On 25-Jun-01, Joel Dobrzelewski wrote:
>>> It seems to me there is a great deal more information in PI than
>>> just the 2 bytes it takes to convey it in an email message.
>> Not much more. One could express pi by a short program - eg the
>> Wallis formula, that would be a few tens of bytes on most Turing
>> machines. Even expressing it as a pattern on your beloved CA, it
>> would probably not consume more that a few hundred bytes.
> Yes, I see. Juergen pointed this out too, and I think it's a valid
> point to make the distinction between different representations of the
> same mathematical object. You are both correct - Pi can in fact be
> represented nicely (as a program) in a finite way.
> But I don't dispute this, as I wasn't talking about the finite
> representation. I was talking about the infinite process / function
> that pi represents.
> Maybe this is obvious, but my whole point is that we are fooling
> ourselves if we think we can compute physics using expressions that
> consume infinite resources (memory, or computing time). Yes, I
> understand that the universe as a whole may grow without bound
> (infinite history), but at any given moment, it must be a finite size.
> Otherwise we can't compute it!
> For example, if somehow the universe requires computations like the
> x = 0
> x = x + pi()
> print x
> Then we are doomed. We cannot run this kind of program. Yes, I know we
> can find a finite representation like this:
> x = 0
> x = x + 1
> print x; " pi"
I find myself agreeing with you on your general point that any
computational theory of everything must be strictly finite - not just
On the other hand I think you are missing the point about pi. Pi can be
represented by a short program that will compute pi to however many
decimal places are required in any given calculation. That program,
not the decimal representation, can be inserted in place of pi in any
calculation where we would write "pi". So in that sense it has a
finite, even small, representation.
> But does this REALLY make use of the details of pi? I don't think so.
> I'm simply trying to get people to confront the truth that we humans
> are incapable of devising Theories of Everything that are NOT run on a
> universal computer. That's all.
> Many will say, "Of course! We know that!".
> And then they go on, as if nothing happened, talking about the
> probabilities of items in infinite sets, and "independent tosses of a
> fair coin", and "quantum indeterminacy", and "the continuum of the
> real numbers", as if these things exist!
I agree. Infinities are convenient abstractions for talking about
things that are very large without saying exactly how large - but they
shouldn't be taken too seriously.
> If we cannot program it... it's not a Theory of EVERYTHING. It's just
> a description.
I'm not so sure about this. A program is also just a description.