Not Even Wrong

There’s a new popular book out about number theory by the team of Avner Ash and Robert Gross, entitled Summing It Up: From one plus one to modern number theory. This is the third such book that they have written, and I’m embarrassed that it seems that I never reviewed the other two here. All three are highly recommended for anyone who wants a popular book-level introduction to some of the central topics of modern number theory.

• Summing It Up is the latest of the three, but it’s also the most elementary. It’s an introduction to the subject of modular forms, starting at the very beginning. The first half of the book covers in detail some basic ideas about number theory that can be understood in elementary terms, including things like the problems of counting the ways an integer can be a sum of squares or higher powers, or partitioned as a sum of smaller integers. The second half of the book tries to explain in as simple and concrete terms as possible what a “modular form” is, and what some of the properties of such objects are.

  I’ve just finished teaching a graduate course on representation theory, and ended up the course with a short discussion of the representations of the group SL(2,R) (two by two real matrices of determinant one), and what this had to do with modular forms. The relation of modular forms to representation theory (not discussed in the book) is roughly the following. The action of SL(2,R) on itself by left multiplication induces an action on functions on the quotient space SL(2,R)/SL(2,Z) (elements of SL(2,Z) are matrices with integer entries) and one can ask how this representation decomposes into irreducible representations, with modular forms providing part of the answer. How this works is quite basic to our modern understanding of how representation theory and number theory are related.

  In the last few chapters the authors try to explain how modular forms answer concrete counting questions raised in the first half of the book. A problem with trying to do this kind of thing is that while the questions may be straightforward to state, and the basic definitions of modular forms relatively accessible, connecting the two requires invoking some subtleties (half-integral modular forms). The authors several times apologize for not being able to explain exactly what is going on. I’ve always been quite fascinated by these particular subtleties, since they appear in the basic relationship of representation theory and quantum mechanics. The half-integrality here is related to the half-integer that appears in the ground-state energy of the harmonic oscillator. For quite a bit about how this is related to representation theory, see the book I’ve been working on. I’d love some day to write about the relationship of the ideas that appear in quantum mechanics to the ones that appear in modular forms, but first of all I need to understand myself much better what is going on.

• The first of the three books by Ash and Gross was Fearless Symmetry: Exposing the Hidden Patterns of Numbers, published in 2006. It’s perhaps the best place to start for a popular introduction to the Langlands program and what it says about the relationship of number theory and representation theory. Modular forms make a brief appearance in that book, where an explanation of them had to be skipped over due to a lack of space. The newest book makes up for that omission.
• The second of the three books was the 2012 Elliptic Tales: Curves, Counting, and Number Theory. It covers in detail the topic of elliptic curves and their role in number theory, aiming at a description of one of the main open problems of the subject, the conjecture due to Birch and Swinnerton-Dyer. This is a very active subject of current research, with significant progress being made. If one had to guess which of the Millenium Problems will be the next to fall, this might be a good bet.

While there’s a long tradition of popular books about number theory, these typically emphasize elementary methods for solving problems, ones that can be understood without a lot of the modern machinery. The first two Ash-Gross books do a good job of trying to give some insight into this modern machinery, even though a popular book can only do this in a very limited way (just as popular physics books can only give a very limited explanation of quantum field theory). The new one is different in that it makes a serious effort to explain exactly what is really going on, although following this path means that the book can only cover the first steps in the direction of modern techniques for understanding number theory.

I’m about to head out on vacation to commune with nature in the Pacific Northwest, so there’s likely to be no more blogging here until after May 17th. That’s just one reason I’ll leave comments closed here, another is that I’m sure you can find other, better, places on the internet to discuss the issues raised in this posting.

A little while ago I bought a copy of Sean Carroll’s new book The Big Picture, which is now reaching the bookstores. This posting is not really a review of the substance of the book, but more a reaction to its basic conception. The first point to make is that I mostly agree with what Carroll has to say, to the extent that one reason for not writing a more usual sort of review is that I didn’t bother to do more than skim a lot of the chapters, since the theme seemed both so familiar and so unobjectionable. One exception would be a small number of pages about the multiverse, which he contrasts with religion, ending with (referring to religion)

This is the problem with theories that are not well-defined.

He’s got the problem right, but doesn’t notice that it applies equally well to this particularly dubious bit of “science”.

