Philosophy of Mathematics: Randomness, Part I

One of the most fundamental principles within mathematics is randomness. The study of random processes is the direct subject matter of probability and statistics, from which a broad assortment of additional fields arise, including game theory, and the theory of stochastic processes. Applied mathematics is heavily populated with applications of statistics, in finance – from market dynamics to insurance models, in physics – from thermodynamics to quantum mechanics, and throughout engineering in the form of error budgeting, tolerancing, safety factoring, and risk analysis.

What, precisely, is meant by randomness? A common definition is that a process is random if no order can be discerned in its manifestation. A more mathematical definition may include the necessity of unpredictability, a lack of bias or correlation to other processes, and may require that though “random”, the process output must follow a particular probability distribution. In a higher-order of randomness, which has been referred to as an “arbitrary” process, no probability distribution in particular applies to the process output. A more philosophically-oriented description may assume a-causality and non-determinism in the events depicted as “random”.

In the quest for the proper philosophical foundations for the fields of mathematics arising from the concept of randomness, two main questions require consideration:

1)What form of randomness exists in Reality?
2)What is the relationship between the philosophical understanding of randomness, and the nature of randomness required by the fields of probability and statistics?

It is an axiomatic, self-evident truth that all physical events are caused by prior events. The Law of Causality therefore forbids the existence of random physical processes exhibiting a-causal and non-determined behavior. On the other hand, there can exist causal, but unpredictable behavior.

Unpredictability can take two forms. What I will term “soft” unpredictability arises from a lack of knowledge on the part of the entity attempting a prediction. Soft unpredictability is not inherent in a process, but lies in the current state of the predictor. By educating the predictor, the unpredictability can be removed. Hence, I cannot predict my company’s cafeteria’s lunch menu for tomorrow only because I am not in communication with the manager of the cafeteria service. My ability to perform this prediction could be altered with an appropriate phone call (if I knew the name or number of the cafeteria manager).

By prediction, I mean specifically the ability to indicate the future state of a process at a time prior to the time at which the process will achieve that state (the prediction must also be for a particular future time). This is an essential distinction to consider in the description of what I term “hard” unpredictability.

There is a vast category of phenomena in which infinitesimal changes in starting conditions lead to exponentially divergent outcomes as the process evolves. This is the (approximate) mathematical definition of a chaotic process. For purposes of illustration, consider a simple mathematical series: x’=Gx(1-x), with G=3.7. On each iteration, take x=x’ and repeat the calculation. If we run two series A and B, with the first value in A being 0.1, and the first value in B being 0.100001, we get the following results:

Start 0.1 0.100001
20 iterations0.256724484 0.256693475
30 iterations0.682868078 0.68171437
40 iterations0.767514785 0.750851267
50 iterations0.608389182 0.924135972

Beyond this point, the two series are completely uncorrelated (as will be very obvious if you plot the series). In this example, an accurate prediction of the 50th iteration of the series requires knowledge of the starting point to better than 1 part in 100,000. As the series progresses, the maximum allowable error of knowledge in the starting condition to allow accurate prediction decreases exponentially. This characteristic of chaos is what I am terming “hard unpredictability”.

(There is much, much more that can be said about our example, which is known as the “logistic series”. For the adventurous: run various examples varying the value of A from 0 to 4 to see a variety of behaviors, particularly right around A=3.57. If you then think you’ve got a handle on it, try A=3.82.)
– To be continued

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