![]() He gives his whole train of thought and it looks obvious. If Gauss is the prince, then Euler is the king. People talk of Gauss as the, I don’t know what they say, the prince of mathematics or something. As far as I’m concerned, he’s the… It’s sort of silly to rank people. When I was a kid, they were many, many volumes and they were going to be more. Gregory Chaitin: Yeah, they may still be publishing his collected works. Do you consider him - I consider him maybe - if not the greatest, certainly the most prolific mathematician in history. You said you had your hands on some of the original works of Euler (pictured). Marks: I guess, especially if you’re interested in that sort of stuff. The discussion turned to 18 th-century Swiss mathematician Leonhard Euler (1707–1783), a pioneer in geometry, trigonometry and calculus, among many other fields: It was a nice time to be a kid growing up in Manhattan. I had piles of books at home and I was reading, reading, reading, physics, mathematics. I had permission to take out books from the adult section in the New York City Public Library. Gregory Chaitin: And then in 1958, a book called Godel’s Proof by Ernest Nagel and James R. Our system of reasoning is incomplete, because some truths are unprovable.” – What is Gödel’s proof? ( Scientific American, February 19, 2006) Therefore, it is in fact both true and unprovable. The only alternative left is that this statement is unprovable. “Now let’s consider ‘This statement is unprovable.’ If it is provable, then we are proving a falsehood, which is extremely unpleasant and is generally assumed to be impossible. To see how the proof works, begin by considering the liar’s paradox: ‘This statement is false.’ This statement is true if and only if it is false, and therefore it is neither true nor false. His proof achieves this by constructing paradoxical mathematical statements. Note: Gödel’s Proof: “Kurt Gödel’s incompleteness theorem demonstrates that mathematics contains true statements that cannot be proved. I remember vividly an article on Gödel ’s Proof with a fantastic photograph of Gödel where he angrily leaned at the camera with a blackboard behind him. There was a course given by a nice professor and they let the kids run programs on big mainframes, which is what we had then. So I started programming in an assembly language. They had a course where the kids could get access to computers. ![]() They were just starting at the Columbia Science Honors Program. Gregory Chaitin: I was always very interested in computers. And so what led that led you down the path of computer science and to do the founding of Algorithmic Information Theory? Marks: I know in your work that you looked at things like relativity, this was in your teens for Pete’s sake, and quantum mechanics. “The mathematically provable idea that something exists but is unknowable has clear philosophical and theological implications.” But we can prove that even though Chaitin’s number exists, we can also prove it is unknowable. It’s a feature of your computer programming language. If you write programs in C++, Python, or Matlab, your computer language has a Chaitin number. Marks): “Chaitin’s number is an intellectually stunning piece of mathematics, ranking with Cantor’s model of the infinite and Shannon’s theory of information in terms of mind-bending brilliance. Note: To get some sense of Chaitin’s work, see “Things exist that are unknowable: A tutorial on Chaitin’s number” (Robert J. Professor Chaitin is also the recipient of a handful of honorary doctorates and a medallion for his landmark work. ![]() Professor Chaitin is a co-founder of the Field of Algorithmic Information Theory that explores the properties of computer programs. We are really fortunate today to talk to Gregory Chaitin (pictured) who has that distinction. Marks: There are few people who can be credited without any controversy with the founding of a game changing field of mathematics.
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