The largest part of the book (from my rather quick read) is a very conventional argument for science as opposed to religion, of a sort that has existed for centuries, been common since the 19th century, and very common in recent years as part of the “New Atheism”. One reason I can’t focus on this is that I just don’t see any evidence that science needs this sort of defense against religion, it seems to me to be doing extremely well without it. Our culture valorizes science and scientists very highly these days (much more so than ministers or theologians), and I just don’t see what some other people see as a need for books arguing the case for science.

The really striking thing about this particular book though is that Carroll has a much more unusual and ambitious goal than just arguing for science. He wants to promote what he calls “poetic naturalism“, which as far as I can tell is a term of his own invention (“naturalism” by itself is now a conventional term for the “science, not religion” viewpoint). Beyond the “science instead of religion” idea though, “poetic naturalism” seems to me to simultaneously lack any real content, while claiming to address the deepest human questions of meaning and morality. Asked in this interview the question

Your book, The Big Picture, roams far beyond cosmology and physics, into consciousness, philosophy and the meaning of life. What do you hope to achieve?

he answers

Well, this is the book that should accompany the Gideons Bible in all hotel rooms in the world – that would be a nice achievement!

I’m not sure what Carroll might have had to do with this, but poetic naturalism is now listed on Facebook as a possible choice for one’s religion.

Perhaps the strangest thing in the book is a chapter devoted to Carroll’s replacement for the Ten Commandments, which he calls the “Ten Considerations”, since they’re not commandments. They are:

• Life Isn’t Forever.
• Desire Is Built Into Life.
• What Matters Is What Matters To People.
• We Can Always Do Better.
• It Pays to Listen.
• There Is No Natural Way to Be.
• It Takes All Kinds.
• The Universe Is in Our Hands.
• We Can Do Better Than Happiness.
• Reality Guides Us.

It’s hard to argue with such sentiments, but also hard to understand what they have to do with the author’s expertise as a theoretical physicist.

The last chapter of the book begins with a description of Carroll’s early experiences in the Episcopal church, which he was quite fond of. I also had such experiences (I was an altar boy for several years at an Episcopal church, the American Cathedral in Paris). Unlike Carroll, I was never a believer, but just figured this was one of quite a few mystifying things that adults got up to, and that it seemed I had to go along with it until I got older. Thinking back to those days, I was struck by the realization that I recognized the tone and a lot of the content of Carroll’s writing. It very much sounds like a sermon, one evangelizing the good news not of Jesus, but of science, and is aiming for much the same effect: “I want to shiver with awe and wonder at the universe”.

My own point of view on all of this is that I just don’t think theoretical physicists have anything useful to tell the average person about meaning and morality, other than that it’s a mistake to search for it in our discoveries about physics. I don’t understand why we’re increasingly seeing texts promoting physics as inspiration for how to live (for another recent example among many see here). I’m with Steven Weinberg, and his famous line

The more the universe seems comprehensible, the more it also seems pointless.

Given that, the best advice to people who come to physicists looking for the meaning of life seems to me to politely tell them that they’re looking in the wrong place and asking the wrong person.

While I deeply value and respect my many friends and colleagues among physicists and mathematicians, I can’t imagine why anyone would think they have any unusual insight into the great questions of meaning and morality. I’m afraid that to some extent the opposite is true. My background, career and circumstances are quite similar to Carroll’s, and when I think about them my main thought these days is that I lead an extremely lucky and privileged (and not just in the white/Anglo/male/hetero sense) life, well-isolated from many of the challenges that most people have to deal with. There are a lot of beautiful, wonderful, and useful things one can learn from physicists and mathematicians, but our expertise is in something very far-removed from the question of how to live a good life in the face of significant challenges.

It seems likely that one motivation for books with this defensive attitude about science is the current ugly environment of our politics and culture. This ugliness I believe is driven by the economic disaster that has been inflicted on a significant fraction of our citizenry over the last few decades by the privileged and well-educated, both Democrats and Republicans. While Carroll and I were enjoying our respective times at Harvard and similar places, and have ended up turning our upbringing and Ivy League experiences into a very pleasant and cosseted lifestyle in the wealthy enclaves along either coast, things have not been going so well for many others. They’re now in bitter rebellion against what has happened to them, with an anger sadly turned against other racial groups, but even more so against self-satisfied elites. I don’t think a book like this has much hope of speaking to such people, to their view of science or their experience of religion. Scientists who want more respect should stick to what they know, and avoid the temptation of “science-splaining” to the public. In particular they should avoid preaching about meaning, morality, and other issues that they know no more about than anyone else.

