Not Even Wrong

Lisa Randall has an Op-Ed piece in today’s New York Times entitled Dangling Particles. The title seems to have little to do with the piece, but I suppose it is a play on words on “dangling participle”, a term for a sort of faulty grammar. Randall’s topic is the difficulty of communicating scientific topics, and her comments on the problems caused by scientist’s different use of words and by the complex nature of much science are true enough and unobjectionable.

But I still find the sight of a string theorist lecturing the public on how to properly understand science to be a bit jarring. Randall tries to claim that the difference between the colloquial usage of the word “theory” and the way it is used by scientists is a source of problems with the public understanding of science. She writes

For physicists, theories entail a definite physical framework embodied in a set of fundamental assumptions about the world that lead to a specific set of equations and predictions – ones that are borne out by successful predictions.

Yet she keeps on referring to “string theory”, although the subject is distinctly lacking in specific equations and predictions (she does note that “theories aren’t necessarily shown to be correct or complete immediately”, but the problem with string “theory” is not that we don’t know whether it is correct or complete, but that it isn’t really a theory, rather a hope that one exists).

Instead of devoting their time to writing for the public about the scientific status of issues that they’re not really experts in (e.g. global warming), it seems to me that string theorists would do better to first address the outbreak of pseudo-science now taking place in their own subject. When the intelligent design people get around to noticing how much of the highest level of research in one of the traditionally most prestigious sciences is now being conducted without any concern for falsifiability or traditional norms of what is science and what isn’t, the fallout is not going to be pretty.

Update: Sean Carroll has a posting about the Randall Op-Ed piece over at Cosmic Variance. He quotes approvingly Randall’s claim that Intelligent Designers don’t make a distinction between the colloquial usage of “theory”, meaning an idea not necessarily better grounded than a hunch, and the way real scientists use the term. As for whether string theory deserves to be called a “theory”, here’s a quote from Gerard ‘t Hooft (from his book In Search of the Ultimate Building Blocks):

Actually, I would not even be prepared to call string theory a “theory� rather a “model� or not even that: just a hunch. After all, a theory should come together with instructions on how to deal with it to identify the things one wishes to describe, in our case the elementary particles, and one should, at least in principle, be able to formulate the rules for calculating the properties of these particles, and how to make new predictions for them. Imagine that I give you a chair, while explaining that the legs are still missing, and that the seat, back and armrest will perhaps be delivered soon; whatever I did give you, can I still call it a chair?

Update: Lubos Motl has some comments about Randall’s Op-Ed piece and about my posting. As usual, I come in for a fair amount of abuse, but at least this time I’m in good company (‘t Hooft’s views are characterized as “just silly”).

Update:John Baez points out that the article is now up at the Edge web-site. Over at Pharyngula, there’s a posting about Danged physicists. Evidently biologists are not amused at all about Randall’s comments about evolutionary biology. They seem to think that string theorists are arrogant and prone to going on about things they don’t really understand.

Mathematician Dave Morrison is giving a colloquium talk tomorrow at the KITP with the provocative title How Much Mathematics Does A Theoretical Physicist Need To Know? It should soon be available for viewing on the KITP web-site, and I’m looking forward to seeing what he has to say.

I’m not at all sure myself how much mathematics a theoretical physicist needs to know, it certainly depends on what they’re trying to do. But there does seem to me to be a well-defined list of what mathematics goes into our current most fundamental physical theories, and anyone who hopes to work on extending these should start by learning these subjects, which include (besides the classical mathematical physics of PDE’s, Fourier analysis, complex analysis):

Riemannian geometry
More general geometry of principal and vector bundles: connection, curvature, etc.
Spinor geometry
Lie groups and representation theory
deRham cohomology

I’m sure others have different ideas about this….

Update: Dave Morrison’s talk is now on-line here. He began his talk my noting that it had been advertised here on “Not Even Wrong”, and he put up a slide of my posting and people’s comments as an example of people’s lists of what mathematics theoretical physicists should know. He did say that that his talk wasn’t intended to provide such a list, but rather various comments about how physicists can fruitfully interact with mathematicians.

He began by giving several examples of people who had to construct new mathematics to do physics: Newton, Fourier, Heisenberg, and Gell-Mann. David Gross correctly objected that SU(3) representation theory was already known before Gell-Mann started using it, even though at first Gell-Mann wasn’t aware of this. As for more recent interactions, he mainly mentioned the connection between the index theorem and anomalies, as well as various math related to the quantum hall effect. For some reason he decided not to go into the relation of string theory and mathematics, which has been quite fruitful. He did say that he still believes there is some unknown more fundamental way of thinking about string theory that will involve now unknown mathematics. His general advice to physicists was that they should be willing to acquire mathematical tools as needed, but should be aware that if they ask a mathematician questions, they are likely to get answers of too great generality. He ended his talk early, opening the floor to a long discussion.

