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.