There was in Munich last week a three-day workshop on the soul of science. According to Quanta Magazine, the conference was driven by George Ellis and Joe Silk who wrote a cri de coeur in Nature about defending the integrity of physics.
The crisis, as Ellis and Silk tell it, is the wildly speculative nature of modern physics theories, which they say reflects a dangerous departure from the scientific method. Many of today’s theorists — chief among them the proponents of string theory and the multiverse hypothesis — appear convinced of their ideas on the grounds that they are beautiful or logically compelling, despite the impossibility of testing them. Ellis and Silk accused these theorists of “moving the goalposts” of science and blurring the line between physics and pseudoscience. “The imprimatur of science should be awarded only to a theory that is testable,” Ellis and Silk wrote, thereby disqualifying most of the leading theories of the past 40 years. “Only then can we defend science from attack.”
Now there is much to discuss about this conference and about Ellis and Silk’s paper. But for today, let’s focus on one small item. There is this Joe Polchinski who is a “staunch” string theorist, who had a paper read for him in Munich. According to the magazine:
Polchinski concludes that, considering how far away we are from the exceptionally fine grain of nature’s fundamental distance scale, we should count ourselves lucky: “String theory exists, and we have found it.” (Polchinski also used Dawid’s non-empirical arguments to calculate the Bayesian odds that the multiverse exists as 94 percent — a value that has been ridiculed by the Internet’s vocal multiverse critics.)
The critic is Peter “Not Even Wrong” Woit. Woit quotes Polchinski on the this 94%-calculation:
To conclude this section, I will make a quasi-Bayesian estimate of the likelihood that there is a multiverse. To establish a prior, I note that a multiverse is easy to make: it requires quantum mechanics and general relativity, and it requires that the building blocks of spacetime can exist in many metastable states. We do not know if this last is true. It is true for the building blocks of ordinary matter, and it seems to be a natural corollary to getting physics from geometry. So I will start with a prior of 50%. I will first update this with the fact that the observed cosmological constant is small. Now, if I consider only known theories, this pushes the odds of a multiverse close to 100%. But I have to allow for the possibility that the correct theory is still undiscovered, so I will be conservative and reduce the no-multiverse probability by a factor of two, to 25%. The second update is that the vacuum energy is nonzero. By the same (conservative) logic, I reduce the no-multiverse probability to 12%. The final update is the fact that our outstanding candidate for a theory of quantum gravity, string theory, most likely predicts a multiverse. But again I will be conservative and take only a factor of two. So this is my estimate for the likelihood that the multiverse exists: 94%.
Whew! Everything so far was merely introductory, both for today’s meat and for discussion later. Without taking any opinion on the existence of the multiverse, let the theory, i.e. the very complex set of premises, which include a vast array or metaphysical, physical, and mathematical propositions, from which we can deduce the multiverse be called T. T is a complex proposition, and we are interested in whether T itself is true. Why? Because we know the multiverse is true if T is: the multiverse is a deduction or theorem of T. Polchinski wants to bring in Bayesian theory to answer whether T is true. That was mistake number one.
Mistake two is this statement, “I will start with a prior of 50%.” This makes no sense. Theories do not have probabilities. And since theories are nothing but (complex) propositions, neither do propositions have probabilities. Indeed, no thing has a probability. Probabilities are measures of knowledge, therefore they have to come equipped with gauges, i.e. conditions. In other words, all probability is conditional.
Many think one natural gauge is the proposition W = “T might be true”, which is logically equivalent to W’ = “T is true or it is false”. Both of these are tautologies, which we know are true conditional on our knowledge of logic and understanding of English grammar. But it makes no sense to say, as Polchinski said, Pr(T | W) = 50%. Tautologies are non-informative. The best we can do, as I pointed out earlier, is to deduce T’s contingency, which gives it a interval probability (0,1). Of course, Polchinski may not have had the tautology in mind, but some other gauge. Call this G, which relates to some complex proposition in Polchinski’s head. Then it might be true that Pr(T|G) = 50%.
But what would this G have to look like? Well, it would have to be directly probative of T itself, which means of the propositions of which T is composed. And if Polchinski really had such a G, it is more plausible these G-propositions would already be in T to give it support. Why withhold from T knowledge relevant to multiverses? It doesn’t make sense. But then G might have nothing probative to say about T except its contingency like W, in which case 0 < Pr(T|G) < 1.
According to the rules of probability, Pr(T false | G) = 1 – Pr(T| G). But what does it mean to say T is false? Just that at least one of propositions within T is false. And if we knew that, then we would never entertain T. We would instead modify T (which really means making a brand new T) to remove or transform these troublesome propositions. If G told us which part of T was wrong we would fix it.
Put all this another way. If all we had in contention for the multiverse was T, then T is all we have. We can’t judge its truth or falsehood because we have nothing to compare it to. T is it. It’s T or bust.
I’m sure (though I didn’t check) Polchinski’s numerical calculations are on the money, but the end result is meaningless. T has to be compared not against some internal gut reaction, because there is no such thing as subjective probability, but against the predictions T makes or against rival theories, which provide the only natural comparators. That is, Polchinski might have some alternative theory M in mind, a rival to T such that, given M, the multiverse is not a theory of M. Now Polchinski’s G makes a little more sense.
There may be, and almost certainly are, overlapping elements of T and M, sub-propositions which they share. Nevertheless, T is not deducible from M, nor vice versa, else they would be the same theory. We’ve already seen that it makes little sense to have in G propositions which duplicate the multiverse-predictability propositions in T, and the same objection applies to M. That means G is something else. The simplest would be the “freshman” G, which is “There are two rival theories, T or M, and only one of which can be true”. Therefore, Pr(T|G) = Pr(M|G) = 1/2, via the statistical syllogism. But that’s as far as we can go without additional evidence, such as observations of the multiverse (which won’t be had) or via observables deducible from T or M. Other G are possible, but it is easy enough to see, since there is no such things as subjective probability, we’re up against the unquantifiable. Gut feeling as a decision takes the place of probability. In other words, it’s better to go out and find proof of T or M.
Naturally, everything said holds for theories of any kind. Clever readers will see in the criticism of Polchinski the standard argument why hypothesis testing, whether by wee p-value or Bayes’s rule, is based on the fallacy of the false dichotomy.
More to come…Incidentally, none of what I wrote has any bearing on whether the multiverse actually exists.