The question whether the question `How to draw a natural number at random?’ can make sense, has been occupying a very small part of my curiosity for a long time.
This started actually already in Probability 101 (by which I mean: the first serious course in probability that I took at university, in the first year of my math studies).
Often enough during my math studies I came across preferred pathways (taken to address mathematical problems), where I did not quite really get all the reasons for this preferential treatment. Nor a good feel for the real motivations behind the mathematical problems, I’m afraid. In hindsight, it appears to me now that this is also a question of time management: subjects are usually difficult enough. To discuss and compare different ways of approaching them, and to motivate theories extensively by linking them to understandable problems frequently, these things take up time and effort. But it is also due to my own slowness in understanding…since I tend to only understand mathematics when I can see every little brick which was used to build it.
Probablity 101 left me with a vague uneasiness: what exactly are the foundational assumptions that we make when dealing mathematically with probability, and why are these assumptions the chosen ones?
I have since come to the conclusion, in a very slow and roundabout way, that these freshman’s questions merit some more attention. I don’t presume to bring much new thought, although there is always a chance (pun intended) that the perspective has some new angle. Simply, I would like to write down some of my loose thoughts on the subject, for whatever little they’re worth, and maybe some other (ex)freshman might benefit somewhere. There is a link with my publications on the foundations of constructive mathematics, through the question `can Nature produce a nonrecursive sequence?’.
This last question intrigues me in several ways. Apart from its intrinsic value, I believe it also yields a good mathematical framework to understand the philosophical problem of determinism. I was of course pleasantly surprised to rediscover that one of the pioneers of probability theory, Pierre-Simon Laplace, also thought about determinism.
(to be continued)