Raymond Smullyan

The Tao Is Silent Is Raymond Smullyan's beguiling and whimsical guide to the meaning and value of eastern philosophy to westerners. "To me," Writes Smullyan, "Taoism means a state of inner serenity combined with an intense aesthetic awareness. Neither alone is adequate; a purely passive serenity is kind of dull, and an anxiety-ridden awareness is not very appealing." This is more than a book on Chinese philosophy. It is a series of ideas inspired by Taoism that treats a wide variety of subjects about life in general. Smullyan sees the Taoist as "one who is not so much in search of something he hasn't, but who is enjoying what he has." Readers will be charmed and inspired by this witty, sophisticated, yet deeply religious author, whether he is discussing gardening, dogs, the art of napping, or computers who dream that they're human.

This article is written for both the general mathematican and the specialist in mathematical logic. No prior knowledge of metamathematics, recursion theory or combinatory logic is presupposed, although this paper deals with quite general abstractions of standard results in those three areas. Our purpose is to show how some apparently diverse results in these areas can be derived from a common construction. In Section 1 we consider five classical fixed point arguments (or rather, generalizations of them) which we present as problems that the reader might enjoy trying to solve. Solutions are given at the end of the section. In Section 2 we show how all these solutions can be obtained as special cases of a single fixed point theorem. In Section 3 we consider another generalization of the five fixed point results of Section 1 and show that this is of the same strength as that of Section 2. In Section 4 we show some curious strengthenings of results of Section 3 which we believe to be of some interest on their own accounts.