Why Goedel was Important to Mathematics

Jordan Ellenberg is a genius mathematician who wrote this 2005 Slate essay:

Goldstein calls Gödel’s incompleteness theorem “the third leg, together with Heisenberg’s uncertainty principle and Einstein’s relativity, of that tripod of theoretical cataclysms that have been felt to force disturbances deep down in the foundations of the ‘exact sciences.’ “ ...

In his recent New York Times review of Incompleteness, Edward Rothstein wrote that it’s “difficult to overstate the impact of Gödel’s theorem.” But actually, it’s easy to overstate it: Goldstein does it when she likens the impact of Gödel’s incompleteness theorem to that of relativity and quantum mechanics and calls him “the most famous mathematician that you have most likely never heard of.” But what’s most startling about Gödel’s theorem, given its conceptual importance, is not how much it’s changed mathematics, but how little. No theoretical physicist could start a career today without a thorough understanding of Einstein’s and Heisenberg’s contributions. But most pure mathematicians can easily go through life with only a vague acquaintance with Gödel’s work. So far, I’ve done it myself.

He has this backwards. He thinks Einstein invented relativity!

If numbers are real things, independent of our minds, they don’t care whether or not we can define them; we apprehend them through some intuitive faculty whose nature remains a mystery. From this point of view, it’s not at all strange that the mathematics we do today is very much like the mathematics we’d be doing if Gödel had never knocked out the possibility of axiomatic foundations. For Gödel, axiomatic foundations, however useful, were never truly necessary in the first place. His work was revolutionary, yes, but it was a revolution of the most unusual kind: one that abolished the constitution while leaving the material circumstances of the citizens more or less unchanged.

No, Goedel did not knock out the possibility of axiomatic foundations. He showed, more than any other single person, that mathematics could be founded on axioms.

He showed that first order logic was strong enough to prove statements that are true in every model. He showed how set theory axioms could help answer questions like the continuum hypothesis. Before him, we did not know that first-order logic would suffice for math foundations. After him, there was a consensus that ZFC works.

Before ZFC, we did not have rigorous constructions of the real number line, or a good concept of a function. And certainly not manifolds or vector fields or Banach spaces. Mathematicians take these things for granted today, but only because of foundational work done in the early XX century. Logicism did not fail.

It is not true that the axiomatic foundations are not necessary. It was not true for Goedel, and not true for the rest of Mathematics. Perhaps Ellenberg has managed to avoid logical subtleties in his papers, but that is only because others have done the foundational work that he built on.

Another way in which Goedel's work has transformed Math is that he invented computability for his famous theorem. It depends on the axioms being recursively enumerable. This became a core concept for theoretical computer science. It is important for math also. I would say that all pure mathematicians should have a basic understanding of first-order logic, ZFC, and computability.

Others do say similar things about Goedel, such as this 1915 book:

John von Neumann, who was in the audience immediately understood the importance of Gödel's incompleteness theorem. He was at the conference representing Hilbert's proof theory program and recognized that Hilbert's program was over.

Hilbert's program was to axiomatize mathematics. That was not over. It had just gotten started. Only a very narrow and unimportant part of it was over. That is, self-consistency could not be proved, and would not help even if it could be.