A Theory of Freedomby John Valley Chapter 1. Randomness.Mathematics in the nineteenth and twentieth centuries has substantially complicated the whole subject of freedom and predetermination. All our thoughts of the subject are viewed through the lens of mathematical theory, concepts which weren’t even available until modern times. Among these concepts are probability theory, statistics, chaos theory, and fractals. Applications of this sort of mathematics include opinion polls, life insurance policies, computer games, and the science of physics (if I may be allowed to speak of physics as an “application” of mathematics). The idea of the random variable is central to them all. I will now take a moment to introduce the mathematical concept of a random variable in fairly exact terms. Any text on probability and statistics, though, will provide an equivalent and more detailed discussion of these ideas. In mathematics, the applicable term is the stochastic variable, which differs slightly from the casual idea of randomness we meet in loose conversation. A stochastic variable is, first of all, a variable in the usual algebraic sense. Represented by a simple letter or symbol, such as X (often capitalized), a stochastic variable has a value which it contributes to the expression in which it occurs. The value is constrained to fall within specified limits which is the range of the variable. A fully unconstrained variable is not really amenable to analysis, so it rarely occurs in analytic work. Instead, we define a range for the stochastic function (for example, its value lies within 0 and 1, as a decimal fraction, say 0.0315599216), and a distribution. The distribution function, of which there are several kinds, establishes the probability that the observed value of the variable will fall within a particular part of its range. There are several commonly-encountered distribution functions, of which the most familiar are the uniform distribution, and the poisson, normal, or binomial distribution (all the terms mean the same thing, to wit, the much-discussed bell curve). Having said that, the mathematically sophisticated reader will know that variables of all types have range limits and value distributions (for example, in finite math theory, a variable may be constrained to have only positive values, or only integer values in the range 1-5). It isn’t the fact that stochastic variables have a range and a distribution that makes them random, although it is very important to note that random variables are like other variables in having these limitations. A random variable in mathematics is not a wild shot out of the blue; we can say something in general about the values it can take on. The key quality of a stochastic variable is that, other than its range and its distribution function, we will say nothing about how it takes on its value. Every “observation” of the variable is a different use. Each use may select one of the allowable values by any means. You should think of the variable as a black box. You can peek into it, but each time you do, you get a new draw, a new selection, from the possible choices. It is not part of probability theory, or of statistics, to say how the values of the stochastic variable are generated. The only requirement we enforce is that a statistical analysis of the values will converge toward the distribution function as a limit. This is the part of the idea that causes difficulties for people. Probability and statistics, as mathematical methods, simply do not specify how random variables get their value. It isn’t important. Mathematics doesn’t care how it’s done. The whole point of the concept is that it’s not specified. By definition, all you know about the values is the limits of its range, and their distribution densities. This leads to a peculiar property of random sequences that is counter-intuitive, namely, that there is no characteristically random sequence. You can’t tell whether a sequence of numbers is random or non-random simply by examining their relationships. In particular, the sequences (1,2,3,4,5) and (1,1,1,1,1) are just as random as (1,4,3,3,2) or (5,4,1,3,2). Let me emphasize this point. In casual usage, the term “random” seems to suggest “disordered” or “without obvious pattern.” In mathematics, that’s not the idea of randomness at all. A random number sequence may indeed appear to be without an obvious ordering principle, but even an apparently highly ordered sequence is still random if it was generated without being forced to show that pattern. This brings us to the key point of mathematical randomness: randomness is not an intrinsic property of numbers or sequences. It’s only about their generative restrictions. A mathematically random sequence of numbers does not have a required ordering. Any ordering it may have is just cleverness of the observer in inventing some pattern which seems to fit the numbers. As long as the pattern wasn’t used to select the sequence, the pattern has nothing to do with the sequence, and hence it will be considered entirely random. Let’s take a look at some examples. In the first case, consider the task of flipping a coin-a common experiment in basic probability theory and statistics. This experiment consists of one or more trials. In each trial, the outcome is either a Head, if the obverse of the coin is showing after it comes to rest, or Tails, if the reverse of the coin is showing. (The unusual case where the coin stands on its edge won’t be discussed, which is quite typical in probability theory, where we define both the experiment and all of its possible outcomes.) We can log the results of our experiment by listing the results of each trial, thus: H, T, T, H, H, H, … Now, clearly, there are only two possibilities. We have, in fact, defined that there are only two possibilities. Yet there will be a great many trials. Hundreds or thousands. What is the likelihood of throwing two heads in a row? Actually that’s quite computable, using the principle of markov chains: it’s ½ (the chance of throwing a head) times ½ (the chance of throwing a head), or ¼. In other words, if we define a trial as two coin tosses, the odds of throwing HH are one in four. Similarly, we can compute the odds of throwing one hundred H’s in a row. The odds are small (2-100) but not zero. What does this mean? Just that throwing a repeating sequence, such as HHHHHHHH, doesn’t void the randomness of the experiment. The repetitions are just accidental. Another interesting example of randomness can be found in the decimal expansion of the number pi, 3.1415926535897932384626433832795…. It has been said that the digits of pi contain all the plays of Shakespeare if you search far enough, and by arranging substrings of the digits in a matrix to yield raster scan lines, you could find images of the American landing on the moon. Random repetitions of digits can give rise, if extended far enough, to generate almost any pattern. Yet no one would suggest that pi is a truly random number. Exactly the same series of digits are generated by any of the different possible methods for calculating it, and you get the same digits with every trial. So in what sense are the digits of pi random? They are random because there is no shortcut to computing them. They don’t follow any pattern other than that they are the digits of pi. They are random, then, with respect to a generating pattern, which there isn’t any (at least none known), and not in some deeper ontological sense. So, this exploration of randomness, which hasn’t touched in any depth on the wider subjects of probability theory and statistics, or the fascinating qualities of fractals, or the mathematical theory of chaos, has nevertheless revealed some of what randomness is, and some of what it is not. Most importantly, randomness is not an intrinsic property of any sequence of numbers. It is only the rule that we will make no rule about the sequence, just as we force no rhyme or reason of pattern onto the digits of pi. Now, having arrived at a sort of definition of randomness, let’s turn to another idea that bears centrally on the question of freedom: causality.
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