Provability and numerical experiments

A tempting view of mathematics is that mathematicians are discovering not facts about what is true, but about what is provable from what.

But proof is not the only way mathematicians have of getting at truth. Numerical experiment is another. For instance, while we don’t have a proof of Goldbach’s Conjecture (each even number bigger than two is the sum of two primes), it has been checked to hold for numbers up to 4 ⋅ 10¹⁸. This seems to give significant inductive evidence that Goldbach’s Conjecture is true. But it does not seem to give significant evidence that Goldbach’s Conjecture can be proved.

Here’s why. Admittedly, when we learned that that the conjecture holds for some particular number n, say 13, we also learned that the conjecture can be proved for that specific number n (e.g., 13 = 11 + 2 and 11 and 2 are prime, etc.). Inductively, then, this gives us significant evidence that for each particular number n, Goldbach’s conjecture for n is provable (to simplify notation, stipulate Goldbach’s Conjecture to hold trivially for odd n or n < 4). But one cannot move from ∀n Provable(G(n)) to Provable(∀n G(n)) (to abuse notation a little).

The issue is that the inductive evidence we have gathered strongly supports the claim that Goldbach’s Conjecture is true, but gives much less evidence for the further claim that Goldbach’s Conjecture is provable.

The argument above is a parallel to the standard argument in the philosophy of science that the success of the practice of induction is best explained by scientific realism.

How likely are the laws of nature?

This is one of those annoying loosey-goosey big-picture posts.

Consider the Newtonian law of gravitation: F=Gmm'/r². What should be the prior probability of that law?

Humean line of thought: Zero. After all, consider the continuum of laws of nature of the form F=Gmm'/r^p, where p is some real number. The case where p=2 is just one case out of a continuum. And of course the schema F=Gmm'/r^p is just one schema out of a continuum of schemata (consider, for instance, replacing the multiplication operations on the right hand side with a continuum of other operations). So the prior probability of F=Gmm'/r² is simply zero.

Complexity line of thought: Moderate. After all, the elegant formula "F=Gmm'/r²" is by far simpler than the vast majority of its alternatives (most laws of the form F=Gmm'/r^p have no finite expression, since in most cases the number p will be a real number with no finite mathematical description).

If the Humean line of thought is right, Bayesianism has no hope as a model of how scientific reasoning works. The Complexity line of thought allows for a Bayesian picture of scientific reasoning.

The Humean and Complexity lines of thought come with different pictures of how probabilities are to be assigned to situations. The Humean picture is based on the idea that you've got a bunch of fundamental physical entities, say particles, and then you generate situations by assigning them random fundamental physical properties. From that point of view, clearly the Newtonian law of gravitation has probability zero.

The Complexity line of thought presupposes a different picture. The picture is that probabilities are tied to linguistic expressions. That to generate the probability of a situation, you generate a random complete linguistic description of a world, and identify the probability of a situation with the probability that such a random description entails the situation.

But what a strange thing the Complexity line of thought is! It is as if our picture of the world was that the really central thing about the world wasn't the physical stuff and its fundamental physical properties, but the descriptions. It is as if reality were fundamentally linguistic, or at least explained by something linguistic, as if the cosmos came from a being who said: "Let it be the case that s", and we then assigned probabilities to different values of "s".

In other words, the Complexity line of thought is at heart not naturalistic. But of the two lines of thought, it is the one that is needed for a Bayesian picture of scientific reasoning.

The Complexity line of thought has technical problems, too. Suppose I perform some experiment and the result can be any real number between 0 and 1. The Complexity line of thought will, I think, assign probability one to the hypothesis that the result of the experiment is a finitely describable real number. But surely other real numbers are possible. So what is to be done? It is, I think, to move from the Complexity line of thought to a theistic line of thought focused on value (and a certain autonomy in nature can then a value, and that could allow for randomness and hence for indescribable real number outcomes).

A hypothesis about inductive reasoning

It is normal to talk of "inductive logic", as if non-deductive reasoning formed a branch of logic, with discoverable rules. But what if it is not so? What if the rules of inductive reasoning, unlike the rules of deductive logic, are merely "subjectively necessary", to use Kant's phrase? It is perhaps simply the case that our minds are hard-wired to think in certain ways inductively. This hard-wiring is truth-conducive, not for any deep logical reason (as in the case of deductive logic, where the validity of modus ponens, and the truth of excluded middle, etc. are all necessary truths), but simply because God created us with minds hard-wired to reason inductively in ways that match the arrangement of large segments of the world that he has created.

One can say some of this with natural selection in place of God, but natural selection will only yield the result that our minds' functioning matches the structure of the world in those respects that are relevant to the fitness of our evolutionary forebears—it will give us little or no reason to think that things will work out when we do cosmology or quantum mechanics.

If this is right, then we should not be surprised if one particular formalization of inductive logic—say, the Bayes-Kolmogorov probabilistic account—yields doxastic rules that some of our doxastic practices break, and are right to break. (See the previous several days' posts.) For the theistic story gives us reason to think that our inductive reasoning will get us to the truth, but does not give us much reason to think that our inductive reasoning can be formalized. If this is right, then working scientists may very well do better than ideal Bayesian epistemic agents, say, and be unable to explain their successes.

Probably, the epistemology that would go along with a view like that would have to be some sort of proper-function epistemology. But I am happy to leave that to the epistemologists—I am just a probability theorist.

[Edited. -ARP]