I’ve used this gimmick many times, so regular readers, please, shhh.
I’m thinking of a number between 1 and 4. What is it?
To you the number is random, because you don’t know the cause, or causes, of my picking the number. To me, it is fully determined, because I picked it. There is nothing random to it—to me. So randomness depends on context.
Now regular readers have seen this stunt before, so they have a bit more information than newcomers. It’s up to them how they use this information. But, if they use it right, they can make a better guess. To these clever characters, the answer is less random, because it is more predictable. And it is more predictable because they have more information on the cause. So randomness has a size. (The answer will be revealed at the bottom.)
Given these two characteristics, we deduce randomness is a measure of ignorance: it is epistemological, a thing in your mind, and not in things.
All this is introduction to a small article about the number pi (π): “Pi might look random but it’s full of hidden patterns”, by Humble.
He correctly notes “we will never be able to calculate all the digits of pi because it is an irrational number, one that continues forever without any repeating pattern.” And the reason we cannot is time: we’d need infinite amounts of it in actuality, and not just potentially.
Now there has to be a reason, a cause or causes, why pi equals (or is in base 10) “3.14159…”, where that “…” stands for an infinite number of fixed digits. But not entirely known digits.
But then Humble makes a mistake. He says, “The reason we can’t call pi random is because the digits it comprises are precisely determined and fixed. For example, the second decimal place in pi is always 4. So you can’t ask what the probability would be of a different number taking this position. It isn’t randomly positioned.”
I say pi is random because we don’t know all the causes why it is “3.14159…”. But he’s quite right in insisting that the proposition “The second decimal place in pi is 4” is, given the relevant premises, certain. However, this proposition is random: “The googol-th decimal place in pi is 4”, recalling a googol is 10^100. We only know the values up to a few trillions of decimal places.
Suppose, somehow and per impossible, we knew all the digits of pi. God wrote them on a rock, say, while also providing an infinitesimal microscope to read them all. Given this observation, the proposition “pi is 3.14159…”, where the ellipsis is filled in, is certain.
But unless we knew why it took that infinite precise value and not another, it is still random in part. That is, any proposition containing proposals or the cause, the reason, for this value, are still unknown, and thus random.
Humble then says “‘Is pi a normal number?’ A decimal number is said to be normal when every sequence of possible digits is equally likely to appear in it, making the numbers look random even if they technically aren’t.” Technically they are, if we don’t know them. But what about this “normal” business? (Not to be mistaken for “normality” or normal distributions.)
Well, he gives a small table given by Yasumasa Kanada of the first trillion decimal digits of pi:
Digit Occurrences 0 99,999,485,134 1 99,999,945,664 2 100,000,480,057 3 99,999,787,805 4 100,000,357,857 5 99,999,671,008 6 99,999,807,503 7 99,999,818,723 8 100,000,791,469 9 99,999,854,780 Total 1,000,000,000,000
So the first trillion digits aren’t normal, since these entries don’t all equal 100 billion. But perhaps, once we reached infinity, uniformity appears. Somebody might even be able to prove it (if they have, I haven’t heard about it).
What’s the probability of the proposition “pi is normal”? It doesn’t have one until we specify premises. Like “First trillion are normalish, so why not the whole thing?”, then the probability of “pi is normal” is high. Probability of “Pi in first trillion is normal” given Kanada’s table is 0.
But it we had some other irrational number that gave a more obvious non-uniformity (and there are some), like having twice as many 1s as 2s, and twice as many 2s as 3s, and so on roughly in the first trillion, then you not only more information about the non-normalness of the number itself, but also its remaining digits. You could use what you knew learned from the first trillion as a premise for the likelihood of the trillionth and one number, which won’t be 1/10.
As above, there is the sense these numbers are less random, because more is known about them.
But we’d still need to know the entire why of the number—why it was this value and none other—before it loses all its randomness.
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