The n-Category Café

In a word, amen!

I imagine that the success of SUSY QCD depends on which quantity you are looking at. The susceptibility seems only to be off by a few percent, whereas the beta-function is off by a factor infinity, since N=4 SYM remains scale invariant after quantization.

The problem with AdS/CFT in this context is that it is an uncontrolled approximation, AFAIU. That it involves infinitely many colors is not a problem, since you typically can calculate n-color corrections as a power series in 1/n, and 1/3 is close to 1/infinity.

In contrast, a theory with SUSY, especially four SUSIES, is qualitatively different from a theory without SUSY. I have at least never heard of somebody considering N = 4-ε SYM, work out the corrections as a power series in ε, and set ε = 4 in the end, which one would expect to do if one could turn N=4 SYM into a starting point for a controlled approximation.

So Atiyah’s comment is rather odd from your perspective?

The part of the quote that I have seen, taken by itself – yes.

Necessarily, any simply principled ‘theory of everything’ will require complicated techniques to extract the way our messy universe is.

I would think so.

Simple consequences of simple principles would be too, well, simple?

And rather unlikely to describe a highly non-symmetric world.

I think the expression “pool of rich mathematics” can only be taken here as a (subjective) statement made by someone relying on his/her past experience rather than a claim that “math is endless” as you ask (which I think isn’t).

About your other questions, complicated mathematics may not have anything to do with reality, just like it’s possible to imagine what the life of a bunch of blue elephants who speak seven languages would be like: pretty complicated but not real. Since maths is always purely imagined (in the sense of not exerting any influence on non-humans, like a cat or a rock) the same non-relevance argument applies.

Christine,

Mathematics tells us about the world precisely because it tells us about the nature of our minds. I wrote about this in an essay in Sica’s book The Language of Science

Is Math endless? Probably. Is math about the Universe endless? Possible, but less probable, in the following sense.

Richard Feynman, most of the time, exulted in how simple assumptions could lead to robustly complicated behaviors. He loved solving problems on the fly, and, in his famous course “Physics X”, even liked to solve audience-supplied problems in public. Well, in a classroom of grad students, a few undergrads, and visitors.

Yet once in a while, he entertained an alternative view. He discussed with me more than once (1968 until a few months before his death) the possibility that there are actually an infinite number of “natural laws”, each expressed by its own equations, with no more fundamental meta-equation. Perhaps, he speculated, some of these infinite number of natural laws only occur at very high energies, or very weird combinations of parameters, or at different ages of the universe.

He always claimed that there were plenty of people, even at Caltech, who were better at Math than him. He gave some priority to his “physical intuition.” He even told me that he tolerated my relative weakness in Math because I did have good physical intuition. He was skeptical of String Theory as perhaps pretty Math, but not established to have any connection to Physics as he saw it.

How many “knobs” are there which we can tweak to make gravity not emerge? Closed strings obeying special relativity yield graviton states upon quantization. I can change the string length, or I can ramp up the dilaton expectation value to vary the coupling, but just how hard is it to make the basic form of gravitation go away?

Etcetera: one can imagine all sorts of scenarios where we’re bored stiff by an event of type 1, or fantastically thrilled by an event of type 3.

Case in point:

I seem to have lent my copy of Intellectual Impostures to a friend, so I’d have to look up the references from scratch, but I recall that Weinberg among others has pointed out that the “type 1” prediction of GR, the deflection of light during a solar eclipse, was in fact a worse test than the perihelion of Mercury. Once all the error bars are figured in, etc., the eclipse measurement was more likely to be wrong. So, it’s the type 2 prediction which gives us more reason to perk up our ears.

A true pedant might want to call the perihelion measurement a type 3 prediction, because it could be “predicted” (or at least “explained away”) by a dark matter hypothesis: an intra-Mercurial planet, which the people of the time called Vulcan. (Insert your own Star Trek joke here.)

Of course, now we have type 1 predictions — gravitational redshifts, to begin with — making the whole shebang more than a little academic.

Blake Stacey wrote:

A true pedant might want to call the perihelion measurement a type 3 prediction, because it could be “predicted” (or at least “explained away”) by a dark matter hypothesis: an intra-Mercurial planet, which the people of the time called Vulcan.

Le Verrier predicted the existence of Neptune in the 1840s, due to anomalies in the motion of Uranus. It was found in 1846. Attempting to repeat his success, in 1859 he predicted the existence of a new planet to explain the precession of the orbit of Mercury. Between 1859 and 1878 there were some reported sightings of this planet, and it was dubbed ‘Vulcan’. But these sightings were never reliably confirmed. I’ve heard that by the time Einstein came along, fans of this hypothesis were reduced to positing a gaseous Vulcan to explain the precession of Mercury.

Shades of dark matter indeed!

Anyway, the true pedant could argue that almost every really interesting type 2 prediction is really a type 3 prediction, because someone, somewhere, has some nutty theory that already predicts this number. For example, there are certainly plenty of crackpot numerologists who can ‘explain’ the masses of elementary particles!

So, perhaps type 2 should be defined as:

2. The theory lets us calculate some quantity whose value we previously knew, but could not calculate using previous accepted theories.

But I don’t have the patience to fill all the other loopholes one can dream up, so please don’t point out more!

