# Atiyah and Witten in Nature

This week’s issue of Nature has short articles by Atiyah and Witten, both addressing the issue of the current state of string theory.

Atiyah’s piece is an interesting review of Lawrence Krauss’s new book Hiding in the Mirror entitled Pulling the strings, and it concentrates on what Krauss has to say about the relation of mathematics and physics. Krauss ends his book quoting the mathematician Hermann Weyl as choosing beauty over truth, remarking that physicists don’t have this luxury. Atiyah points out the story of Weyl’s work on gauge theory, which Weyl published over Einstein’s strong argument that it was physically wrong. The idea was just so beautiful that Weyl felt there had to be something to it, an opinion that turned out to be amply justified as the concept of a gauge theory has turned out to be among the most fundamental ideas in theoretical physics.

Witten’s piece is entitled Unravelling string theory and it tells the story of how he got interested in string theory and offers a defense of its continued study despite the lack of progress during the past 21 years in using it to come up with a unified theory. His defense consists of three points:

1. It appears to be a consistent generalization of QFT, and is worth study on that grounds alone.

2. It incorporates general relativity and provides a “rough draft” of particle physics.

3. Research on string theory has led to all sorts of spin-offs: insights into confinement, black-holes, mathematics.

Those are certainly the strongest arguments for working on string theory, but I find it disappointing that Witten chooses to ignore much of what has been happening in string theory over the last few years. He addresses only by indirect allusion the whole issue of the landscape and the strong possibility that the string theory framework for unification is inherently incapable of predicting anything. Witten would do particle theory a huge favor by at least acknowledging that if the string theory landscape really exists, it is not, as many seem to think, a new paradigm for how to pursue theoretical physics, but instead the end of hopes for this idea about how to achieve unification.

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### 24 Responses to Atiyah and Witten in Nature

1. Dumb Biologist says:

I have a question for you, Dr. Woit, if you don’t mind:

What’s so great about beauty? It seems one clear distinction between your assessment of strings, and that of, say, Witten, is the opinion regarding its purported “beauty”. Obviously Witten and those who share his belief find their perception of the beauty of strings so compelling they’ve devoted their professional lives to it. You don’t their aesthetic judgement on the matter.

Either way, I have to wonder if the beauty criterion has produced as many red herrings as it has fruitful research programs, given that “beauty” is, in the end, in the eye of the beholder (with many philosophers arguing as only they can about its significance).

I don’t remember the precise wording, but Feynman asserted essentially that “beauty” pales next to experimental verification. I tend to agree, only because I distrust the aesthetic judgements of others, as well as my own, when it comes to describing what is vs. what I’d like reality to be. I wouldn’t want to deny anyone the pleasure of deriving human pleasure from the perceived elegance or gorgeousness of theoretical models any more than I’d want to tell them their joy in experience art or music is worthless. However, it seems to me, in science, evidence trumps everything and anything else, precisely because good experimental data is not generated as a matter of oppinion.

If the theory does it’s job the way a scientific theory should, whether it strikes some as beautiful and others hideous is completely irrelevant. Isn’t it?

2. Juan R. says:

Some comments on Witten’s defense of its continued study despite the lack of progress during the past 21 years in using it to come up with a unified theory:

1. Witten says “It appears”. Yes, i agree, only that…

Is Witten ignoring that there are others more interesting, consistent and realistic ways for generalizing standard QFT: via direct curved spacetime, nonlinear, NC-QFT, TFD (generalized vectors on doubled space), time-symmetric AAAD (Hoyle/Narlikar), via Gelfand triplets (Brussels School), etc.

2. String theory does not incorporates general relativity. This is one of more dishonest claims of string theorists i know. In fact, the supposed background-independent version is not a string theory and is called M-theory. M-theory is not formulated and nobody know what is, if it really exists. M is for mistery…

3. The comment that string theory provides a “rough draft” of particle physics can be easily replied via Woit’s appeal to Pauli

one is reminded of another quote from Pauli. Annoyed by Heisenberg’s claims that modulo some details he had a wonderful uniffied theory (he didn’t), Pauli sent his friends a postcard containing a blank rectangle and the text “This is to
show the world I can paint like Titian. Only technical details are missing.”

