Subject: Re: Randomness
Date: Wed, 01 Jan 1997 14:42:43 -0500
From: Alex Bogomolny

Dear Rick:

Seems to me you came across a very interesting question. However, in its generality I can easily think of a couple interpretations at least. Can you please give me more of the Talbot's context? Must confess I have not read the book.

Just a couple of thoughts that popped up in association with yor question:

1. Proving something positive is, in general, easier than proving something negative. This is because when you prove a positive statement, your subject is concrete (mathematically concrete, of course). It may be hard to prove that an object such and such has such and such properties. However, one talks specifics. (There are exception. For example, proving, say, transcendentality of specific numbers was/is notoriously difficult.)

Intuitively, it's like juxtaposing finite and infinite. Once you can describe an object through its (even unproven) properties it automatically falls into the limited category of human perceptions. The immensity of everything beyond human ability remains largely unaffected. Negatives attempt to prove something about this immensity.

2. Look at number Pi. It's a tricky question to ask whether its digits form a random sequence. Being random implies not subject to a law. But is it not a law to be an n-th digit of the decimal expansion of Pi? However, it can be shown that the sequence is, in some sense, pseudorandom.

Now that I mentioned it, being random implies not subject to a law. The sentence (and your question along the way) is highly ambiguous. Indeed, Theory of Probability would not be a theory unless it were establishing theorems and laws about random quantities.

As I said, just two cents. Let me know if the above somehow matches the Talbot's context.

Sincerely,
Alexander Bogomolny