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When Is It Rational to Believe a Mathematical
Statement?
Jendrik Stelling
Traditional philosophy of mathematics has long held that the fundamental business of
mathematics is to ascertain the truth of mathematical propositions by finding suitable proofs
for them, where “proof” means a chain of logico-deductive steps, starting from accepted
premisses or axioms, and terminating in the proposition to be proved. Under this view, we
are justified to believe in a mathematical statement if and only if there is a proof for it. My
goal here is to call this view into question. To this end, we will discuss two examples of
mathematical argumentation that suffer, prima facie, from the same shortcoming. We will
then attempt to extract the significant differences and use them to elucidate the concept of
proof that actually informs mathematical practice.
1. Oracles and Probabilities
Imagine that, in some remote mountain cave, there lives a mysterious mathematical oracle. If
one manages to make the dangerous trek there, one is allowed to ask a single question of the
form “is mathematical statement p true?” The oracle knows the truth value of all
mathematical statements and will neither lie to us nor try to deceive us in any way.
Everything the oracle says can be taken at face value. “Mathematical statement” is meant to
be understood as something that might be considered a result in some publication; e.g., we
might ask the oracle about the truth value of “P=NP” but not whether set theory or category
theory is the more appropriate foundation of mathematics. 1 Let’s also, for the moment, ignore
questions about things that are in some sense indeterminate, such as the Continuum
Hypothesis.
Imagine then that one day we make the trip and stand before the oracle. “Oracle,” we ask, “is
it true that all even numbers greater than 2 can be expressed as the sum of two primes?” The
oracle pauses for a moment, and gravely intones, “Goldbach’s conjecture is true if and only if,
on randomly drawing a real number, the number you draw is not a natural number.”
Given what we know about the oracle, what do we do with this answer? Obviously we cannot
simply carry out the experiment and draw a random real number. So we need a mathematical
approach: When we ask for the probability of drawing a given number from the reals, we need
to come up with an account of probabilities that can handle the enormous amount of real
numbers we face. Ordinarily, we would perhaps try to define probabilities as a quotient, the
positive outcomes divided by all possible outcomes. But since there are infinitely many real
numbers, and within them infinitely many natural numbers, this won’t do. But we can use
some concepts from measure theory to help us circumvent this problem. The Lebesgue
measure, for instance, works by comparing smaller slices of real numbers, and seeing how
many naturals we can find in them, and then extending these results across the real number
system. Important in this regard is the notion of a zero measure set (meaning an interval of
1
I presuppose that the future won’t show either framework to be inconsistent; the decision between set
theory and category theory is not a valid question precisely because I understand it to be a judgement
call and not amenable to mathematical proof.