4. Research on string theory has led to all sorts of spin-offs: insights into confinement, black-holes, mathematics.

Yes, but like lot of other disciplines with minor publicity in mass media. Or has only string theory provided advances? For example, i estimate the impact of string theory on math less than 1%. The real impact of string theory on disciplines as chemistry, biology, or geology for instance, is 0%.

Bad numbers for the so-named TOE; really bad numbers for the theory that, in words of Brian Greene, was revolutioning our views…

Juan R.

Center for CANONICAL |SCIENCE)

3. woit says:

Dumb Biologist,

One problem with beauty is that it is not precisely defined, so one has to be very careful about what exactly is being claimed to be beautiful, and what isn’t. In Witten’s article, note that he isn’t actually claiming that string theory is beautiful. I’ve written extensively elsewhere about string theory and what about it is or isn’t beautiful, so won’t here.

By “beauty”, theorists often mean the property of getting a huge amount of explanatory power out of a very small number of assumptions, or if you like, a simple equation or equations. Most people agree that the Dirac equation is “beautiful” for this reason. This is the sort of thing many theorists are hoping to find again. They haven’t found this in string theory, since no one knows what the equations are. Experimental evidence is of course ultimately the arbiter of what is true and what isn’t, and it can also often provide hints to theorists about what directions to be looking in. Beauty is another source of hints, especially important when experimental guidance is lacking. Atiyah’s example of gauge theory is an excellent one of an idea that was so beautiful there had to be something right about it. But in its initial form, the idea wasn’t quite right, input from another direction (QM) was needed to get it right.

4. Dumb Biologist says:

OK.

Being myself a mathematical unsophisticate (with none of the virtue of earnest purity that word might imply), it’s clearly a property of mathematically deep theoretical frameworks that is lost on me, but for the qualitative features you describe. Semantics may be as much the issue as any other, I suppose. “Beauty” here seems more-or-less interchangeable with notions like “elegance”, “economy”… To the extent that this more specialized use of “beauty” is understood with broad consensus, then the problem could very well be, as you say, not one of naive reliance upon beauty as a “hint”, but rather its misidentification or abandonment.

Again, the beauty, as defined, of the Dirac equation is over my head, and perhaps I ought to do my best to understand the basics more (buy Penrose’s book?) before I get too suspicious. Because I know I’m no genius, I’ve never trusted my own impressions enough to have ever said to myself, in my work, “that’s just so darn beautiful it’s GOT to be…” I suppose that allows me to always be pleasantly surprised when I’m right, though perhaps it also means I lack “common sense”. I guess I’ve always been far more impressed with, say, the accuracy to so-many decimal places of QED and GR than any other feature those theories posess…probably because I’m too dumb to understand those other features well enough. But, because even the brightest among us can be misled, it seems, by personal biases, it’s always seemed best to toss aesthetics if it looks too much like intellectual baggage.

If one cannot observe, even in principle, my (perhaps ignorant) suspicion turns to alarm, and what little trust I have goes out the window. I find it difficult to resist the urge to dismiss considerations of “every possible universe” that lies beyond our horizon out of hand, no matter how beautiful the theory that describes them might be. If the theory tells us nothing specific about our own universe (or vacuum, or whatever), my dismissive reflex turns to a sense of betrayal of the scientific method.

5. Dumb Biologist says:

I must say, after re-reading Witten’s essay, in light of the notion of beauty meaning “getting a huge amount of explanatory power out of a very small number of assumptions, or if you like, a simple equation or equations”, he certainly seems to be saying ST is beautiful, in not-so-many words. Witten says String Theory “leads in a remarkably simple way to a reasonable rough draft of particle physics unified with gravity”, and that it “has proved to be remarkably rich, more so than even the enthusiasts tend to realize”, citing the insights it provides to quark confinement, BH entropy, and so forth. He outright asserts that its equations are profitable (so to speak), in this physically “beatiful” fashion, since that they give back more than what is put in. This seems, allegedly, to be congruent with what is said of the Dirac equation (which looks simple enough, I suppose, though it’s been a long time since I’ve had to contend with diffy-q’s and matrices), as it explains electron spin, magnetic moment, charge quantization, (where earlier work had to incorporate those properties “by hand”), predicts the existence of positrons, as well as agrees with all the experimental data of the day.

