Gregory Chaitin

“"The Omega Man" New Scientist magazine, March 10, 2001.”

“A one-page proof of Chaitin's theorem from Ron Garret”

“Gregory Chaitin @academia.edu”

“Buildable/Runnable Lisp code from Chaitin's book, The Limits of Mathematics (in C)”

“Information theory”

“Leibniz, Gottfried”

“History of mathematics”

“Mathematicians are coming up with {{w|proof checker}}s that they use in their actual mathematical research. And these, like {{w|Lean (proof assistant)|Lean}}, I think is the name of one of them ... these are actually like {{w|formal system}}s that have been engineered in a way that they can actually be used by working mathematicians to check the work they're doing.”

“[A]ccording to Weyl, complexity is essential in understanding the concept of a law of nature. If laws of nature may be arbitrarily complex, he argued, the very concept... becomes vacuous. What difference would remain... if the laws meant to explain them were as complex as the phenomena they are meant to explain?Laws of nature must be simple.”

“I'm interested in the computer as a new idea, a new and fundamental philosophical concept that changes mathematics, that solves old problems better and suggests new problems, that changes our way of thinking and helps us to understand things better, that gives us radically new insights...”

“Why do I think that Turing's paper "On computable numbers" is so important? Well, in my opinion it's a paper on epistemology, because we only understand something if we can program it, as I will explain in more detail later. And it's a paper on physics, because what we can actually compute depends on the laws of physics in our particular universe and distinguishes it from other possible universes. And it's a paper on ontology, because it shows that some real numbers are uncomputable, which I shall argue calls into question their very existence, their mathematical and physical existence.”

“Are there mathematical propositions for which there is a considerable amount of computational evidence, evidence that is so persuasive that a physicist would regard them as experimentally verified?”

“At first it might seem that quantum mechanics (QM), which began with Einstein's photon as the explanation for the photoelectric effect in 1905, goes further in the direction of discreteness. But the wave-particle duality discovered by de Broglie in 1925 is at the heart of QM, which means that this theory is profoundly ambiguous regarding the question of discreteness vs. continuity. QM can have its cake and eat it too, because discreteness is modeled via standing waves (eigenfunctions) in a continuous medium.”

“...Once you entomb mathematics in an artificial language à la Hilbert, once you set up a completely formal axiomatic system, then you can forget that it has any meaning and just look at it as a game that you play with marks on paper that enable you to deduce theorems from axioms. You can forget about the meaning of the game, the game of mathematical reasoning, it's just combinatorial play with symbols! There are certain rules, and you can study these rules and forget that they have any meaning!”