I just finished reading an interesting new book by astrophysicist Mario Livio. It’s called The Equation That Couldn’t Be Solved, and the subtitle is “How Mathematical Genius Discovered the Language of Symmetry”. Livio’s topic is the idea of a symmetry group, concentrating on its origins in Galois theory.

The first part of the book contains a wonderful detailed history of the discovery of the formulas for the roots of third and fourth order polynomials, and the much later proofs that no such formulas existed for general fifth order polynomials. The romantic stories of the short and tragic lives of Abel and Galois are well-told, in much more detail than in other popular books that I’ve seen. Galois was the one responsible for first really understanding the significance of the concept of a group, and using it to get deep insights into the structure of the solutions of polynomial equations.

The latter part of the book deals with the important role of symmetry in modern theoretical physics, and this is a topic treated in many other places in more detail. Livio gives the standard party-line about string theory, but he does do one very interesting thing. He notices that while string theory implies various sorts of symmetries, e.g. supersymmetry, it lacks a fundamental symmetry principle itself, and this leaves open a very important question. Does physics at its most fundamental level involve a symmetry principle, or are symmetry principles an artifact of our throwing out complexity and only focussing on simple situations that we can understand? Perhaps symmetry is not fundamental, but only an artifact of our limited abilities to understand things. Livio asks several people this question, and gets the following answers:

Weinberg: symmetry might not be the most fundamental concept in the ultimate theory, and “I suspect that at the end the only firm principle will be that of mathematical consistency”. (I don’t think I really understand what Weinberg has in mind here)

Witten: “there are still missing, or unknown ingredients in string theory” and “some concepts, such as Riemannian geometry in general relativity, may prove to be more fundamental than symmetry.”

Atiyah: “We come to describe nature with certain spectacles… Our mathematical description is accurate, but there may be better ways. The use of exceptional Lie groups may be an artifact of how we think of it.”

Dyson: “I feel that we are not even at the beginning of understanding why the universe is the way it is.”

There is one interesting thing that Livio gets wrong. He explains Klein’s Erlangen program of identifying the notion of symmetry with the notion of a geometry, but then says that this is precisely what Riemannian geometry is. This isn’t really right, since the non-Euclidean geometries Klein was using are basically homogeneous spaces of Lie groups, whereas Riemann’s notion was more general, just insisting that the geometry be locally Euclidean. To unify these two points of view, you need the later ideas of Elie Cartan about Cartan geometries and connections. A related distinction is that Klein was considering finite dimensional symmetry groups, whereas in Riemannian geometry you don’t have a global symmetry group. You do have infinite dimensional groups of local symmetries, e.g. the diffeomorphism group, and the gauge group of frame rotations. By the way, a nice article about the early history of gauge theory has just appeared on the arXiv.

My main problem with Livio’s book is that he only discusses the groups themselves, and doesn’t even try to explain what a representation of a group is. For the applications to quantum mechanical systems and to particle physics, it is this notion of a representation of a group that is absolutely crucial.

I realize that this is a low form of entertainment, but reading Lubos Motl’s blog today has definitely livened up my birthday, which in recent years has been a rather sad occasion. It’s hard to say what is the funniest thing there since it’s all great stuff, including:

1. Crazed, heavily ideological attacks (here and here) on climate scientists, who unlike Lubos, actually know something about the subject. The comment sections feature mathematician Greg Kuperberg, who has the hilarious idea that it’s possible to try and have a rational discussion with Lubos on this subject.

2. Kuperberg’s attempts to endear himself to Lubos by attacking the evil Peter Woit, announcing that even though he doesn’t understand string theory (something he has shown a perverse interest in demonstrating publicly, besides his comments on Lubos’s blog, see here and here) he believes it because “string theorists seem credible, seem talented, and have appointments at top universities.”

3. Lubos’s response to said attempts, comparing Kuperberg to some of his more “out-there” commenters.

4. Lubos’s claims that neither Lee Smolin nor I know what we’re talking about when we point out that perturbative finiteness of the superstring is not yet proved beyond two loops, followed by his claim that QFT perturbation series are Borel-summable, nonsense that Jacques Distler then writes in to correct.