I can’t help but interpret Witten’s words as an endorsement of String Theory’s “beauty”, and that this beauty is the very reason why it is to be taken seriously. Perhaps I’m just reading it all wrong.

Again, its distant relation to empirical inquiry seems not to bother.

6. woit says:

I think Witten is being extremely careful in what he says, trying to find the absolute best argument he can make for string theory, without going too far and saying something that can’t be supported. I don’t think he is saying string theory is too beautiful to not be true, largely because he is saying we don’t yet know what string theory is. He is marshaling evidence for the idea that there is something beautiful going on, but we don’t yet know what it is. Until he actually knows what string theory is, he can’t claim that it is beautiful, and I think he is being careful not to do this in his article.

You’re quite right to be suspicious of arguments based on universes we can’t observe, but this has nothing to do with questions about beauty. Susskind forcefully makes the point that the string theory landscape picture of the multiverse is a quite ugly one. He doesn’t say you should believe because it is beautiful, quite the opposite.

7. Chris W. says:

D. B.,

I think the use of the word “beauty” in science is confusing precisely because the meaning summarized by Peter—“getting a huge amount of explanatory power out of a very small number of assumptions”—does not play much of a role in discussions of aesthetics in other fields (with the possible exception of music, given an appropriate recasting of “explanatory”). When non-scientists see the word used they naturally tend to think of the uses of it that are familiar to them, in connection with the visual arts, architecture, music, etc. Those uses tend to emphasize subjective responses of appreciative non-specialists to works of art.

The best analogy in biology to the deep principles of physics are the deep structural principles of biology, and of course the principles of evolutionary biology. (Needless to say many of these principles have important connections to physics.) These principles place the staggering body of detailed empirical knowledge in biology in an intellectual framework without which much of that empirical knowledge would be far less significant, ie, far less worth knowing.

Of course one might be inclined to say the same of various mythologies. The problem with myths is that, considered objectively and critically, they don’t stand up to close scrutiny as explanations. Seeking after beauty in physics only becomes pernicious if it is used to argue for the irrelevance of testability and empirical verification in science. The existence of this potential should not be turned around and used to argue that theoretical beauty (in the sense given above) is itself irrelevant. We not only want but need more than an enormous catalog of verified but largely disconnected facts.

In short, great theories (and for that matter, great engineering) elicit a kind of objective relief that a cacophony of disconnected and apparently contradictory facts (or problems) can be reconciled and understood in a unified, lucid, and robust way. To a scientist that is beautiful. Mathematical formulation can be be tremendously helpful, and sometime essential, in achieving this relief, although without key informal insights into the relevant problems formalization may simply augment the confusion.

That said, I think we are still in the process of figuring out what beauty in scientific theories ought to mean. The word is merely a label for a quality we are struggling to identify and articulate with ever increasing depth and clarity. That quality is best conveyed by lucid explanations of well-chosen examples, and the fertile insights they have provided.

8. Dumb Biologist says:

Oy. I must simply give up and say beauty gives me a migraine, especially when I try to contemplate what it ought to mean. I guess I’ll never be comfortable with it, and must leave substantive discussion of the matter to those wiser and smarter than myself.

I guess it’s a small comfort that I’ve got comparisons with objective reality to compensate for my philistine lack of aesthetic sense or interest. And no, I don’t mean that sarcastically.

9. pax says:

Justifiying millions of (brilliant) man hours based on beauty alone is definitely questionable. After all, the very notion of beauty is most likely a result of evolution. The human mind has been selected to differentiate between ugly and beautiful as a survival mechanism. Harsh, dry desert terrain – ugly. Waterfall and greenery – beautiful. To now say that the key to understanding the way the world is put together lies in concepts like beauty and its younger sibling symmetry is, at the very least, a circular argument.

