Clippings

“I don’t see any reason,” [Gödel] wrote, “why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories.” According to Gödel, since the continuum is a real object, it was only a matter of time before new axioms would be discovered that would settle the continuum hypothesis, axioms that would “force themselves upon us as being true.”

In the possible worlds governed by these new cosmological solutions, the so-called rotating or Gödel universes, it turned out that the space-time structure is so greatly warped or curved by the distribution of matter that there exist timelike future-directed paths by which a spaceship, if it travels fast enough—and Gödel worked out the precise speed and fuel requirements, omitting only the lunch menu—can penetrate into any region of the past, present or future.

Gödel’s theorems traded on crucial distinctions such as truth versus proof, semantics versus syntax, and completeness versus formal consistency, distinctions that, though in the air, became fully clarified for the first time only after Gödel’s proofs had appeared. It was not that Hilbert, the founder of formalism, distinguished carefully between truth and proof and simply opted for the latter. Rather, as Gödel himself put the matter years later, “formalists considered formal demonstrability to be an analysis of the concept of mathematical truth and, therefore, were of course not in a position to distinguish the two.” In the realm of mathematics, proof, for the formalist, was indistinguishable from truth, and so any attempt to draw distinctions between them was simply incomprehensible.