Mathematics and Philosophy by Daniel Parrochia

Author:Daniel Parrochia
Language: eng
Format: epub, pdf
ISBN: 9781119528074
Publisher: John Wiley & Sons, Inc.
Published: 2018-05-30T00:00:00+00:00

7.4. Chance, coincidences and omniscience

According to the theory of probability, unexpected encounters, lucky breaks or misfortunes, the most astonishing “coincidences” (the apparently rare but significant conjunction of circumstances that we – wrongly - assume have deviated from the natural order of things) can be perfectly explained by the law of large numbers. Their exceptional character is mitigated by statistical neutrality. This also has its own philosophical consequences, beginning by challenging all theories of destiny (from Stoics to Schopenhauer) [PAR 15a].

At the same time we must recognize that the concept of “chance” or, in its mathematical determination, that of randomness, remains partly mysterious. Although it did not completely clarify things, a breakthrough was made in the 20th Century with the algorithmic theory of information, which attempted to clarify the idea of a random series by likening it to an algorithmically incompressible series. This would be the case with the decimal series in π, for example, or, again, any series of numbers which could not be generated by a computer program smaller in size than the one that displays the series itself.

In accordance with Greek thought, especially Platonic thought, philosophy, like science, had earlier wagered that the universe was comprehensible, which was the same as saying that the information it contained, regardless of how it was defined, was “compressible”. Every theory had to be economical, as Leibniz remarked in his Discourse on the Metaphysics (V), observing that “the decrees or hypotheses” formulated by philosophers “hold good inasmuch as they are more independent than one another: because reason desires that we avoid a multiplication of hypotheses or principles, much as in astronomy we always prefer the simplest system”. It is, therefore, to our advantage to construct well-integrated theories that are greatly simple or that have the lowest complexity (which amounts to the same thing).

In formal language, this signifies that a theory in which the information content is smaller, or whose complexity is lower than the reality it seeks to explain, is capable of rendering that reality in both the most elegant as well as the most efficient manner. The “theories of everything”, founded on quantum physics, had, until recently, the same ambition.

This hope now seems to clash with the incompressibility or randomness of certain realities that we encounter both in physics as well as in mathematics.

That which overtakes us or transcends us, therefore, is not another world, as we believe - it is already here. Thus, for example, the Borel number or the ‘know-it-all number’ or, again, Chaitin’s famous number Ω.

In 1927, in a letter to the Revue de Métaphysique et de Morale [BOR 27, pp. 271–275], Emile Borel shared the idea that it was possible to inscribe, within a real number, all the responses to all the problems that could be formulated in a given language and which could not be answered by a simple yes or no. The list of problems in questions is infinite but countable and it can be ordered. We can, thus, compose a number, unique and clearly specified, whose nth binary decimal is the response to the nth question.

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