10. Chris W. says:

D. B.,

There is another issue here, I think. “Comparisons with objective reality” in physics have probably been more problematic* in physics than in biology for much of the last hundred years, if not before. Theories in physics set the very terms of the discussion to a much greater extent; they tell us what we can expect to observe, and how to go about observing it, which may require tremendous effort and technical skill. Many (most?) physicists’ conception of beauty in a theory is also associated with its consistency, clarity, and comprehensiveness when used in this role.

Trying to follow the Baconian ideal and observe the world without preconceptions or prejudices—whatever you see in the microscope or the telescope,* as it were—is hopeless. You always have to assume something and have some problem in mind; the question is what these presuppositions are, how one would know if a given assumption or problem formulation was leading one astray, and what presuppositions are absolutely central to the enterprise. It obviously helps to have several alternatives in play, which may cut up the alleged objective reality in different ways. Some of these alternatives may be more or less “ugly”—unappealing on logical, philosophical, or “aesthetic” as well as empirical grounds—while still contributing something valuable to critical discussion of a problem. (Consider the case of MOND.)

(I should point out that philosophers of science have discussed this issue at great length, starting in the first third of the 20th century.)

(* Actually, I wonder if this is really the case, that is, if biological observation is really so unguided by theoretical presuppositions.)

11. Anonymous says:

pax: you have to consider that people are not just making that argument from out of the blue. look at it more as a historical obervatoin: physical theories have an uncanny tendency to be beautiful/symmetrical.

12. anon says:

Dumb biologist:

The Feynman quote on the benefits of beauty versus those of experiment is:

‘You can recognise truth by its beauty and simplicity. … The inexperienced, and crackpots, and people like that, make guesses that are simple, but you can immediately see that they are wrong, so that does not count. … We have to find a new view of the world that has to agree with everything that is known, but disagree in its predictions somewhere, otherwise it is not interesting. And in that disagreement it must agree with nature.’ (Character of Physical Law, 1965, p171.)

13. sunderpeeche says:

Dumb biologist:
Chris W gave you an excellent reply. (“The best analogy in biology to the deep principles of physics are the deep structural principles of biology, …”) — this is right on the money. Think of the DNA double helix (coupled with what I believe is called the “Standard Dogma” in biology), how much it has explained based on a few simple ideas. That is beauty. The double helix is an icon of popular culture now — it is on the cover of Dec issue Scientific American and many other places.

14. Juan R. says:

i find really interesting the discussion about the supposed beauty of the Dirac equation addressed above.

Of course, beauty is a subjetive concept and whereas some theoreticians as Kaku claim that the Dirac equation is specially beatiful, others theoreticians such as the own Dirac (i already cited his words several times in this blog) or myself think that the equation is NOT beatiful.

Since string theorists like a wrong concept of ‘beauty’ and are rather unrigorous in her/his academic discussions, it is not so strange the three decades stringy failure for finding a consistent theory of nature.

Still more interesting is that nobody here has still expressed that she/he really understand by “Dirac equation”:

– The original Dirac wavefunction, which is wrong and was ABANDONATED in modern quantum field theory by a Schrödinger equation for funtionals.

– The modern Dirac operator identity appearing in QED, which is not fundamental since does not apply to two-electron systems (in fact, still nobody know what is the full two-electron equation). I would note that also Dirac rejected the modern QED view in his last publications.

– Dirac-like equation obtained on string-brane theory when $$D_{10}$$ is splinted into “compact” and “non-compact” dimensions $$D_{6}\psi_{n} = m_{n} \psi_{n}$$.

I see also some incongruencies about the supposed ‘predictions’ of the original Dirac equation. For example, it is simply untrue that Dirac equation predicted the correct magnetic moment for electron and the prediction of antiparticles would be discussed because the concept of antiparticle on the original Dirac-hole theory is NOT the antiparticle of the modern QED.

