A philosophical question on randomness

Question

I have read in some book the following "philosophical" statement : "Introducing randomness we can make unstable things stable". Is there any practical example of this statement.

Answer 1

Practical example: In the absence of randomness, we can balance an egg standing up. This is unstable. The tiniest breeze or shake of the table will cause it to fall. Then it will be stable, lying on its side.

Answer 2

Hmmm. I guess it depends on what is meant by ''randomness' here and stability.

Do you mean an event that is utterly chaotic, in for example $(1)$

$(1)$- an in-deterministic event, but to the extent that we cannot even grade possibility of the event's occurrence with a qualitative/numerical degree of modality (probability). Such as the limiting relative frequency of a 'free will event occurring', or when sampling events from a set, and we want to know the relative frequency of such sampled events which happen to be picked/sampled from a non-measurable set. For which we can say, a priori, nothing at all

(2) Where we can only uses upper and lower probability values; such as imprecise or p-adic probabilities where there is something that we can say (where there is no ) unique limiting relative frequency, but which oscillates between say two values. Such HTTHHHHTTTTTTTTTHHHHHHHHHHHHHHHHHHHHHHHHHHH ($2^{n}$ heads followed by $2^{n+1}$ tails

(3) standard (often -in-deterministic) events for which we nonetheless put numerical grades of modality as to their possibility of occurrence(probability values). Such as in Quantum mechanics,or in many uses or probability theory

(4) the same as (3) but where events have equal probability as well.

I presume you mean something like (3)

If say, you are talking about random-ness in the context of probability theory and independence.

That is, the outcome of a coin toss (say heads) being, probabilistic- in-deterministic, but whose outcome probability, is independent of the outcome of the previous coin toss .

And if by stability, you mean in the context of 'stable outcome relative frequencies in a collective of such (random) trials'.

That is, a collective of such trials, whose outcome relative frequency limits to a precise value in an orderly fashion. That is (surely, or almost surely, in the infinite), and which begin to approach that value nonetheless, after a finite, albeit relatively large finite number of trials. Then there are some examples.

Then independence and the strong law of large numbers is a case in pt; it requires in effect, that the events are random, and independent of the last trial in order for the stable frequencies to limit to some value almost surely. Otherwise if not, one might wind up continually getting heads, if the first n trials are heads (as a matter of luck); if the conditional probabilities of heads given the last trial of heads is very high. And likewise if the same follows for tails

For example in $(2)$ is a case in point. The events are hardly independent and the relative frequencies are not completely stable (in some sense they are too orderly). It was often thought that within Von Mises style frequentism, that the axioms of convergence, and axioms of random-ness (immunity to place selection rules, after-effect, gambling plots, after effect) were incompatible with each other.

Ie that is, to produce orderly relative frequencies, some kind of place selection scheme/deterministic or non-independent algorithm (that a gambler could exploit), would be required to produce such frequencies. Otherwise on the contrary, nothing is guaranteed.

However, many now believe that the opposite is true: that the axiom of relative frequency convergence is in some sense helped by the axiom random-ness (immunity to place selection rules), as without independence/randomness. See https://plato.stanford.edu/entries/chance-randomness/, and see the work of Terrence fine on this view pt in

Fine, Terrence L., A computational complexity viewpoint on the stability of relative frequency and on stochastic independence, Found. Probab. Theor., stat. Inference, stat. Theor. Sci., Vol. I, Proc. int. Res. Colloq., London/Canada 1973, 29-40 (1976). ZBL0364.60009.Terence Fine

Answer 3

Another good source by T Fine on this is http://ieeexplore.ieee.org/document/1054453/ and his book 'Theorys of Probability, An examination of Foundations, 1973; which you find here http://www.sciencedirect.com/science/book/9780122564505

In the non random case, where place selection rules are in place. Often one can find any sub-sequence converging to any value that one wants, depending on the order in which one look at the sequence. So that the stability of relative frequency data is arguably just appearance. It depends on the a-priori chances of such and such particular sub-sequence being selected. And if its deterministic, there arguably no sense to made of such a statement.

• at any pt the initial conditions could determine that the next million heads are all tails; so whilst there might convergence in the infinite limit to rel frequency =1/2 its not particularly relevant.

  It would be as if the initial conditions/universe/time slice were continually repeated or sampled from an urn themselves (as Feller 1968 puts it), and we use that relative frequency instead instead; and that indeterminism kreeps back in at this meta-albeit now irrelevant level.

It may be one, in every sub-sequence ordered by the even numbers, and zero otherwise. That may not be something that one can control

-On the one hand the sequence will look too orderly. It may be something that one can control and thus and manipulate, and knowing the overall relative frequency (in the collective)is not completely relevant the probabilities and the frequencies come apart

-On the other hand independence will be violated and once cannot make great use of the laws of large numbers for IID events (not that the frequentist really makes use or a need for such theorem). One could find any sequence one wants.

So some times random-ness and stability go hand in hand, when it comes to stable and homogeneous frequency data. HTHTHTHTHT is stable in one sense, but not in another- its not homogeneous. One wants the frequency to be 1/2 in every collective if possible. Otherwise its arguably irrelevant.