The discussion centers around Gregory Chaitin's concept of Omega, a non-computable number related to the halting problem and its implications for the foundations of mathematics. Participants explore the nature of mathematical discovery, the relationship between mathematics and other scientific fields, and the interpretations of Chaitin's work.
Participants express differing views on the interpretation and implications of Chaitin's work, with some agreeing on the significance of non-computable numbers while others question the media's representation and Chaitin's own enthusiasm. The discussion remains unresolved regarding the clarity and implications of the concepts presented.
Some participants note the potential for misunderstanding in popular science writing, and there is uncertainty regarding the specific mathematical concepts mentioned, such as Diophantine equations.
Gregory Chaitin, a mathematics researcher at IBM's T. J. Watson Research Center in Yorktown Heights, New York, has shown that mathematicians can't actually prove very much at all. Doing maths, he says, is just a process of discovery like every other branch of science: it's an experimental field where mathematicians stumble upon facts in the same way that zoologists might come across a new species of primate.
The reason for Chaitin's provocative statements is that he has found that the core of mathematics is riddled with holes. 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."
Decades later, in the 1960s, Chaitin took up where Turing left off. Fascinated by Turing's work, he began to investigate the halting problem. He considered all the possible programs that Turing's hypothetical computer could run, and then looked for the probability that a program, chosen at random from among all the possible programs, will halt. The work took him nearly 20 years, but he eventually showed that this "halting probability" turns Turing's question of whether a program halts into a real number, somewhere between 0 and 1.
Chaitin named this number Omega. And he showed that, just as there are no computable instructions for determining in advance whether a computer will halt, there are also no instructions for determining the digits of Omega. Omega is uncomputable.