While we do sometimes use ‘probability’ to mean an empirical (and therefore finite) proportion, that is hardly a full account of its scientific uses—e.g. the gap shows up as the following 2 sorts of problems for Finite Frequentism (each considered via 3 of Hájek’s 1997 arguments). The first sort of problem concerns singular cases.
(1a) Hájek’s (1997: 221) 9th argument is essentially as follows. It is surely possible that only one atom of a certain radioisotope exists, in the entire history of the universe, and that it happens to have a lifetime of 1 day. But if so then, according to Finite Frequentism, the chance of that atom having another lifetime would be 0. And yet presumably its lifetime would (were it to exist) be just as chancy as all the other lifetimes appear to be.
(1b) Hájek’s 3rd argument is that all chancy events are essentially single cases, insofar as their probabilistic behaviour is independent of what actually happens in similar cases, as follows.
Here is a radium atom; its probability of decaying in 1500 years is ˝. If radium atoms distant from it in space and time had behaved differently, this probability would still have been ˝. […] The chance that this coin lands heads now does not depend on how the coin will land in the future—as it were, the coin doesn’t have to ‘wait and see’ what it happens to do in the future in order to have a certain chance of landing heads now. (Hájek 1997: 216)
(1c) Hájek’s 4th argument is that such individual chancy events are often our primary concern. Although this argument concerns our priorities, rather than our physical intuitions, it poses a profound problem for all forms of Frequentism because von Mises (see §3; and similarly for Reichenbach see §5) is typical when he asserts the following:
We can say nothing about the probability of death of an individual even if we know his condition of life and health in detail. The phrase ‘probability of death’, when it refers to a single person, has no meaning at all for us. This is one of the most important consequences of our definition (von Mises 1957: 11)
And that is surely therefore too weak a definition, e.g. consider decays, or emissions, or (more graphically) how empirical evidence indicates that dietary methods might be able to reduce the risk of heart disease.
A man on such a diet is not worried about “a measure associated with a population”; he is concerned to change that property of himself which would otherwise be displayed in an excessive chance of his death during the years ahead. (Mellor 1971: 91)
The statistics about others like him may give him good evidence as to his own chance of dying, but the fact that he ultimately cares about, is a fact about himself—one expressed by a meaningful ‘probability of death’ statement that refers to a single person. (Hájek 1997: 218)
The second sort of problem (Hájek 1997: 222-5) pits Finite Frequentism against even stronger intuitions:
(2a) Hájek’s 12th argument is that Finite Frequentism implies that any coin tossed an odd number of times cannot have prob(H) = ˝, whereas it is surely not an analytic truth that any coin tossed only an odd number of times is biased.
(2b) His 14th argument is, similarly, that it is implausible that quantum-mechanical probabilities must be rational numbers.
(2c) His 13th argument concerns spurious correlations—attributes A and B (e.g., A might be ‘dies by age 50’, and B ‘wears pink shirts’) are correlated if they are not mathematically independent. Such correlations are clearly evidence for a physical (e.g. a causal) connection between A and B.
But given Finite Frequentism, if the smaller population size (e.g., p = the number wearing pink shirts) does not divide the larger one (e.g., s = the number in the sample)—which is very likely to be the case with large sample sizes—then prob(A | B) (e.g. d/p, where d = the number of those who die by age 50) will not equal prob(A) (e.g. d/s; as though the probability of dying by age 50 were not causally independent of the wearing of pink shirts).
To avoid such problems, Frequentists usually define prob(A | B) to be the limit (as the number of the Bs tends to infinity) of the ratios that are the number of Bs that are As divided by the number of Bs—section 3.