[Alamo-Girl]

Thank you so very much for your post and all the excellent examples! The last one about the billards table is particularly engaging to me. You've given me much to research! Thank you!

But focusing narrowly on algorithmic randomness, is there any reason not to use the Chaitin/Kolmogorov definition?

[Doctor Stochastic]

There is nothing wrong with the Chaitin/Kolmogorov definition of algorithmic randomness. It isn't too useful in the sense that no computable random sequence can exist. Knuth's Volume 2 (Chapter 3) has a discussion of randomness that is similar but one could generate a "random" sequence. (Of course, one cannot generate all such sequences nor could one tell if a given sequence was random.)

There are a couple of places where intrinsic "randomness" arises in physics. One is in Brownian motion. If trying to measure the Brownian particle more and more closely, the measuring instrument becomes subject to Brownian motion itself. Thus, even were the physical laws moving a particle deterministic, in practice, one would still have to average over the smallest measurement sizes to get results. This is a practical limit on measuring, not a claim that things such as position or velocity don't exist.

Another place is in the uncertainty relations (Heisenberg). These arise from the theory of Fourier transforms and the wave nature of particles. This is quite another problem from the Brownian motion "randomness." There doesn't seem to be anything to average over nor can one assume that the particle even has a sharply defined velocity or momentum.

I prefer in most things to avoid the term "random" because the concept isn't clearly defined. I like to talk about things like "idependent identically distributed" events. These are well defined. Of course "random variables" are not random nor are they variables. Since the 1920's, "random variable" means "measurable function."