What is the difference between an ape and the number pi?

My message in French of December 19th had to do with tomorrow's newspaper being hidden somewhere in the number pi. Express pi in base 26 and write it down with our alphabet. You obtain an infinite series of random letters, in which any given finite sequence of letters (including e.g. Hamlet, on any text yet to be written) appears with a probability equal to 1.

A related paradox is that of an ape rewriting Hamlet by just typing randomly on a keyboard. The difference though is that it will take the ape a time longer than the age of the universe before scene 1 is finished. In pi, it seems to be there from the beginning. Just say "the ratio of a circle's perimeter to its diameter" and here is Hamlet!

The ape's paradox was invented by Borel, who solved it in a quick-and-dirty way by arguing that small probabilities ought to be considered as impossibilities, and large probabilities as certainties [1]. Such a pragmatic approach is so surprising under the pen of a mathematician, that I cannot but understand it as a real upset from Borel's part. Apparently, he stopped working on infinity in the twenties [2].

The difference between the ape and the number pi has to do with the difference between actual and potential infinity. The ape may potentially write Hamlet if he keeps typing for some time, while Hamlet is actually in pi. The possibility of actual infinities drove mathematicians nuts, at the end of the nineteenth and early twentieth century. Quite literally for a few of them [2] who - unlike Borel - kept working on these problems.

Another reading of mine sheds other light on that issue [3]: actuality/potentiality is a dichotomy, just like empiricism/speculation, creation/evolution, etc. We love opposing things because it helps us understand; a dichotomy is a simplified model that puts order in our mind, not in our world. We are constantly shrinking the complexity of the world to make it fit in the much smaller space of what we can understand. We most often shrink a hyper-dimensional reality to a 1-dimensional segment, and eventually we ignore the segment and consider only its two ends. Potentiality and actuality are two such ends, just as finite and infinite as a matter of fact.

Choosing one of the two options in a dichotomy cannot be right or wrong. Dichotomies are fruitful because they pave the way towards a better understanding of our fundamentally non-dichotomous world. When dichotomies are no longer taken too seriously, this generally means that the phenomenon they describe is becoming better understood: solid/liquid, rational/intuitive, good/bad, wave/particle, etc. Other dichotomies are quite lively but they will no doubt end up in the same way. I would love to be there when this happens for deterministic/random, past/future, matter/spirit and dead/alive.

[1] E. Borel, Probabilité et Certitude, Presses Universitaires de France (1950);
[2] L. Graham & J.-M. Kantor, Naming Infinity, A True Story of Religious Mysticism and Mathematical Creativity, Harvard University Press (2009);

[3] S.J. Gould, Time's Arrow, Time's Cycle: Myth and Metaphor in the Discovery of Geological Time, Harvard University Press (1987);