Some may object that it’s highly unfair to use the fact that some of its practitioners and supporters are out of their gourds to make fun of string theory, but, hey, it’s my birthday, so I can do what I want today, right?

The October issue of the Notices of the AMS is now available on-line. It has an interesting historical article about Henri Poincare, and a short expository article called WHAT IS… a Pseudoholomorphic Curve by Simon Donaldson. Counting these pseudo-holomorphic curves is what topological sigma models do, and they have turned out to have many different kinds of mathematical applications, including the new field of so-called Gromov-Witten theory, as well as several others.

There’s also an extensive interview with Fields medalist Heisuke Hironaka. I’ve heard that Hironaka is a celebrity in Japan, with one of my colleagues once telling me that during a trip to Japan he was surprised to see Hironaka on a billboard selling something or other.

The latest issue of Astronomy magazine has two articles hyping the landscape/multiverse/anthropic principle and cosmic superstrings. Many well-known theorists are quoted supporting the anthropic principle and the multiverse, including Joe Polchinski, Nima Arkani-Hamed, Martin Rees, Max Tegmark, Alexander Vilenkin, Alan Guth and Lenny Susskind. The only negative quotes are from Paul Steinhardt and David Gross (whose quote is just one word: “virus”, that he used to refer to the anthropic principle a couple years ago).

Max Tegmark is quoted as saying the kind of thing that motivated John Horgan’s recent NYT Op-Ed piece: “I fully expect the true nature of reality to be weird and counterintuitive, which is why I believe these crazy things.” Funny, I thought scientists were supposed to believe things, crazy or not, because of experimental evidence for them. At a Templeton Foundation sponsored conference on the multiverse, supposedly Martin Rees “was confident enough of the multiverse’s existence to stake his dog’s life.” And Andrei Linde “went further, claiming he would put his own life on the line.” Horgan might point out that neither Linde nor Rees’s dog are in much immediate danger since no one has any plausible idea of how one could ever show that there is no multiverse.

Linde, Vilenkin and Susskind acknowledge that they don’t know how to use the anthropic principle to predict anything, with Linde noting the problem of an infinite number of possible vacuum states: “There are many different ways of counting infinities, and we don’t know which methods are preferable.” Vilenkin is quoted as saying the problem has to do with the lack of the right “statistical techniques”, which is kind of misleading, since the problem isn’t one of mathematical technique. Susskind actually sounds the most sober of the lot, saying it will be a long time before the anthropic principle can be used to predict anything and “At the moment, it’s telling us more about what not to do than what to do.” Polchinski on the other hand, goes for maximum hype value, claiming that “The value we now measure for the cosmological constant is precisely what Weinberg predicted.” Of course, by “precisely”, he means “off by one to two orders of magnitude, much more if you allow not just the cosmological constant to vary.”

The article on cosmic superstrings also contains quite a lot of hype from Polchinski, who not only is pushing the idea that the “CSL-1” object is a galaxy lensed by a cosmic string, but that “We’re likely to go from one event to 1,000 events in 10 years” and “We’re really at the dawn of a new era of science.” For more about this, see the latest posting on Lubos Motl’s weblog.

Over at Cosmic Variance, Clifford Johnson has a posting about an article in the Guardian about wacky science stories in the media by Ben Goldacre, who runs the Bad Science weblog devoted to this topic. Clifford is quite critical of media coverage of science, but the only examples he gives are ones related to health scares. String theory inspired wacky science stories like the ones in Astronomy aren’t mentioned, neither is the fact that here the problem may not be incompetent science journalists, but the fondness for hype of some of his prominent colleagues.

In other popular science magazine news, I learned from David Appell’s weblog that the New York Times is reporting that Discover magazine is being sold by Disney to Bob Guccione Jr. Guccione says that he intends to add a humor column to Discover and to create two new print magazines devoted to science. He claims that scientists are kind of like rock stars: “a bunch of people with strong egos and God complexes. That sounds like rock ‘n’ roll to me.” I guess he liked Michio Kaku’s recent cover article.

During the last couple days, some interesting commentary on quantum gravity has appeared at a couple places on the web. One is at John Baez’s latest edition of his proto-blog This Week’s Finds in Mathematical Physics. John is mainly writing about operads, but he begins by saying a bit about why he’s working on pure math rather than quantum gravity these days:

Work on quantum gravity has seemed stagnant and stuck for the last couple of years, which is why I’ve been turning more towards pure math.

