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“…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!”

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Gregory Chaitin

1999 Lecture—"A Century of Controversy over the Foundations of Mathematics" at U. Massachusetts at Lowell, quoted in [2012, Conversations with a Mathematician: Math, Art, Science and the Limits of Reason, Springer, https://books.google.com/books?id=DczTBwAAQBAJ&pg=PA15] p. 15

“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.”

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Gregory Chaitin

How real are real numbers? https://arxiv.org/abs/math/0411418 arXiv:math/0411418v3 (2004). p. 12

“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?”

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Gregory Chaitin

Thoughts on the Riemann hypothesis http://link.springer.com/article/10.1007/BF02985392 The Mathematical Intelligencer (December 2004) vol. 26, issue 1, pp. 4–7, quote on p. 4

“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.”

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Gregory Chaitin

"Epistemology as information theory: From Leibniz to Omega." https://arxiv.org/abs/math/0506552 arXiv preprint math/0506552 (2005). p. 3

“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…”

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Gregory Chaitin

Meta Maths!: The Quest for Omega https://books.google.com/books?id=ZACLDQAAQBAJ&pg=PA11. Vintage Books (2006). p. 11