Checking out Timothy’s blog reveals that he is hoping to instigate some large scale math collaborative research. Good luck with that. Here is a problem: Whoever writes up a math problem clearly enough so that a lot of people understand it well enough to contribute, can probably do most of the work with a couple of helpers. Does anyone know of a problem where this reasoning does not apply?

  2. asdf Says:

    Cool! I’ve been wanting to get a copy about this for a while. Terence Tao has written some articles in it and mentioned it several times in his own blog, which is why I’d heard about it before.

  3. Dominic Says:

    Raoul: Fermat’s last theorem? Goldbach’s conjecture? Both are easy to present clearly; both are hard to crack.

  4. matt Says:

    And when many high energy theorists don’t know the CHSH inequality, many quantum info theorists couldn’t calculate electron-electron scattering to 2 loops in QED, and neither of them them could make a dilution fridge work, I don’t think it’s fair to say that it’s any easier to understand what’s going in the different subfields of physics than it is in mathematics.

  5. Peter Morgan Says:

    It’s a good book.

  6. Scott Says:

    Matt: I’d certainly shell out cash for a Princeton Companion to Physics in exactly the same style. (The only recent physics book I can think of with similar audacity is The Road to Reality, but that’s just fundamental physics, and just Penrose’s view of it.)

  7. KaoriBlue Says:

    “…I don’t think it’s fair to say that it’s any easier to understand what’s going in the different subfields of physics than it is in mathematics.”

    It’s both nice and surprising to me that the mathematical community was able to put a book like this together. I think it would be a great idea to have, say, the NSF encourage and fund similar projects in the physical sciences.

  8. panos Says:

    That sounds like the “Mathematics, Its Content, Methods and Meaning”, a 3-Volume collected work of top Russian mathematicians. It seems nice, I already ordered it

  9. John Sidles Says:

    Scott: “I’d certainly shell out cash for a Princeton Companion to Physics in exactly the same style.”

    Two-point-five words: Landau & Lifshitz.

    Surprisingly readable if you skip over the equations … remarkably challenging if you rederive them … extraordinarily satisfying to work at extending them.

  10. John Sidles Says:

    For those Shtetl Optimized fans whose mathematical understanding tends to progress “from the concrete to the abstract” (a phrase from Tao’s blog), one good stepping-stone is Daniel Martin’s not-too-expensive paperback Manifold Theory: Introduction for Mathematical Physicists.

    The book might be enjoyable, too, for students whose understanding is evolving in the opposite direction: “Oh wow … these scientists are using coordinates … that sometimes have decimal places.”

  11. asdf Says:

    Scott, in case this is of interest:

    Claims to be about some experimental evidence of the holographic principle.

  12. senderista Says:

    _Mathematical Physics_, by Robert Geroch, is the best math book I’ve ever read and an excellent introduction to the kind of math used in theoretical physics at the first-year grad level. Category theory fans especially will love it, but the approach is anything but abstract. Every concept and every proof is motivated and intuitively explained, many with hand drawings ala Penrose. Unfortunately, there are quite a few typos.

  13. Martin Says:

    asdf, great article!

  14. John Sidles Says:

    Senderista Says: “Mathematical Physics, by Robert Geroch, is the best math book I’ve ever read.”

    I wish to second that notion! My database includes the following quote from Geroch’s book (I attended Geroch’s lectures as a grad student—they were great):

    It takes a certain amount of effort to learn mathematics. Fortunately, two circumstances here intervene. First, the mathematics one needs for theoretical physics can often be mastered simply by making a sufficient effort. This activity is quite different from, and far more straightforward than, the originality and creativity needed in physics itself. Second, it seems to be the case in practice that the mathematics one needs in physics is not of a highly sophisticated sort. One hardly ever uses elaborate theorems or long strings of definitions. Rather, what one almost always uses, in various areas of mathematics, is the five or six basic definitions, some examples to give the definitions life, a few lemmas to relate various definitions to each other, and a couple of constructions.

