(I suddenly doubted the previous post, and had to look and think twice again. I discovered that I need to look more in detail at Benford’s law, because certain aspects of the relation between the solution of the previous post and Benford’s law are still unclear to me. This is a drawback of writing on the fly. On the other hand, for me the process of searching for direction and answers is always very error-prone, hit-and-miss with a preponderance of the latter category ;-), and perhaps for readers this is more fun than the usual dry mathematical prose. Addendum January 2014: no need for doubt, there is perfect correspondence with Benford’s law.)
Anyway, we were still discussing Question 1, whether our physical universe is finite or infinite. There is much to be philosophize about in this regard, but to maintain a certain pace I will cut a large number of corners. My personal view is: a truly finite universe doesn’t make sense. At least not to me, I cannot speak for others. On the other hand, an infinite universe doesn’t make sense either…(be it actual or potential infinity). All variants suffer in my limited mind from the subquestions:
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?
And you can easily add to this list yourself.
With respect to Question 1.4: I personally have not seen any good arguments why the arrow of time should be irreversible. Last year there was a conference at the IMAPP institute at `my old’ Radboud University of Nijmegen on this subject, and I was asked as an artist to make an attractive design for a poster, leaflets, advertisements. (Perhaps I will post some of my designs below, in a later addition). Of course it set me thinking about this so-called arrow of time.
And one major argument for the existence of such arrow, is what is perceived as irreversible increase in entropy in certain processes. I personally think this argument is flawed, but the idea of entropy and its relation to time is very interesting. Bluntly put, it seems to me that one could define time as entropy…but this leads to other problems which I cannot oversee. (Addendum January 2014: for instance, what does anything mean if we do not already have a time concept…so it seems more logical to simply equate time with (some monotone injective function of) entropy, which is closer to what I meant.)
Still, like I said, I think that entropy provides a nice way to connect Question 1 with Question 2 (What is the role of information in probability?). So I will continue in the next posts along that line.