was the definition of chance informally proposed by Borel (1909): the proportion of times that a finite sequence is in an infinite (even very long) sequence corresponds (approximately) to the probability that such a sequence occurs during a particular test.12 12 The British mathematician and logician Frank P. Ramsey studied conditions under which order must appear.
13 13 The adjective “large” has precise definitions for both finite and infinite sets.
14 14 Consider a gas particle and its momentum: the average value of the momentum over time (the time integral) is asymptotically assumed to coincide with the average momenta of all particles in the given, sufficiently large, volume (the space integral).
15 15 It is not unreasonable to hypothesize that pseudo-randomness reflects its creators’ subjective “understanding” and “projection” of randomness. Psychologists have known for a long time that people tend to distrust streaks in a series of random bits; hence, they imagine a coin flipping sequence alternates between heads and tails much too often for its own sake of “randomness”. As we said, the gambler’s fallacy is an example.
16 16 Incidentally, the same conference at which Gödel presented his famous incompleteness theorem.
Chapter written by Cristian S. CALUDE and Giuseppe LONGO.
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