# Hilbert's Sixth Problem: Is Kolmogorov's solution the last word?

The demand for axiomatization of probability was put forward by Hilbert at the very beginning of the past century: it was the sixth problem in his famous twenty three problems he deemed of high importance.

Hilbert's sixth problem regarded the axiomatization of probability as part of axiomatization of the physical sciences. And before the acceptance of Kolmogorov's axioms of probability (1933), there were other attempts at defining probability in an axiomatic fashion; for example, von Mises' theory of Collectives (1919). Von Mises believed that probability is an empirical science describing mass phenomena. He seemed to believe that he is in line with Hilbert's demand to axiomatize physics; but on the other hand - he also did not really provide purely formal system for probability per se (but rather for the notion of Collective).

Is axiomatization of probability regarded today as a closed issue in virtue of Kolmogorov's axioms?

There is no general agreement on the axiomatisation of probability. Kolmogorov was a frequentist and his approach proceeds by supposing the existence of an event space, or possibility space, defining the probabilities of propositions in terms of their frequency relative to the total size of the space, then defining negation, conjunction and disjunction in terms of these, and finally defining conditional probability in terms of conjunction.

Many since have found this approach unsatisfactory and have looked for others. Popper developed an axiomatisation in which conditional probabilities are considered as fundamental, because he came to think of probabilities as dispositions and these are more naturally interpreted as conditionals.

John Maynard Keynes and Rudolf Carnap developed an approach to probability in which it is a logical relation: a kind of degree of partial entailment. This approach was taken further by Richard Cox (The Algebra of Probable Inference) who derived axioms for probability based on fundamental postulates about what qualifies as a plausible inference. Edwin Jaynes developed the idea further (in the first few chapters of his book, Probability Theory: the Logic of Science) and showed how the concept of epistemic probability can be combined with information theory to extend the scope of the Bayesian approach to probabilistic reasoning.

Bruno de Finetti showed how the axioms of probability theory can be derived from decision theory if we understand probabities to be degrees of rational belief. His approach takes a bet as a paradigm example of a decision made under uncertainty, and shows that if we take as a minimal criterion of rationality that it would be bad if you allowed a Dutch book to be made against you (i.e. a combination of bets such that you are bound to lose whatever happens) then if you are to avoid this state of affairs, your bets must conform to the calculus of probability theory.

There is a good summary of this material in Alan Hajek's article, Probability, Logic and Probability Logic. http://philpapers.org/rec/HJEPLA Hajek also wrote an interesting paper called What Conditional Probability Could Not Be. http://philpapers.org/rec/HJEWCP in which he argues against the Kolmogorov axiomatisation and in favour of the view that conditional probabilities are fundamental.

• Kolmogorov was sympathetic to frequentism, but "his only attempt to formalize this philosophy came in 1963". His 1933 axiomatization had nothing to do with it, it is measure-theoretic and formal, and compatible with any philosophy. For better or worse it is almost universally accepted today by mathematicians, your "many" are few and are more of philosophical interpretations of probability rather than different formalizations of it. It makes no difference to a mathematical structure if one axiomatizes it starting from sample spaces or from conditional probabilities. Feb 16 '16 at 23:14
• @Conifold, and Bumble: I wonder in light of the answer and the comment given: does philosophy of probability center on providing interpretations to merely Kolmogorov's axiomatization? Does it make sense to at all assess someone's interpretation - say - Keynes', with regard to Kolmogorov's axioms, despite the fact that Keynes had different axioms and different approach to mathematics? (Keynes seemed to belong to Logicism rather than Formalism) Feb 16 '16 at 23:28
• PS: There seem to be much confusion I think in the field of philosophy of probability - or at least I myself get confused in virtue of the attempt of philosophers to assess interpretations of probability under implicit assumption of mathematical Formalism - some early proposed interpretations or views of probability rejected Hilbert's formalism, and so it seems to me wrong to assess them through formalist prism. But maybe I'm missing something. Feb 16 '16 at 23:43
• @student: One cannot assess Keynes' interpretation with respect to Kolmogorov, since Keynes' book was published earlier. Some defenders of the logical concept of probability dispense with categorical probabilities altogether and use only conditionals, so they can hardly be said to be following Kolmogorov. Also, Hajek shows that conditional probabilities can have defined values in situations where Kolmogorov does not permit them. Apart from that, the difference between the intepretations is one of explaining or justifying the axioms. Feb 16 '16 at 23:52
• @student Mathematics of probability is different from philosophical interpretation of probability. The issue of representing conditional probabilities is philosophical, just like the frequentist interpretation of probability values, even Haijek writes "there is nothing problematic about introducing the symbol | into the language of the probability calculus, and defining it in terms of the ratio formula". One can reject Kolmogorov's formalism, but since most probability work in mathematics and statistics presupposes it, it is more common to reinterpret it according to one's philosophical wishes Feb 17 '16 at 1:45

It is important to distinguish between axiomatisation and interpretation. The mathematical formalism is nailed down by Kolmogoroff's axioms, but this only puts slight restrictions on the philosophical interpretation. All three major philosophical interpretations are consistent with Kolmogoroff's axioms.

Secondly, what Kolmogoroff did was not actually an axiomatisation, but a definition. He defined that any measure that satisfied certain conditions was to be called a "probability measure". And the concept of measure was already defined as a function satisfying certain conditions. Now, it is traditional to call such conditions "axioms" but they are not axioms like the axioms of Logic or Set Theory. Kolmogoroff did not add any real axioms to the usual foundations of mathematics, he simply added a definition.

Kolmogoroff was not a completely committed frequentist, and he criticises the usual presentations of the frequency interpretation in his popular contribution to Mathematics etc. of which a translation was reprinted by MIT Press. He felt the frequency interpreation was useful in praxis but was logically circular and so could not be considered logically valid. Littlewood published a similar criticism.

BTW, "formalism", and axiomatisations are formal, never includes "interpretation". Klein and Hilbert famously insisted on this. Hilbert joked that the axioms of geometry should make equal sense if interpreted in terms of "tables, chairs, and beermugs" instead of planes, lines, and points.

In the 60's Kolmogoroff proposed an "algorithmic complexity" interpretation of probability and randomness. Eventually his proposal was worked out by Finnish mathematicians and is also discussed by Prob. von Plato, of Helsinki Univ., a famous logician and computer scientist.

See, for a careful discussion of the physical interpretation of probability, http://rev-inv-ope.univ-paris1.fr/fileadmin/rev-inv-ope/files/35214/pdf_35214-09.pdf an open source journal, hosted at the Sorbonne, and, but less relevant, see also Prof. Leo Corry's several publications on Hilbert's Sixth Problem in general, e.g., https://www.jstor.org/stable/41134029
Hmmm... I guess jstor is not open access, is it. Prof. Corry's contribution to the ICM can be found for free on the Internet somewhere or other.