Many sources make the case that evolution is not random, and to call it random is a horrible offense. Richard Dawkins, for example, wrote
The belief that Darwinian evolution is 'random' is not merely false. It is the exact opposite of the truth. Chance is a minor ingredient in the Darwinian recipe, but the most important ingredient is cumulative (natural) selection which is quintessentially nonrandom. (The Blind Watchmaker, p. 49)
Kenneth Miller wrote
Evolution is not a "random" process, and to characterize it so seriously misleads students. Natural selection, the most important force driving evolutionary change, is not random at all, but an observable, verifiable process that fine-tunes variation in populations of a species to the demands of the environment in which they live. It is true, of course, that variation in a species arises from sources such as mutation and sexual recombination, which are inherently unpredictable. Therefore evolution, like any historical process, can be influenced by random forces. (http://www.millerandlevine.com/km/evol/disclaimer.html)
These comments about randomness playing a role in evolution but evolution not being random, and similar comments from others, seem imprecise. It is true that natural selection is not random, but evolution is more than natural selection.
Mathematically, I conceptualize evolution as:
where NS is a non-trivial function and is not random, RM is random mutations and is random, and OS is other stuff, some random, and some that is not random.
That is, following Dawkins and many others, evolution is the non-random selection of random variation.
In probability theory, it is known that if X is a random variable, then for a non-trivial function f, f(X) is a random variable. Therefore, because evolution is a non-trivial function of variables, some of which are random, evolution is a random variable, and therefore it is correct to call it random.
To summarize, while evolution could reasonably be considered and described as biologically or practically non-random, it is technically, mathematically random. To say that calling evolution random is "the opposite of truth" and "false" could itself be viewed as the opposite of truth and false.
The "counter example" I most often receive is a function that has random inputs, but the output is always constant, for example f(u) = 1+u^0, where u is a random number between 0 and 1. The output of f is always 2. However, a function that is always constant, for example, constant allele frequencies, cannot be realistic even for a toy model of evolution because it would imply all of the following (paraphrasing from Biology, Life on Earth, 5th Edition, p. 291):
All of these at once is simply something we do not observe in nature. The function has to be capable of producing change that we observe.
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