He mentions the “landscape” and the problems it is causing for string theory, suggesting a reason Susskind’s “anthropic” nonsense is getting attention:

perhaps it’s because nobody really knows how to get string theory to predict experimental results! Even after you chose a vacuum, you’d need to see how supersymmetry gets broken, and this remain quite obscure.

But instead of spending time bashing string theory, John admirably also has a critical take on his own side of the LQG/string theory controversy, noting that

it has major problems of its own: nobody knows how it can successfully mimic general relativity at large length scales, as it must to be realistic! Old-fashioned perturbative quantum gravity failed on this score because it wasn’t renormalizable. Loop quantum gravity may get around this somehow… but it’s about time to see exactly how.

Jacques Distler also has an interesting posting about quantum gravity, based on his introductory lecture to the string theory class he is teaching this semester. He explains what some of the generic problems with quantum gravity are, from an effective field theory/renormalization group point of view, and how string theory gets around them. There are also some interesting comments about observables in quantum gravity and the signficance in this context of non-trivial gauge transformations at infinity. Unfortunately, unlike John, Jacques doesn’t believe in being very explicit about the problems his side is having (to be fair, maybe that’s the topic of another lecture). He does mention background independence and refers to discussion elsewhere, where students could learn about the lack of a non-perturbative formulation of the theory. But his claim that string theory “provides a unique, or nearly unique UV completion” seems to me seriously misleading, and deserving of elaboration lest the uninitiated get the wrong idea.

Jacques does deal in a somewhat peculiar way with a commenter named Jason who is happy with the idea of a quantum gravity theory that can’t predict anything at all at the Planck scale. Instead of making the obvious point that believing in a theory that can’t predict anything is not what scientists do, Jacques writes

Careful, Jason. A certain self-anointed String Theory gadfly might hear you.

Perhaps Jacques meant to write “self-appointed”, since I’d never thought of myself as a “gadfly” until Sean Carroll recently referred to me as such. If I were the sort to self-anoint, I suppose I’d prefer something more serious sounding than “String Theory gadfly”, maybe “String Theorist’s worst nightmare”…..

I recently got a copy of a very interesting new textbook entitled A First Course in Modular Forms by Fred Diamond and Jerry Shurman. Fred was a student of Andrew Wiles at Princeton, and came here to Columbia as a junior faculty member at the same time I did. He now teaches at Brandeis.

The title of the book is a bit deceptive, what it is really about is what used to be called the Taniyama-Shimura-Weil (or some subset of those names) conjecture, but now is often known as the Modularity Theorem. Most of this theorem was proved by Andrew Wiles (with help from Richard Taylor), who famously used his result to prove Fermat’s last theorem. More recently, the proof of the full theorem was completed by Fred, together with collaborators Christophe Breuil, Brian Conrad and Richard Taylor. Stating the modularity theorem precisely requires some serious mathematical technology, an imprecise statement is the “All rational elliptic curves arise from modular forms”. This fits into the Langlands program of establishing a correspondence between arithmetic objects (in this case elliptic curves over the rational numbers), and analytic objects (in this case modular forms). If one can do this, typically the fact that the analytic objects are pretty well understood allows one to get a vast amount of very deep information about the more mysterious arithmetic objects (e.g. being able to count solutions to equations over the rationals or integers).

The book takes an interesting approach to the Modularity Theorem, not trying to actually prove it. The proof involves highly sophisticated mathematical technology, and really understanding it is still the province of experts. If one wants to try and learn this technology, two places to look are the volumes Modular Forms and Fermat’s Last Theorem and Arithmetic Algebraic Geometry, which are the proceedings of two different instructional conferences. Instead of trying to give a proof, Diamond and Shurman’s book explains exactly what the various related versions of the Modularity Theorem say. This covers a range of beautiful mathematical ideas, much of which hasn’t before had a particularly readable exposition. Until now, the main reference for some of this material has been Shimura’s Introduction to Arithmetic Theory of Automorphic Functions, a famously difficult text.

The book is advertised as “A First Course” and attempts to minimize the prerequisites necessary to read it, making it conceivable to even use the book with advanced undergraduates. This is a worthy goal, but may be a bit over-ambitious. I suspect most people will get more out of the book if they already have had exposure to some of this mathematics at a slightly more basic level. One place to get this is Neal Koblitz’s Introduction to Elliptic Curves and Modular Forms. But this really is a wonderful book, making accessible parts of the really beautiful mathematics which mathematicians have been making great progress in understanding over the last decade.