Lambda the Ultimate

inactiveTopic "Maths is true by accident."
started 4/5/2001; 11:00:12 AM - last post 4/10/2001; 5:08:07 AM
Oleg - "Maths is true by accident."  blueArrow
4/5/2001; 11:00:12 AM (reads: 360, responses: 2)
This is the topic from a recent article in the New Scientist

devoted to philosophical implications of existence of uncomputable real numbers. The article introduces Chaitin's Omega, the probability that a (Turing machine) program, chosen at random from among all the possible programs, will halt. This is a real number between 0 and 1, but it is uncomputable. Given a well-known connection between Turing machines and Diophantine equations, Chaitin discovered (a very long) Diophantine equation with a parameter N. It turns out that fixing N and asking a question if the equation has an infinite number of solutions gives the N-th bit of Omega. Given that Omega is uncomputable, there does not exist an algorithm that can tell if the equation has an infinite number of solutions as a function of N. Thus in general the question about the number of solutions of a Diophantine equation is undecidable. Even in the number theory, there are "simple" questions that simply do not have any answer.

As the article says, "Chaitin has shown that there are an infinite number of mathematical facts but, for the most part, they are unrelated to each other and impossible to tie together with unifying theorems. If mathematicians find any connections between these facts, they do so by luck. "Most of mathematics is true for no particular reason," Chaitin says. "Maths is true by accident."

andrew cooke - Re: Maths is true by Accident  blueArrow
4/5/2001; 11:20:40 AM (reads: 373, responses: 0)
I wonder whether New Scientist is trying a bit hard here. For example, it might be that (all?) the interesting mathematical facts are related together and form unifying theorems (for a defintiion of "interesting" that is "describes the physical world around us", for example).

But yes, the bits I understand of this are really interesting.

[As you may have noticed, your post has only appeared in discussions - not the front page. To get access to the front page in future, email Ehud - it's in the FAQ (no obligations involved - you need never post again! - but I prefer the front page because I think more people read it). But check we've not covered the link before (I think we've mentioned Chaitin, but not this article)]

Ehud Lamm - Re:  blueArrow
4/10/2001; 5:08:07 AM (reads: 407, responses: 0)
... Prove that the is an infinite number of interesting numbers etc.