Party trick for you. I’m thinking of a number between 1 and 4. Can you guess it?
Two? Nope. Three? Nope. And not one or four either.
I know what the number is, you don’t. That makes it, to you, truly random. To me, it’s completely known and as non-random as you can get. Here, then, is one instance of a truly random number.
The number, incidentally, was e, the base of the so-called natural logarithm. It’s a number that creeps in everywhere and is approximately equal to 2.718282, which is certainly between 1 and 4, but it’s exactly equal to:
The sum all the way out to infinity means it’s going to take forever and a day, literally, to know each and every digit of e, but the only thing stopping me from this knowledge is laziness. If I set at it, I could make pretty good progress, though I’d always be infinitely far away from complete understanding.
Now I came across a curious and fun little book by Donald Knuth, everybody’s Great Uncle in computer science, called Things a Computer Scientist Rarely Talks About whose dust flap started with the words, “How does a computer scientist understand infinity? What can probability theory teach us about free will? Can mathematical notions be used to enhance one’s personal understanding of the Bible?” Intriguing, no?
Knuth, the creator of TeX and author of The Art of Computer Programming among many, many other things, is Lutheran and devout. He had the idea to “randomly” sample every book of the Bible at the chapter 3, verse 16 mark, and to investigate in depth what he found there. Boy, howdy, did he mean everything. No exegete was as thorough; in this very limited and curious sense, anyway. He wrote 3:16 to describe what he learned. Things is a series of lectures he gave in 1999 about the writing of 3:16 (a book about a book).
It was Knuth’s use of the word random that was of interest. He, an expert in so-called random algorithms, sometimes meant random as a synonym of uniform, other times for unbiased, and still others for unknown.
“I decided that one interesting way to choose a fairly random verse out of each book of the Bible would be to look at chapter 3, verse 16.” “It’s important that if you’re working with a random sample, you mustn’t right rig the data.” “True randomization clearly leads to a better sample than the result of a fixed deterministic choice…The other reason was that when you roll dice there’s a temptation to cheat.” “If I were an astronomer, I would love to look at random points in the sky.” “…I thin I would base it somehow on the digits of pi (π), because π has now been calculated to billions of digits and they seem to be quite random.”
Are they? Like e, π is one of those numbers that crop up in unexpected places. But what can Knuth mean by “quite random”? What can a degrees of randomness mean? In principle, and using this formula we can calculate every single digit of π:
The remarkable thing about this equation is that we can figure the n-th digit of π without having to compute any digit which came before. All it takes is time, just like in calculating the digits of e.
Since we have a formula, we cannot say that the digits of π are unknown or unpredictable. There they all are: laid bare in a simple equation. I mean, it would be incorrect to say that the digits are “random” except in the sense that before we calculate them, we don’t know them. They are perfectly predictable, though it will take infinite time to get to them all.
Here Knuth seems to mean, as many mean, random as a synonym for transcendental. Loosely, a transcendental number is one which goes on forever not repeating exactly its digits, like e or π; mathematicians say these numbers aren’t algebraic, meaning that they cannot be explicitly and completely solved for. But it does not mean, as we have seen, that formulas for them do not exist. Clearly some formulas do exist.
As in coin flips, we might try to harvest “random” numbers from nature, but here random is a synonym for unpredictable by me because some thing or things caused these outcomes. And this holds for quantum mechanical outcomes, where some thing or things still causes the events, but (in some instances) we are barred from discovering what.
We’re full circle. The only definition of random that sticks is unknown.