Induction: From Kolmogorov and Solomonoff to De Finetti and back to Kolmogorov

Abstract:     
This paper compares the solutions to "the induction problem" by Kolmogorov, de Finetti, and Solomonoff. Brief sketches of the intellectual history of de Finetti and Kolmogorov are also composed. Kolmogorov's contributions to information theory culminated in his notion of algorithmic complexity. The development of algorithmic complexity was inspired by information theory and randomness. Kolmogorov's best-known contribution was the axiomatization of probability in 1933. Its influence on probability and statistics was swift, dramatic, and fundamental. However, Kolmogorov was not satisfied by his treatment of the frequency aspect of his creation. This in time gave rise to Kolmogorov complexity. De Finetti, on the other hand, had a profound vision early in his life which was encapsulated in his exchangeability theorem. This insight simultaneously resolved a fundamental philosophical conundrumHume's problem, and provided the bricks and mortar for de Finetti's constructive probabilistic theory. Most of his subsequent research involved extensions of his representation theorem. De Finetti was against determinism and celebrated quantum theory, while Kolmogorov was convinced that in every seemingly indeterministic manifestation there lurked a hidden deterministic mechanism. Solomonoff introduced algorithmic complexity independently of Kolmogorov and Chaitin. Solomonoff's motivation was firmly focused on induction. His interest in induction was to a marked extent sparked by Keynes' 1921 seminal book. This interest in induction has never faltered, remaining prominent in his most recent research. The decisive connection between de Finetti and Kolmogorov was their lifelong interest in the frequency aspect of induction. Kolmogorov's solution to the problem was algorithmic complexity. De Finetti's solution to his frequency problem occurred early in his career with the discovery of the representation theorem. In this paper, we try to explain these solutions and mention related topics which captured the interest of these giants.

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