We can imagine three things happening:

Yes! Certainly. I agree. I wrote essentially the same, necessarily in other words, a few comments above:

Technically I think it is quite right to say that string theory predicts gravity. […] we may tend to dislike stating this technically correct statement this way, because we may feel that it does not sufficiently amplify the point that this prediction is rather useless, for our practical purposes. #

So we all agree that “predicts/explains gravity” does not imply “get too excited”.

What I was just trying to point out is that the reverse is also false: “don’t be excited about it” does not imply that “string theory does not predict/explain gravity”, which was the statement I was responding to #.

For me, it is important that I understand what is true and not about the technical aspects of a given theory and know how to distinguish that from assessing what that implies for the relevance of the theory for our endeavor of understanding the universe.

I think it is a true fact that string theory predicts/explains gravity. You and Peter Woit keep emphasizing of how little use this is, for our practical needs (that’s what I meant by the “sociological” implication of this fact). And I agree with that!

I don’t have the relevant books to hand, but I seem to remember an argument to the effect that scientists were more impressed by GR’s retrodiction of Mercury’s behaviour than they were by the bending of star light, and that they were right in this as Mercury tested GR more severely.

The argument was conducted in terms of number of parameters tested. For the sake of argument let’s simplify the situation so that we have a theory which when we ensure it lines up with an old theory in a limit fixes its constants. Then we may imagine that as we look at the expansion as a Taylor series about the classical solution there are a string of predicted coefficients. Now to the point, it may happen that a piece of retrodiction puts more of these coefficients to the test than does a prediction. In which case the retrodiction was a more severe test and has greater confirmatory power.

Of course, if you’re viewing all this from the outside and you see a new phenomenon - like light-bending - predicted and found, you’ll probably be more impressed than by an old phenomenon - like Mercury - explained. Without knowing the details of parameter-fixing this is reasonable.

Internally things are more subtle than I’ve allowed so far. You’d want to have an idea of how the selection of a particular theory from a family is done. Was it a ‘natural’ family, the chosen member of which is a simple choice by say agreement in the limit with an old theory. How much support do the principles guiding the choice of that family have. The fear being that the family has been engineered to look like a simple choice has been made to agree with the old theory.

A final note, those of you who have been reading my posts and/or cake talk on statistical learning theory will know that learning is not just about the number of parameters.

John suggests 3 things that may increase our confidence in a scientific theory:

1. The theory lets us to calculate some quantity whose value we didn’t previously know. We measure the quantity and the theory turns out to be right.
2. The theory lets us calculate some quantity whose value we previously knew, but could not calculate using previous theories.
3. The theory lets us calculate some quantity whose value we previously knew, and could calculate using previous theories.

Item 1. (and only item 1.) is what I understand by the word ‘prediction’. If the quantity predicted and measured had been in some way determined earlier (but unknown to the proponents of the theory, for whatever reason), then you might say ‘retrodiction’ instead, but this is not philosophically signifcant.

I would call item 2. ‘explanation’, rather than ‘prediction’. Item 3. may be an explanation as well, if there is some superiority of the new theory over the old: initially, if it simpler (or background independent, or otherwise preferable, possibly controversially so); later on, if we come to believe (probably through separate evidence of type 1.) that the old theory is simply wrong and the new theory is (more) correct.

Well, this is how I understand the words ‘prediction’ and ‘explanation’, but I won’t fight for them. Now that John has made this list, it’s probably better just to use the numbers 1,2,3., since they get at the heart of the matter, rather than arguing over the meaning of common words.

And here is one point at the heart of the matter: No matter how many successes a theory has of type 3., and even of type 2., in science we require confirmation of type 1. as well. If necessary, we force events of type 1. to occur; that is, even if we have already accounted for all observations, we perform experiments to create new observable phenomena. Without experimentation (or some other source of continually new observations), we do not have falsifiability (to use Popper’s term), at least not in practice. Then we risk degeneration into ‘Greek science’ (if I may be so boldly insulting to Aristoteles and company), flying into the clouds of theory without the ground of observation. (And this problem may occur through no fault of the theory or its proponents, if we simply lack the technological ability to perform experiments! Therein lies the tragedy of this endeavour.)

“Explains” would be fine with me.

My choice, in chapter 6 of my book, was “accounts for”. “Explains” comes with quite a lot of philosophical baggage from the heyday of logical empiricism.

On the other hand, a complicated theory doesn’t really ‘explain’ a simple fact

I take it you are arguing that string theory does not even “explain” gravity (let alone predict it) due to it being more complicated than Einstein gravity. Is that what you have in mind?

This is curiously opposite to how I perceive the situation. String theory may be all wrong and everything, but it sure derives gravity (and a little more, maybe a little too much ;-) from a very simple premise:

The action of the plain relativistic particle (in Nambu-Goto form) is the proper worldvolume of its worldline. Quantizing this gives the Klein-Gordon equation (the relativistic Schrödinger equation). Second-quantizing this gives free field theory.

The single premise of perturbative string theory is to replace the 1-dimensional worldline of the Klein-Gordon particle by a 2-dimensional surface, while naturally keeping the action to be given by the proper volume.

That’s it. Nothing more. Nothing more natural than this generalization.

Quantize this, and you run into a very rich structure, including gravity.