For a simple review of the original Dirac equation and its obvious flaws see pages 13-14 of Weinberg manual.

Juan R.

Center for CANONICAL |SCIENCE)

15. Dumb Biologist says:

Well, there’s a lot of public discussion these days about the beauty of Darwin’s insight, as the very simple program of random variation coupled with selective pressure is all that is needed to explain the complexity and diversity of life. (Of course modern evolutionary theory incorporates other mechanisms like genetic drift, takes into account that mutation isn’t always completely random, etc., but those necessary additions to the theoretical framework in no way diminish the importance of Darwin’s proposed mechanism of natural selection).

Logically, evolution is a slam dunk, there’s no denying. I can’t think of a single good argument against the notion that, given a population of replicating, mutating entities in a dynamic environment, evolution simply MUST happen. Its inevitability does seem to make experimentation to empirically demonstrate the point essentially superfluous. What else could possibly happen? I certainly can appreciate the perception of beauty in a concept so simple, yet so powerful and, probably, _necessary_.

Then I look at the ID debate, and any confidence I might have had in the reliability of human perception crumbles. Platonic solids and epicycles have beguiled the minds of many a brilliant thinker in the past. Who knows what today’s equivalent might turn out to be. Of course, without some initial bias, sensible scientific investigation is impossible; and indeed most of those biases are entirely justified. But by the example humanity has provided time and again, it seems ultimate reliance upon anything but the well-tested hypothesis will yield rapidly diminishing marginal returns, and eventually complete detachment from reality.

16. pax says:

Dear Anonymous: (Out of the blue guy…)

Has it occured to you that you regard the physical theory as beautiful
simply because your thoughts, aesthetic sensibility and yourself are a to some extent a product of that construct ?

It’s like saying “This candy is sweet because it tastes of sugar…”

17. Shantanu says:

Peter, one interesting thing in Witten’s article is that he makes
no mention of LQG or CDT (when he talks of critics of string
theory) but does mention twistor theory, non-commutative
geometry .

18. woit says:

Shantanu,

Witten says something like “when critics of string theory have good ideas, they tend to be absorbed as part of string theory”. His failure to mention LQG and CDT could be interpreted as meaning he doesn’t think they are very good ideas. I found that sentence of his kind of strange: he seems to be claiming for instance that black hole entropy and twistor theory have been absorbed as part of string theory, which is a somewhat peculiar way to phrase the situation of the relation between these subjects.

19. Samuel Prime says:

The string theory bashing here is amazing. Why is this? Are such feelings directed towards researchers of string theory (and what they say) or towards the theory itself? (Or both?) They sound like a hint of jealousy.

Okay, so there is competition going on. What’s the deal about that? Physics has always had competition in its endeavor to find the truth (or the most successful theory to date). From what i have read by Witten he has been more careful than how some seem to portray him. He frequently speaks in tentative terms like “if string theory is correct”. And why would he bother to mention the problems with string theory if he’s that biased? He does mention them, but those who want to bad mouth him and string theory choose not to be fair.

Instead of engaging in negative interpretations of Witten’s not mentioning LQG, isn’t it best to settle the matter by asking Witten himself via a friendly e-mail (how he feels about LQG)? On a couple of occasions I have asked him some questions and Ed politely answered me to clear a speculation of mine. The reason Witten mentions noncommutative geometry is because he used it in some of his papers. And perhaps he doesn’t refer to LQG is because he’s either not convinced of it or does not see the need to use it. Do we do the same for LQG researchers when they don’t use string theory?

20. D R Lunsford says:

Homo biologicus non sapiens,

Beauty involves economy, in all cases. The Dirac equation
$$\begin{eqnarray*} (\gamma^{\mu}\partial_{\mu} + im)\phi & = & 0\end{eqnarray*}$$

is very economical. Another way of saying it is

P psi = M psi

the mass is the eigenvalue of the momentum. This equation is really the generalization of E = m to moving electrons. It explains not only the otherwise incomprehensible phenomenon of spin, but also gives a direct statement about the electron’s magnetic moment. To top it off, this little equation implies the existence of a new kind of matter, that was hitherto unsuspected. Not bad for 8 symbols.

