Definition of randomness

If you don’t know the way the series was generated any guess will be inductive. You might think that

1, 1/2,1/3,1/4,1/5…

Is the series 1/n but that’s just an inductive guess, and there’s nothing wrong with that, physical laws are inductive guesses too. Sure every time I’ve dropped something it’s fallen down (toward the center of the earth), that doesn’t mean it will next time, but that didn’t stop Newton from formulating the universal theory of gravitation.

You mean, like, explains everything? O boy, can’t wait.

But the algorithm itself constitutes a formal means of predicting what the next digit will be ! It’s just silly to limit the meaning of prediction to those things that can be deduced by looking at something without a formal plan of analysis.

(Regarding the digits of Pi not being random in Algorithmic Information Theory.)

Umm, if the digits are Pi, then they are not random, if the digits are not Pi they may or may not be random. Go search about “Kolmorgorov” and such. Read up on it. You are missing the whole concept. By the very existence of an algorithm to generate its digits de facto means it’s not random in this context.

When a computer scientist says that a sequence of digits is random, (s)he means that it’s incompressible–there’s no program/input pair shorter than the sequence of digits that can produce that sequence of digits. [symbol]p[/symbol] is not random, because it’s infinitely long, but can be produced by a finite algorithm. 1 is random, because even the statement “output 1” is longer.

So, would [symbol]p[/symbol] be random, then, because “output [symbol]p[/symbol]” is longer? :slight_smile:


Of course, I’m talking about representations in base 10 here.

I thought you were talking about representing numbers in a compressed form? That’s certainly a different representation, right?

And I thought you meant representing numbers by an algorithm.

Pi can be repressented in compressed form by an algorithm; a “truly random” set of numbers could not be.

Since nobody’s covered this yet, let me point out that this would not be a very useful test. I can write a program that always outputs 0.5, which would converge very quickly indeed, but which would not satisfy any reasonable definition of randomness.

A sine function with an amplitude of 1 would be a little more elaborate way to mess with possibility 2.

Maybe random numbers only exist in the minds of those that cant think (insane), because people think generally in patterns until there mind becomes damaged