Drawing a natural number at random, foundations of probability, determinism, Laplace (2)

I probably ūüėČ chewed off more than I can swallow, certainly in one go…

Still, my general lack of proficiency in probability theory should -I believe- not preclude me from asking certain questions, and offering uncertain answers. Like I stated earlier, it seems to me that we are still largely in the dark about many ‘simple’ issues, which yet frequently are presented as being reasonably cut and dried (I feel). However, a disclaimer is called for: my knowledge and understanding of physics is very limited (I will comment on that later).

Let me pose the first relevant ‘simple’ question:

QUESTION 1   Is our physical world finite or infinite?

The relevance of this question I hope to make apparent later on. But first perhaps I should pause to ask: do I really understand the question itself? 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’.

Before going into this question any deeper, I wish to phrase another `simple’ relevant question/issue:

QUESTION 2   What is the role of information in probability?

(Easy to illustrate the basic problem here, not so easy perhaps to demonstrate why it has such relevance.) Suppose we draw a marble from a vase filled with an equal amount of blue and white marbles. What is the chance that we draw a blue marble?

In any high-school exam, I would advise you to answer: 50%. In 98% of university exams I would advise the same answer. Put together that makes … just kidding. The problem here is that any additional information can drastically alter our perception of the probability/chance of drawing a blue marble. In the most dramatic case, imagine that the person drawing the marble can actually feel the difference between the two types of marbles, and therefore already knows which colour marble she has drawn. For her, the chance of drawing a blue marble is either 100% or 0%. For us, who knows? Perhaps some of us can tell just by the way she frowns what type of marble she has drawn…?

It boils down to the question: what do we mean by the word `chance’? I quote from Wikipedia:

The first person known to have seen the need for a clear definition of probability was Laplace.[citation needed] As late as 1814 he stated:

The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.

‚ÄĒ¬†Pierre-Simon Laplace,¬†A Philosophical Essay on Probabilities[4]

This description is what would ultimately provide the classical definition of probability.


One easily sees however that this `definition’ avoids the main issue. Laplace did not always avoid this main issue however:

Laplace([1776a]; OC, VIII, 145):

Before going further, it is important to pin down the sense of the words chance and probability. We look upon a thing as the effect of chance when we see nothing regular in it, nothing that manifests design, and when furthermore we are ignorant of the causes that brought it about. Thus, chance has no reality in itself. It is nothing but a term for expressing our ignorance of the way in which the various aspects of a phenomenon are interconnected and related to the rest of nature.

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
‚ÄĒPierre Simon Laplace,¬†A Philosophical Essay on Probabilities[37]
Dear reader, these issues are still hotly debated…although we seldom hear about it! From Wikipedia again:

Some (including¬†Albert Einstein) argue that our inability to predict any more than probabilities is simply due to ignorance.[36]¬†The idea is that, beyond the conditions and laws we can observe or deduce, there are also hidden factors or “hidden variables” that determine¬†absolutely¬†in which order photons reach the detector screen. They argue that the course of the universe is absolutely determined, but that humans are screened from knowledge of the determinative factors. So, they say, it only appears that things proceed in a merely probabilistically determinative way. In actuality, they proceed in an absolutely deterministic way. These matters continue to be subject to some dispute. A critical finding was that quantum mechanics can make statistical predictions which would be violated if local hidden variables really existed. There have been a number of experiments to verify such predictions, and so far they do not appear to be violated. This would suggest there are no hidden variables, although many physicists believe better experiments are needed to conclusively settle the issue (see also¬†Bell test experiments). Furthermore, it is possible to augment quantum mechanics with non-local hidden variables to achieve a deterministic theory that is in agreement with experiment. An example is the¬†Bohm interpretation¬†of quantum mechanics. This debate is relevant because it is easy to imagine specific situations in which the arrival of an electron at a screen at a certain point and time would trigger one event, whereas its arrival at another point would trigger an entirely different event (e.g. see¬†Schr√∂dinger’s cat¬†– a thought experiment used as part of a deeper debate).

Thus, the world of quantum physics casts reasonable doubt on the traditional determinism that is so intuitive in classical, Newtonian physics. At the small scales, our reality does not seem to be absolutely determined. Yet this was precisely the subject of the famous¬†Bohr‚ÄďEinstein debates¬†between Einstein and¬†Niels Bohr. There is¬†still no consensus. In the meantime, humans continue to benefit from the fact that reality obeys determined probabilities at the quantum scale. Such adequate determinism (see¬†Varieties, above) is the reason that Stephen Hawking calls¬†Libertarian free will¬†“just an illusion”.[35]¬†Compatibilistic¬†free will (which is deterministic) may be the only kind of “free will” that can exist. However, Daniel Dennett, in his book¬†Elbow Room, says that this means we have the only kind of free will “worth wanting”. For even more discussion, see¬†Free will.

So. I have argued in my article and book that Church’s Thesis gives an unambiguous mathematical way to interpret determinism, but so far I have received very little reaction, probably because this is a very blunt way of looking at things, and from a rather different angle than traditional physics too. However, the statistics involved in the physical experiment that I propose, inescapably lead to questions such as: can we assign meaning to the question of drawing a natural number at random?

(to be continued)


About fwaaldijk

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