# Logical uncertainty and Mathematical uncertainty

1y26th Jun 20186 comments

# 39

There is a significant difference between uncertainty about mathematical truths in cases where there isn't a known procedure for checking whether a mathematical claim is true or false, versus when there is but you do not have the computational resources to carry it out. Examples of the former include the Collatz and twin prime conjectures, and examples of the latter include whether or not a given large number is a semi-prime, and what the first decimal digit of Graham's number is.

The former should not be called logical uncertainty, because it is about what is true, not about what can be proved; I'll call it mathematical uncertainty instead. The latter really is uncertainty about logic, since we would know that the claim is either proved or refuted by whatever theory we used to prove the algorithm correct, and we would just be uncertain as to which one.

It's well-known that standard probability theory is a poor fit for handling logical uncertainty because it assumes that the probabilities are logically coherent, and uncertainty about what the logical coherence constraints are is exactly what we want to model; there are no possible outcomes in which the truth-value of a decidable sentence is anything other than what it actually is. But this doesn't apply to mathematical uncertainty; we could study the probability distribution over complete theories that we converge to as time goes to infinity, and reason about this probability distribution using ordinary probability theory. Possible sources of evidence about math that could be treated with ordinary probability theory include physical experiments and black-boxed human intuitions. But another important source of evidence about mathematical truth is checking examples, and this cannot be reasoned about in ordinary probability theory because each of the examples is assigned probability 1 or 0 in the limit probability distribution, since we can check it. So just because you can reason about mathematical uncertainty using ordinary probability theory doesn't mean you should.

Logical induction, taken at face value, looks like an attempt at handling mathematical uncertainty, since logical inductors assign probabilities to every sentence, not just sentences known to be decided by the deductive process. But most of the desirable properties of logical inductors that have been proved refer to sequences of decidable sentences, so logical induction only seems potentially valuable for handling logical uncertainty. In fact, the logical induction criterion doesn't even imply anything about what the probabilities of an undecidable sentence converge to, except that it is not 0 or 1.

Another reason not to trust logical induction too much about mathematical uncertainty is that logical induction gets all its evidence from the proofs of one formal system, and there isn't one formal system whose proofs completely account for all sources of evidence about mathematical claims. But it seems to me that correctly characterizing how to handle all sources of evidence about mathematical truths in a way precise enough to be turned into an algorithm would be a quite shockingly huge advance in the philosophy of mathematics, and I don't expect it to happen any time soon.