There is no formal definition of ‘randomness’ within the field of mathematics and therefore none within theoretical physics. As far as practical physics is concerned then, there can be no quantitative study of randomness or even recognition of ‘random’ events. Any philosophical musings on ‘randomness’ have no foundation in anything meaningful.
No individual event then can be said to be ‘random’. However, there do exist processes that generate multiple events whose collective outcome can be usefully studied and characterised by the techniques of statistics and probability theory.
Wikipedia: “The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space”
Wikipedia: “A random variable is a mathematical formalization of a quantity or object which depends on random events. The term ‘random variable’ can be misleading as its mathematical definition is not actually random nor a variable,“
Wikipedia: “A random sequence is a vague notion… in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians” – D H Lehmer
So randomness is defined in terms of ‘random variables’ which are not really random but nevertheless depend upon ‘random events’. We have a somewhat circular definition with experts not even liking the whole idea anyhow.
What is happening is that intuitive notions of randomness and unpredictability of individual events are impossible to formalise and therefore have no relevance to mathematics or physics. What is important however is the predictable patterns that emerge from the repeated outcomes of such events.
The language gives the impression that one thing is true whereas it is the precise opposite. So called ‘random’ events are characterised by the predictable emergent properties of collective outcomes and not by the unpredictability of individual events.
Coin tossing example. A coin was tossed 32 times and the number of heads was noted. The experiment was repeated 50000 times and the results plotted on the chart below. Unsurprisingly, 16 was the most likely number, with other outcomes being progressively less common as shown.
The resulting curve is a probability distribution, in fact a Gaussian distribution. The shape of the curve becomes more precise the more coin tosses there are and it is this shape that can be used to make predictions and to do some practical science.
Coin tosses are unpredictable as individual events but highly predictable as collective events. We have a sort of reversal of entropy whereby the more ‘noise’ (randomness) is put into the system, the more ordered the output becomes.
The coin tossing procedure therefor is characterised by the outcome curve and not by the unpredictability of individual events and not by the physical mechanism of the coin toss.
Unpredictability. Attempts to define randomness as something to do with being ‘unpredictable’ fail because:
- Once an ‘unpredictable’ sequence of events has been observed it is no longer unpredictable but is nevertheless the same sequence of events
- The whole concept of ‘unpredictability’ is therefore not well-defined
- Probability distributions are characterised by their long term outcomes and the predictability or otherwise of a single event has nothing to do with this anyhow
Random number generators. Attempts to define randomness by the complexity or opaqueness of the generation method fail because:
- Probability distributions are characterised by their long term outcomes and the method by which these outcomes are generated has nothing to do with this
The digits of π are distributed evenly throughout the range 0 to 9 and may said to be occur ‘randomly’, but they are certainly not unpredictable and are, furthermore, generated by a mechanism that is completely deterministic, well-defined and reproducible.
The Galton board. In the video, it may be said that each ball follows a ‘random’ path which is superficially unpredictable but actually predictable if you have enough information. However, the outcome is always predictably the same – another Gaussian curve. Emergent properties arise from multiple events.