    In short, what one needs from mathematics is a general idea of what areas of mathematics are available and, in each area, enough of the flavor of what is going on to feel comfortable. This broad and largely shallow coverage should in my view be the stuff of “mathematical physics.”

    The sole reason that I recommended Martin’s Manifold Theory instead of Geroch’s Mathematical Physics is that Martin discusses a topic that has central importance for quantum information science: the Riemannian geometry of Kählerian manifolds (which is considerably more elegant than the Riemannian geometry of real manifolds).

    Hmmm … Geroch’s quote suggests a provocative question: is modern quantum information science one of those disciplines in which “the mathematics one needs is not of a highly sophisticated sort?”

    Seriously, to what level of simplicity can the mathematics of quantum information science be distilled, and still be useful in practical calculations and in research?

    Of course, Bob Geroch’s notion of what constitutes “highly sophisticated” mathematics may differ from other folks! 🙂

  15. Jonathan Vos Post Says:

    So far as I’m concerned, as (one time or another) a professional Mathematician, Computer Scientist, Engineer, Economist, Physicist, and teacher, the primary purpose of Mathematics is INSIGHT.

    The Princeton Companion to Mathematics has an absolutely astonishing amount of insight. As with broadcasting Bach to extraterrestrials, this is putting our best foot forward.

    I restrain myself from a tangent about the famous Biologist and the famous Mathematician who discussed what fraction of the current literature they could understand if they had to. The Mathematician confessed that this was about 10%. The Biologist, amazed, said that he could, if he had to, follow 90% of current Biology papers. “Why is the difference so great?” they wondered. I’ve never seen a completely satisfactory answer.

  16. -Job- Says:

    One nice feature of Wikipedia: there’s no first page/chapter. A reader can start with the topics he/she find interest in and work down through the requirements (which are conveniently linked to) without getting too caught up in the details unless they are willing and prepared to do so – personally i am fine with doing multiple passes to pick up the requirements i missed.

    The sequential/structured nature of Math/Physics books, with the heavy referencing and cryptic text i think is what makes them unattractive to read without an instructor – even when you have a genuine interest in the subject.

  17. asdf Says:

    Martin, I remember hearing that the holographic principle implies that no quantum computer larger than 400 or so cubits can ever be built in the universe. Is that article trouble for quantum computation then?

  18. asdf Says:

    Yeah, here’s a mention of that 400 qubit number. Can anyone tell if this paper is bogus?

  19. Ian Durham Says:

    PCM sounds terrific. Another fantastic book that I often list as a supplemental text in many of my classes (and once required as a secondary text in a History of Mathematics class) is Jan Gullberg’s tome Math: From the Birth of Numbers.

  20. Scott Says:

    asdf: We’ve discussed this question before on this blog. Paul is just completely, totally, flat-out wrong about the holographic bound imposing an upper limit of 400 qubits in the observable universe, even though he keeps repeating it. The limit imposed by the holographic bound is about 10120 qubits. That’s because the holographic bound is a bound on the (log of the) Hilbert space dimension, not on the amount of classical information needed to specify a quantum state.

  21. Michael Nielsen Says:

    Scott – Try Iain Lawrie’s “A Grand Unified Tour of Theoretical Physics”. I doubt it’s as good as the PCM, given my limited exposure to the PCM, but overall I preferred it to Penrose’s book (which is excellent, but uneven).

  22. John Sidles Says:

    Mathematical books that are clear, comprehensive, and inspiring to students obviously are good. However—by Bohr’s Opposites Principle—it follows that mathematical books that are dismally opaque, grossly incomplete, and hugely intimidating to students, are good too.

    “Yeah,” you’re thinking “but what are some examples? Aside from pretty much everything by Grothendieck, I mean.”

    Well, the Clay Mathematical institute has two on-line proceedings Strings and Geometry, and Mirror Symmetry that (IMHO) are pretty good examples.

    In aggregate, these two proceedings are 1,324 pages of cutting-edge mathematical excellence, in which dozens of mathematicians do their very best to explain to students (and to each other) what’s going on.

