# Littlewood’s Dilemma

The Dilemma of Probability Theory

in:

J.E. Littlewood A Mathematicians Miscellany, Methuen, 1953

There is a solid body of propositions of the theory, and no one dreams of doubting their practical applicability. … The question now is about the foundations of the subject.

Mathematics … has no grip on the real world; if probability is to deal with the real world it must contain elements outside mathematics; the meaning of ‘probability’ must relate to the real world … . We will suppose … that there is just one primitive proposition, the ‘probability axiom’, and we will call it A for short. Although it has got to be true, A is by the nature of the case incapable of deductive proof, for the sufficient reason that it is about the real world … .

There are 2 schools. One, which I will call mathematical, stays inside mathematics, with results that I shall consider later. We will begin with the other school, which I will call philosophical. This attacks directly the ‘real’ probability problem; what are the axiom A and the meaning of ‘probability’ to be, and how can we justify A? …

I said above that an A is inherently incapable of deductive proof. But it is also incapable of inductive proof. …

We come finally, however, to the relation of the ideal theory to the real world, or ‘real’ probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: ‘If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician.’ In practice he is apt to say: ‘try this; if it works that will justify it.’ But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.