Drawing a natural number at random, foundations of probability, determinism, Laplace (7)

We repeat the solution given in post 4 of this series, for the question of how to assign meaning to `drawing a natural number at random’, in order to explain the details of this solution, while building on the previous posts.

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}.

Cutting corners once again, let’s proceed from the standpoint that even a truly finite universe has an enormous cardinality (from the physics’ point of view). This would seem to imply that it is hard to devise a physical experiment which distinguishes a truly finite universe from an expanding universe (potentially infinite) or even an actually infinite universe.

In practice, the difference between a large unknown cardinality N and potential and actual infinity would seem hard to substantiate. For our solution, all situations lead to the same problem: from our analysis and assumptions (especially on entropy) we cannot adequately assign an absolute chance P(n) of drawing a natural number n at random.

-In the case of an unknown finite cardinality N of our universe, we also do not know the relevant quantity log(N) (or equivalently \Sigma_{i=1}^{i=N}\frac{1}{i}).

-In the actually infinite case the integral log(x) of the density function p(x)=\frac{1}{x} diverges (or equivalently, if you prefer, the series (\frac{1}{n})_{n\in\mathbb{N}} is not summable).

-In the potentially infinite case, we are always at some finite point in an infinitely expanding time (I don’t know how to put this better, in physics’ terms, but it is inadequate, I realize that). This would imply that the absolute chance of drawing a natural number at random will decrease as time increases.

However, in all three cases this does not affect the relative chances \frac{P(n)}{P(m)}, which is already an interesting characterization. It leads directly to Benford’s law (exercise), for instance.

The role of the natural number 0 remains mysterious. One possible interpretation which I would like to mention is that, in the speculative analysis given so far, the number 0 might represent all the failed projects of Nature, all the `possibilities not come true’. Mathematically, this would lead to the idea that just as many things fail as which succeed…

Even more caveats meant just to set us thinking:
-In the (unknown cardinality) finite case, why would there be entropy issues?
-In the potentially infinite case, we assume that the entropy assumption remains valid in the course of time. This seems optimistic, to me it looks more likely that laws of nature vary with time and even space. This is linked to the next caveat:
-There is no reason to assume that what holds for small cardinalities will also hold for larger cardinalities. The conjured density function p(x)=\frac{1}{x} can be justified somewhat for small cardinalities, but for large cardinalities perhaps p(x)=\frac{1}{x^s} for some s>1 becomes more accurate. Notice that for s>1 very close to 1, we cannot in practice distinguish between the density functions p(x)=\frac{1}{x} and p(x)=\frac{1}{x^s}, but their respective `infinite’ behaviour is rather different. Perhaps far more radical changes occur beyond some practically hardly-accessible cardinality.
-Numbers are assigned by us, humans. There seems to be no conceivable way in which we can really assign physical meaning to any cardinality beyond 10^{10^{10}}. The way in which we count and measure strongly affects the meaning of `drawing a natural number at random’.

Well, at least I think I have achieved as much mysticism as the average cosmological physicist ;-). Sometimes I thank heavens for mathematics, in which at least some things can be more or less sharply defined, I believe ;-).

(to be continued)


About fwaaldijk

mathematician (foundations & topology in constructive mathematics) and visual artist
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