Originally published in The Reasoner Volume 10, Number 5– May 2016
The concept of Probability is interesting, among other reasons, for the variety of ways in which we may be talking about distinct things and yet, in the end, still talking about probability. From the philosophy-of-mathematics point of view, this is vividly illustrated by the fact that, except possibly for one’s views on `finite vs. countable additivity’, one axiomatisation serves a great number of largely incompatible interpretations of the concept being axiomatised. Chapters 1-3 of J. Williamson (2010. In Defence of Objective Bayesianism. Oxford University Press.) offer a wide angle picture which I recommend to those who are unfamiliar with the landscape of probability interpretations.
Viewed at a relative coarse grain, the axiomatisation of probability developed by following a similar path to other mathematical concepts until at the turn of the twentieth century the key motivation became that of securing its applications against the threat of paradoxical consequences Needless to say David Hilbert played an important role in this. The explicit question appears as number “six” in the list of problems Hilbert posed to the audience of the Second International Congress of Mathematicians, in Paris on 8 August 1900:
Six. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. [] As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.
Various attempts at putting probability on axiomatic grounds followed Hilbert’s call, with a stable answer arriving almost three decades on, in a series of three papers published by Andrej Kolmogorov between 1929 and 1933. There seems to be agreement among experts that Kolmogorov had rather strong objectivist inclinations with regards to the interpretation of the concept of probability. But this is of little consequence, for with his axiomatisation, Kolmogorov emphasised the highly abstract nature of probability, which could then be pursued as a chapter in mathematical analysis.
Hilbert’s own views on the interpretation of probability feature rarely in foundational debates on the subject. However the recent paper (L. M. Verburgt, 2016: “The place of probability in Hilbert’s axiomatization of physics, ca. 1900–1928” in Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 53, 28-44.) shows that this does not reflect a lack of interest on the topic on Hilbert’s side. Quite the opposite appears to be true. In fact, and perhaps surprisingly, Hilbert changed his mind significantly over three decades on the meaning and interpretation of probability, and consequently on how it should be axiomatised. As Verburgt sums it up:
Hilbert understood probability, firstly as a mathematizable and axiomatizable branch of physics (1900-1905), secondly as a vague statistical mathematical tool for the atomistic-inspired reduction of all physical disciplines to mechanics (1910-1914), thirdly as an unaxiomatizable theory attached to the subjective and anthropomorphic part of the fundamental laws for the electrodynamical reduction of physics (1915-1923) and, fourthly as a physical concept associated to mechanical quantities that is to be implicitly defined through the axioms for quantum mechanics (1928).
It is then apparent than Hilbert’s starting point is similar to Kolmogorov’s: probabilities are to be understood as properties of the physical world. Unlike the Russian, however, he moved on to consider other, radically different, interpetations. This is perhaps due to Hilbert’s interest for the applications of probability theory, from statistical mechanics, to what we would now call mathematical finance, as testified by his 1905 lecture notes. Be this as it may, Hilbert embraced a number of distinct positions on the meaning of probability, including a rather extreme form of subjectivism which lead him to consider probability to be “unaxiomatisable.” The details reported by Verburgt are rather involved, but quite fascinating.
One question that I anticipated would be addressed in the paper is an account of how Hilbert reacted to Kolmogorov’s own axiomatisation, and in particular whether he consequently settled down for a definite interpretation. But the author makes no reference to this. Maybe similar works will enlighten us on this side of the story too.
