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.

Comments

The idea that a probability theory is always appropriate is not in itself a mathematical idea. Nor is it a conclusion of science, since – in its usual formulations – it is an assumption. Its justification – if any – would thus seem to fall in the crack between the two. As Littlewood notes in his conclusion, the use of probability theory might reasonably be considered to be more justified where it has a good track record, but this is far from justifying its universal unthinking application, as too often seems to be the norm. There is also the technical difficulty that the assessment of the track record cannot itself rely on probability theory. What is needed is a theory of knowledge that sits between mathematics and science.

Keynes hints at one resolution. Suppose that in some domain we employ a framework in which we proceed ‘as if’ some probability axiom were true, and that the use of the framework amounts to a ‘severe test’ in some sense. Then, while admitting Littlewood’s observation that this is in no sense evidence in favour of our axiom (since the notion of evidence depends on the notion of probability), we might accept it as providing some ‘pragmatic’ grounds for continuing to accept the axiom as a short-term working hypothesis while not overlooking the longer-term need to test it.

Thus science proceeds in two ways: it can gather and analyse more data using the same framework to refine probability estimates, or it can compare and contrast frameworks to replace or refine them. It should always be noted that there are uncertainties associated with both aspects, and probability theory only deals with the former. Sometime in the ‘hard’ sciences, more often in the ‘softer’ sciences and very often in broader life, the probability as assessed using the supposedly appropriate framework is not actually addressing the min source of uncertainty.

Dave Marsay

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