Clippings

She leaned forward a little and her smile became just a little glassy. Suddenly, without any real change in her, she ceased to be beautiful. She looked merely like a woman who would have been dangerous a hundred years ago, and twenty ago daring, but who today was just Grade B Hollywood.

In the first century of the Christian era, the bold atheist Wang Ch’ung denied that the Phoenix was of a distinct species. He said that just as a serpent could change into a fish, and a rat change into a tortoise, and just as the stag, in times of general prosperity, was transformed into a unicorn, so the goose took the form of the Phoenix. He attributed this mutation to the “propitious liquid” which, 2,356 years before the Christian era, had made the garden of Yao, one of the exemplary emperors, grow vermilion grass. As one can see, Wang Ch’ung’s information was faulty — or rather, excessive.

In the underworld there is an imaginary building called the Tower of the Phoenix.

What did it matter where you lay once you were dead? In a dirty sump or in a marble tower on top of a high hill? You were dead, you were sleeping the big sleep, you were not bothered by things like that. Oil and water were the same as wind and air to you. You just slept the big sleep, not caring about the nastiness of how you died or where you fell. Me, I was part of the nastiness now. […]

On the way downtown I stopped at a bar and had a couple of double Scotches. They didn’t do me any good. All they did was make me think of Silver-Wig, and I never saw her again.

“How did you happen to write your books together, with one of you at Lehigh and the other at UConn, and how do you manage to keep collaborating on their successive revisions?” These are the two questions most often asked of our two authors.

The answer to the first question is simple. Russ Johnston’s first teaching appointment was in the department of civil engineering and mechanics at Lehigh University. …

This brings us to the second question: How did the authors manage to work together so effectively after Russ Johnston had left Lehigh? Part of the answer may be provided by their phone bills and the money they spend on postage. As the publication date of a new edition approaches, they call each other daily and rush to the post office with express-mail packages in order to double-check their work. There are also frequent visits between the two families. At one time there were even joint camping trips, with both families pitching their tents next to each other. The Beers were the first to graduate to a trailer, which was used to illustrate a problem in one of the early editions of their text, but was replaced by the Johnstons’ trailer in the next one. Now this trailer has also been replaced, both authors preferring the comforts of a motel and its dining room to those of a camping ground and its fireplace.

“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.