Perfectly Random

Suppose you have a string of random binary digits such as the following:

00111100010101001100011011001100110110010010100111

This string is 50 digits long, and was the result of a single attempt using the linked generator.

However, something seems distinctly non-random about it: there are exactly 25 zeros and exactly 25 ones. Naturally, this will not always happen, but most of the time the proportion of zeros will be fairly close to half. And evidently this is necessary, since if the proportion was usually much different from half, then the selection could not have been random in the first place.

There are other things about this string that are definitely not random. It contains only zeros and ones, and no other digits, much less items like letters from the alphabet, or items like ‘%’ and ‘$’.

Why do we have these apparently non-random characteristics? Both sorts of characteristics, the approximate and typical proportion, and the more rigid characteristics, are necessary consequences of the way we obtained or defined this number.

It is easy to see that such characteristics are inevitable. Suppose someone wants to choose something random without any non-random characteristics. Let’s suppose they want to avoid the first sort of characteristic, which is perhaps the “easier” task. They can certainly make the proportion of zeros approximately 75% or anything else that they please. But this will still be a non-random characteristic.

They try again. Suppose they succeed in preventing the series of digits from converging to any specific probability. If they do, there is one and only one way to do this. Much as in our discussion of the mathematical laws of nature, the only way to accomplish this will be to go back and forth between longer and longer strings of zeros and ones. But this is an extremely non-random characteristic. So they may have succeeded in avoiding one particular type of non-randomness, but only at the cost of adding something else very non-random.

Again, consider the second kind of characteristic. Here things are even clearer: the only way to avoid the second kind of characteristic is not to attempt any task in the first place. The only way to win is not to play. Once we have said “your task is to do such and such,” we have already specified some non-random characteristics of the second kind; to avoid such characteristics is to avoid the task completely.

“Completely random,” in fact, is an incoherent idea. No such thing can exist anywhere, in the same way that “formless matter” cannot actually exist, but all matter is formed in one way or another.

The same thing applies to David Hume’s supposed problem of induction. I ended that post with the remark that for his argument to work, he must be “absolutely certain that the future will resemble the past in no way.” But this of course is impossible in the first place; the past and the future are both defined as periods of time, and so there is some resemblance in their very definition, in the same way that any material thing must have some form in its definition, and any “random” thing must have something non-random in its definition.

 

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