Randomness, necessity, and non-determinism

If we want to talk philosophy then it is necessary to mention Aristotle. Or is it just a likely beginning? For Aristotle, there were three types of events: certain, probable, and unknowable. Unfortunately for science, Aristotle considered the results of games of chance to be unknowable, and probability theory started — 18 centuries later — with the analysis of games of chance. This doomed much of science to an awkward fetishisation of probability, an undue love of certainty, and unreasonable quantification of the unknowable. A state of affairs that stems from our fear of admitting when we are ignorant, a strange condition given that many scientists would agree with Feynman’s assessment that one of the main features of science is acknowledging our ignorance:

Unfortunately, we throw away our ability to admit ignorance when we assign probabilities to everything. Especially in settings where there is no reason to postulate an underlying stochastic generating process, or a way to do reliable repeated measurements. “Foul!” you cry, “Bayesians talk about beliefs, and we can hold beliefs about single events. You are just taking the dated frequentist stance.” Well, to avoid the nonsense of the frequentist vs. Bayesian debate, let me take literately the old adage “put your money where you mouth is” and use the fundamental theorem of asset pricing to define probability. I’ll show an example of a market we can’t price, and ask how theoretical computer science can resolve our problem with non-determinism.
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