Update: Robert Crease has a review of the book in Nature. He also finds odd the “greeting-card-like homilies” that appear in the book.

There’s a new book out in the Princeton “Nutshell” series, Tony Zee’s Group Theory in a Nutshell for Physicists. I liked his Quantum Field Theory in a Nutshell quite a lot, it’s packed with all sorts of insights into that subject. Both books are written in a very light, chatty and entertaining style, full of various sorts of worthwhile digressions. Long ago I tried to use the QFT book in a course I was teaching, and there found that it functions better as a supplement to a standard QFT book. Zee’s treatment is just too short, covering too much, to provide the level of detail most students need to learn the subject for the first time. For people who already have gone through a standard book or course, Zee’s QFT book is great as a follow-on, likely to explain a lot of things they found confusing the first time around.

The new book on group theory has a length much better matched to the amount of material (it’s longer than the QFT book, and the material covered is much less complicated). The level of detail for most topics should be a good amount for students encountering the subject for the first time. The main topics covered are:

• Finite groups and their representations.
• Unitary and orthogonal groups, their representations, and applications to quantum mechanics.
• Classification of simple Lie algebras.
• The Lorentz and Poincare groups and their representations, with a discussion of the Dirac equation and Weyl and Majorana spinors.
• A grab-bag of some other topics, including a little bit about conformal symmetry and grand unified theories.

While for each of these topics there are other good textbooks out there, this is a great selection for an advanced undergraduate/graduate physics course. I expect this to justifiably become a popular choice for such courses.

While I liked a lot about the book, I have to confess that there were things about it that did put me off. Some of this likely has to do with the fact that I’ve been working for the last few years on a book (see here) that covers some of the same topics, so I’m hyper-aware of both the technicalities involved, and the issues that arise of how best to approach these subjects. In addition, much of these topics is standard core mathematics, but Zee seems to have consulted few if any mathematicians (at least I didn’t recognize any in his acknowledgements). Unlike some others, this is a subject where mathematicians and physicists really can communicate and teach each other a lot.

Some of the choices Zee makes that I don’t think are good ones are things that input from mathematicians probably would have helped with. Maybe the most egregious is his decision to use the same notation for a Lie group and its Lie algebra, on the grounds that physicists sometimes do this, and to notationally distinguish the two in the usual way (upper vs. lowercase letters) is “rather fussy looking”. Using the same notation for two very different things is just asking for confusion, and I remember struggling with this as a student. Zee is well aware of the problem, on page 79 having his interlocutor “Confusio” say:

When I first studied group theory I did not clearly distinguish between Lie group and Lie algebra. That they allow totally different operations did not sink in. I was multiplying the Js together and couldn’t make sense of what I got.

Please, if you’re using this book to teach students about this subject, discourage them from following Zee in this choice.

There are places in the text where Zee gets things wrong in a way that just about any mathematician could likely have saved him from. One minor example is a footnote saying “Mathematicians have listed all possible finite groups up to impressively large values of n” (actually, they’re classified for ALL values of n) (my mistake, I misread and wasn’t looking at the finite group chapters carefully enough. Zee does get this right).

One place Zee gets things wrong is when he writes down the Heisenberg commutation relations, and says this is an “other type of algebra”, off-topic “since this is a textbook on group theory, I talk mostly about Lie algebras”. Actually those are the commutation relations of a Lie algebra, the Heisenberg Lie algebra, and there’s a group too, the Heisenberg group.

This gets into my own prejudices about the subject, with the story of the Heisenberg group to me (and I think to most mathematicians), a central part of the story of quantum mechanics, something little appreciated by most physicists. Another place where I think Zee goes wrong due to current physics prejudices is in ignoring Hamiltonian mechanics in favor of Lagrangian mechanics. As a result, instead of being able to tell the beautiful story of the Lie algebra of functions on phase space and what it has to do with conservation laws, he just mentions that Noether’s theorem leads to conservation laws and refers elsewhere for a discussion. The connection between symmetry and conservation laws is one of the central parts of the connection between Lie groups and physics, and deserves a lot more attention in the context of a course like this.

So, in summary, the book is highly recommended, with the caveats that you absolutely shouldn’t use the same notation for Lie groups and Lie algebras, and you should supplement Zee’s treatment with that of a certain more mathematically-minded blogger…