(for the mathematics of `random natural number’, see post 8 and before)
It seems to me that Question 1: Is our physical world finite or infinite? (from post 2 of this series) has had a lot of attention so far, when compared to Question 2: What is the role of information in probability theory?.
Of course, to boost the theory proposed so far, I should devote close to times more attention to Question 1 than to Question 2 ;-). Still, the subquestions of Question 1 from post 5 intrigue me, and I wish to write about them just a little bit more, before going into Question 2. Repeated from post 5:
QUESTION 1.1 How did our physical world begin?
QUESTION 1.2 What is the role of time in the origin of our universe? Can we even talk about origin?
QUESTION 1.3 What came before the Big Bang?
QUESTION 1.4 Is the so-called Arrow of Time an anthropic artifact?
QUESTION 1.5 Are the laws of physics constant, or do they vary over (space)time?
QUESTION 1.6 If our physical world is growing (as in potential infinity), where does this growth come from, by what is it fed?
There is an interesting older approach to these questions, which is called the steady state theory. Most physicists today reject this theory, since it is at odds with the Big Bang theory, of which many predictions have been reasonably confirmed experimentally.
Before I continue, let me repeat that my knowledge of physics is very limited. However, this does not prevent me from seeing certain mathematical issues which I believe are involved. I have advocated for a long time now, that physicists should take notice of the foundational issues surrounding constructive mathematics vs. classical mathematics. The usual choice of classical mathematics as the preferred math for physics is very questionable in my eyes, especially in the light of all the unsolved problems in physics.
Right now I’m pondering the possibility of reconciling different types of infinity and even finiteness, using an anthropic approach. An interesting link with the previous: the function plays an important part in this reconciliation. It seems wiser however, to postpone the discussion of this to a new thread, since the relation of this subject to the foundations of probability has been discussed, and the current thread should not be interrupted too much. Therefore I promise to revisit the above in a new series, and leave Question 1 in peace for now.
So from here on, I will link the previous posts to Question 2, and to (Laplacian) determinism as well. This will take some time however. To fully understand the mathematics involved, one should be aware of what is called `recursive mathematics’ and computation theory, Turing machines etc. Once again, I advocate that physicists should be adequately informed of these areas of mathematics. (I know this can seem rather hypocritical, since I myself am not adequately informed of physics. I would like to be thus informed however, very much so. Plus, on reflection, you will perceive the difference in necessity, I hope…).
(January 2014: this thread is continued here)