## Drawing a natural number at random, foundations of probability, determinism, Laplace (4)

(OK, enough apologized for all my shortcomings. From now on mostly math and philosophical thoughts on physics.)

Back to the two fundamental questions which I deem relevant for the question of drawing a natural number at random.

QUESTION 1   Is our physical world finite or infinite?

QUESTION 2   What is the role of information in probability?

After pondering on these questions, and looking at various mathematical and physical scenarios, a clear-cut solution presents itself to me. This solution is in accordance with Benford’s law. The solution seems at slight variance with Zipf’s law, also depending on interpretation. (It is a matter of discrete’ vs. continuous’. It would be interesting to see if the data used to support Zipf’s law would support this solution equally well, better, or worse).

Let’s first see what this would-be solution actually boils down to.

Solution to drawing a natural number at random’:

* We can only assign relative chances, and the role of the natural number $0$ remains mysterious. (All to be explained)

* For $1\leq n,m \in \mathbb{N}$ let’s denote the relative chance of drawing $n$ vs. drawing $m$ by: $\frac{P(n)}{P(m)}$.

* For $1\leq n,m \in \mathbb{N}$, we find that $\frac{P(n)}{P(m)} = \frac{\log{\frac{n+1}{n}}}{\log{\frac{m+1}{m}}}$

* An alternative discrete’ or Zipfian’ case $P_{\rm discrete}$ can perhaps be formulated, yielding:  for $1\leq n,m \in \mathbb{N}$, we find that $\frac{P_{\rm discrete}(n)}{P_{\rm discrete}(m)} = \frac{m}{n}$.

[It occurred to me that if you think the above formula is rubbish, then you will probably not want to read my justifications of it either ;-), which is why I decided to give the solution first.]

(* At first glance another seemingly defendable position would be to have $\frac{P(n)}{P(m)} = \frac{\log{\frac{n+\frac{1}{2}}{n-\frac{1}{2}}}}{\log{\frac{m+\frac{1}{2}}{m-\frac{1}{2}}}}$, but this is at slight-but-noticeable variance with Benford’s law.)

OK. Now for those of you who are curious where the above comes from, we first tackle Question 1 mentioned above.

The relevance of this question comes to the fore almost immediately when we try to clarify what we mean by the set of natural numbers’ $\mathbb{N}$. If one thinks of our world as being truly finite in every sense, then the mathematical concept of $\mathbb{N}$ is flawed from the start. I’m not ruling out that possibility, by the way.

For our world to be truly finite in every sense, some very strange’ things need to hold. (Space)Time becomes finite, for instance, and thus a fortiori (space)time becomes discrete. Well, these very strange things have already been proposed: we have Planck time and Planck length:

The physical significance of the Planck length is a topic of research. Because the Planck length is so many orders of magnitude smaller than any currently possible measurement, there is currently no way of probing this length scale directly. Research on the Planck length is therefore mostly theoretical.
In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it would become impossible to determine the difference between two locations less than one Planck length apart. The precise effects of quantum gravity are unknown; often it is suggested that spacetime might have a discrete or foamy structure at Planck length scale.

It is also instructive to read http://en.wikipedia.org/wiki/Actual_infinity

This is meant to illustrate that it might be harder to understand Question 1, than many mathematicians and physicists think. I repeat my earlier subquestions: What is the physical world made of,that it could possibly be classified as finite or infinite? Do we really understand what is meant by infinity? The classical mathematicians’ view of infinity (for instance in regarding $\mathbb{N}$ as a completed entity) seems hopelessly inadequate to me. Brouwer’s view of $\mathbb{N}$ as a developing-in-time, potentially infinite construction makes more sense. But this latter constructive’ view still ignores the very real-life impossibility to go beyond say $10^{10^{10}}$ in any meaningful way, which is the point of the mathematical philosophy known as (ultra)finitism’.

The relevance of Question 1 to our search for meaning in random natural number’  reveals itself in many subfacets. If our world is truly finite, then perhaps we can apply a sort of vase model, to arrive not only at our relative chance

$\frac{P(n)}{P(m)} = \frac{\log{\frac{n+1}{n}}}{\log{\frac{m+1}{m}}} = ^{\frac{m+1}{m}}\log{\frac{n+1}{n}}$ (showing base invariance for Benford’s law)

but even at absolute chances for $P(n)$…If on the other hand our world is actually infinite’ , like classical mathematicians seem to think (without any support from reality, I believe) then there are still anthropic limitations to be considered…

We will deal with all that in posts to follow, but here I would like to make a connection with Question 2. In any urn model, the usual tacit assumption is that the urn is well-mixed’ …whatever that may mean (I believe one easily has some form of circularity going in the definition of well-mixed: an urn for which the outcomes when drawing blindfolded conform to the theory of well-mixed urns…). To put it bluntly: Question 1 is connected to Question 2 through the notion of entropy.

The solution presented above is derived from a certain interpretation of the entropic state of a random’  natural number. And yes, this is all highly speculative etc etc…but isn’t it nice that it corresponds so well to Benford’s law and Zipf’s law?