    Do they succeed? Well … as the Mirror Symmetry introduction candidly describes it:

    We are at a delicate point in the history of the interaction of [math and physics]: while both fields desperately need each other, the relationship seems at times to be a dysfunctional codependence rather than a happy marriage!

    Why would a practical engineer be looking at these two proceedings? Well, we take a practical interest in the intersection of information theory and geometric quantum mechanics—that intersection being a working definition of the core mathematics of quantum simulation science.

    So I was keyword-searching for informatic-type concepts like “decoherence, entanglement, simulation, tensor networks, Choi, Kraus, Stinespring”.

    Zero, zip, nada. Which is (from one point of view) very good news for students … because it means that fundamental mathematical invariances remain to be explored … and plenty of work remains to be done! 🙂

  23. asdf Says:

    Scott, thanks. I looked further at Paul Davies’ paper and although I still don’t understand it much, I can see that the 400 qubit number is somewhat pulled out of nowhere.

  24. Peter Norvig Says:

    I ordered my copy after seeing your piece. It arrived yesterday, and when I opened to a random page, I saw “NP Completeness (definition of).” I thought that was appropriate to this blog.

  25. Raoul Ohio Says:


    I love Wikipedia, and also math and physics books. They play a different role. In most of Physics and CS, you can work some problems, see how it goes, and move on to other aspects of the subject. In math (and complexity theory) a major role is played by delineating exactly what is proven. Thus some fraction of the books must start with the axioms and inch their way forward. Usually these don’t make light reading.

    Classic examples in math are an analysis book by Landau (same guy who introduced/popularized the O and o symbols?) starting with some axioms (ZF set theory? I forgot) and proving and numbering each theorem through advanced calculus and into real analysis. Along the way, the first place the axiom of choice is truly needed is noted (existence of a non lebesgue measurable set?) I believe Titchmarsh followed up by carrying the numbering on through all the theorems of classical complex analysis (possibly in, still in print).

    I doubt if many are interested in reading these treatments, but it is nice knowing they are there in case you need them. Contrast this with Wikipedia: usually (for tech stuff) pretty good, often interesting, but you are never quite sure that it is right. Also, it might change in the next couple minutes, usually for the better. There is also the problem of topics that nuts and quacks take an interest in.

  26. Greg Kuperberg Says:

    Paul is just completely, totally, flat-out wrong about the holographic bound imposing an upper limit of 400 qubits.

    But let’s be generous with him just as we would be with a Nobel Laureate such as Freeman Dyson who sometimes says wrong things. The holographic bound does say that there is an upper limit of 400 qubits needed to address any qubit in the universe. A computer in the observable universe will never need more than a 400-bit address bus.

  27. Greg Egan Says:

    Greg K, the problem is not so much that Davies says something analogous to claiming that my iMac has “32 bits of RAM”, which could be brushed off as a typo in need of exponentiation, but that he then goes on to infer all kinds of sweeping consequences about the nature of the laws of physics that don’t actually follow at all from the “generous” (i.e. correct) interpretation of his original assertion.

  28. Greg Kuperberg Says:

    Well yes, Greg E (except for the E I feel as if I’m debating with myself), my emendation of Davies was secretly meant to be untenable and backhanded.

    But then, a few of Freeman Dyson’s errors are also irreparable. Part of my real point is that sometimes we are a little too generous with mathematicians and scientists who win big awards.

  29. Martin M. Musatov Says:

    Doron Zeilberger’s 45th Opinion
    Apr 1, 2002 … Every mathematician is trying to solve at least one of the seven notorious … since we mathematicians and theoretical computer scientists care very little … than a mundane `rigorous mathematical proof’ of PNP would have been. … hence it is safe for Mr. and Mrs. Clay to offer big prizes!, … – 6k – Cached – Similar pages –

    Somebody has to speak for those not able to…

  30. Egipt Says:

    Ian Durham, thats right. Very good book. Contains more than 1000 original technical illustrations, a multitude of reproductions from mathematical classics and other relevant works, and a generous sprinkling of humorous asides, ranging from limericks and tall stories to cartoons and decorative drawings.