The Liar’s Paradox, he explains, shows that even mathematics cannot be reduced to a fixed set of axioms:
In the early twentieth century, the mathematical community embarked on a grand quest to understand all of mathematics. This quest is entertainingly documented (with some poetic license) in the comic book Logicomix: An epic search for truth. The driving conceit of the leaders of the quest, the Vienna circle, was that—just as Euclid came up with five axioms to explain all of geometry—the same could be done for all of mathematics…
The fundamental implication is that nothing within math, science, and technology can create information. Yet information is all around us. This problem arises in many areas: evolution, artificial intelligence, economics, and physics. All of these fields investigate various kinds of information yet researchers re flummoxed when trying to explain the origin of the information. For example, it is possible to prove that evolutionary algorithms are incapable of generating information.
Eric Holloway, [Math shows why the mind is not just a formula” at Mind Matters News