When Einstein Walked with Gödel: Excursions to the Edge of Thought

by Jim Holt

Most great mathematicians claim insight into an eternal realm of abstract forms transcending the ordinary world we live in. How do they interact with this supposed “Platonic” world to obtain mathematical knowledge? Or could it be that they are radically mistaken—that mathematics, for all its power and utility, ultimately amounts to a mere tautology, like the proposition “A brown cow is a cow”?

If time travel is possible, he submitted, then time itself is impossible. A past that can be revisited has not really passed. And the fact that the actual universe is expanding, rather than rotating, is irrelevant. Time, like God, is either necessary or nothing; if it disappears in one possible universe, it is undermined in every possible universe, including our own.

But does primate humor ever rise above sheer physicality? The researcher Roger Fouts reported that Washoe, a chimp who was taught sign language, once urinated on him while riding on his shoulders, signing “funny” and snorting but not laughing.

To a mathematician, a “group” is a set of actions or operations that hang together in a nice way. What is meant by “in a nice way” is spelled out in the four axioms of group theory, which define the algebraic structure of a group.

What would it look like if a four-dimensional Stranger—a “hypersphere”—were to pass through our three-dimensional Spaceland? First we would see nothing; then a point-like ball would appear, expand into a sphere, then dwindle to a point and vanish. The spheres of varying sizes that we saw would be 3-D cross sections of the hypersphere, just as the circles that Mr. Square saw were 2-D cross sections of the sphere.