I will start with a key quotation that stands out as reason for the rest of the post. Probability is not a notion of pure mathematics, but of physics or philosophy, as Hardy and Littlewood noted. Somewhat similar valuation, in this context, has been proposed by Gowers, who said Although the prime numbers are rigidly determined, they somehow feel like experimental data. In this post I will focus on the first quotation and try to address the question what probability is.
For the sake of simplification, lets assume that we’re dealing with a coin, ie. with the machine with two different states, ie. 1 and 0. Those two states are recognizable only in the context of their value, not under any other circumstance whatsoever. If we are to choose out of these two states, we tend to say that we will choose one of two so that, in infinite number of choices, we’re going to have the exact same amount of the chosen ones and zeros consecutively. That claim is based on the fact that we’re dealing with two states locally undiscernible, imperceptible.
But these states, even if locally undiscernible, remain potetial to be globally differentiated, therefore one cannot claim such probability distribution, thus the very quotation at the beginning of the post. To give an example, we could assume that probability (Borel) of both states enabled are equal locally; still, globally the states can be discernible, that’s for the starters.
Lets assume that locally undescernible states have a different distribution (but fixed) of being enabled, thus the very states are not undescernible in the global sense. Insofar as the local sense, we could notice that just visible perceptible (that non-fragile one) undescernibility might not be the objective one (and it’s likely it is not actually). That implies much regarding games with random element. See word “random”: http://www.dartmouth.edu/~chance/chance_news/recent_news/chance_primes_chapter2.html
I would even hazard a comparison of this with the combination of two: the integers (and their predictability) as well as the primes (building blocks of integers and their hidden “predictability”, which is not known to the mankind). For more, Riemann, http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf
What I’m focusing on is to find out how to combine the given, the perceived, logic, fast problem-solving in terms of addition and multiplication, and geometry itself- and all that in the context of the R-sequence. Assuming that claim of potential global differentiation of distribution of probability for locally undescernible events is true, we could assume that probability of events comes with normalized p(i)/sum of n-primes or something similar and then play with another kind of probability lessons. In the same time, assuming that everything is assumed locally impossible to differentiate, but indeed differentiable, hence the probability of the even itself in given (fixed) period of time would be fixed; in the same time, the phenomenon of time itself would be somewhat fixed, in this sense that the “time progress” would be handled by the probability distribution handled by primes, which would mean there would be no time in the given sense. That would imply even more, but it’s already time for sleep.
(I am sorry for typos, didn’t have enough energy to fix it)