21. Chris W. says:

Follow-up on “Theories in physics set the very terms of the discussion to a much greater extent; they tell us what we can expect to observe, and how to go about observing it” (in my comment on 12/23): See this recently posted preprint:

Observables in effective gravity (hep-th/0512200)
Authors: Steven B. Giddings, Donald Marolf, James B. Hartle

This paper indicates the extent to which the above statement has been realized in quantum field theory and general relativity, and how its significance is further magnified in quantum gravity. The theme is pervasive in the vast literature on these subjects.

22. woit says:

Samuel Prime,

“why would he bother to mention the problems with string theory if he’s that biased?”

I don’t think Witten actually does mention any of the serious problems with string theory, other than that we’re not quite sure what the theory actually is. By now there is strong evidence of problems with string theory as a framework for unification, and Witten doesn’t address these, instead just making the positive case for continued research into the theory. He’s within his rights to do so in this kind of short essay, but a serious examination of the problems with string theory is not what he is doing.

You’ll not find coming from me personal attacks on leading researchers in string theory. I think there are serious scientific problems with the theory, which I’ve repeatedly explained in detail, as well as problems with how research into the subject has been conducted. Discussing these issues is not a personal attack on anyone. As for jealousy, I have immense amounts of respect for both Atiyah and Witten, for their talents, their hard work and their incredible accomplishments. Sure I wish I could have accomplished a fraction of what they have, but I’m well aware I’m just not as smart and hard-working as they are.

About string theorists in general, I don’t envy them at all their current situation, and, on the contrary am quite happy with my professional life, as well as with my decision many years ago to not go into research in string theory.

23. Samuel Prime says:

Peter, but Witten does mention the serious problems with string theory. For example in his expository article “Reflections on the fate of spacetime” (which is available on his homepage) he states the good news with string theory, and then later asks “what is the bad news?” And he proceeds to discuss them—the first of which he calls “glaringly unsatisfactory” (see page 26, Ibid). He says clearly the concepts are unclear and the principles underlying string theory unknown or not yet understood. Later on in the article (page 28) he says “We are far from cing to grips fully with this paradigm, and one can scarcely now imagine how it will all turn out.” These are a lot more than saying we are not sure what the theory is, but points to problems with it—and certainly leaves the reader (or me, at least) with the impression that it could all turn out wrong (not that it is “not even wrong”—which is actually unclear at this point). (Unless you say that you have a disproof of string theory!)

By the way, and respectfully, I do not think anyone is in a position to say what Witten is not doing. He may be doing privately what we don’t know from his work publically. No one knew Wiles was working on Fermat’s Last Theorem when he was; many mathematicians like to work on some things and keep them to themselves.

It’s okay with me that there are serious problems with string theory. They could reflect that ultimately it is the wrong, but ingenious, approach. Or that there is still a lot of work ahead; work which might point to a different direction to something closer to the truth.

Anyway, in the end, string theory could be wrong in describing nature (or could be right), but whatever the outcome it has left its mark on the history of Physics it seems to me (having been around a couple decades or more). Let’s remember that quantum theory didn’t come to us on a silver platter.

24. biophysicist says:

Dumb biologist,

I think if you stop worrying about popular ideas of what beauty is and focus on the beauty we as scientists find in the remarkable simplicity and necessity of Darwin’s law of natural selection, then you’ll appreciate the physicist’s view of beauty as well. It seems to me that the only difference (right now at least, this could of course change) between beauty in purely biological theory and physical theory is the presence of symmetry principles in the latter. But the psychological satisfaction in the Dirac equation and Darwin’s natural selection are what I at least mean by beauty (both are beautiful in unique ways though, since the former is conjectured and the latter, as DB points out, just seems inevitable – but remember it still took a human being to point out what we take for granted as inevitable, that’s why it is still beautiful!).