Metamagical Themas by Douglas R. Hofstadter

Then came Scott Kim, and soon afterward, the mathematician Benoît Mandelbrot. Then I met the statistician and magician Persi Diaconis and the computer wizard Bill Gosper. He introduced me to the magician Jerry Andrus. First it was Ray Hyman, a psychologist who studies deception. Over the next several years, by a series of total coincidences (which turned out to be not so total), I met one after another of Martin’s friends. The notion of doing such a thing seemed ideal, even dreamlike. It seemed exciting to be able to plunge into almost any topic one liked and to say amusing and instructive things about it to a large, well-educated, and receptive audience. I remember once, years ago, wishing I were in Martin Gardner’s shoes. I never expected to be writing a column for Scientific American. Reflexivity will therefore make many return appearances in this book. Furthermore, although their connection with paradox may make reflexive systems seem no more than intellectual playthings, study of them is of great importance in understanding many mathematical and scientific developments of this century, and is becoming ever more central to theories of intelligence and consciousness, whether natural or artificial. The variety of these loopy tangles is quite remarkable, and the subject is far from being exhausted. In the following four chapters, these strange phenomena are illustrated in sentences and stories that talk about themselves, ideas that propagate themselves from mind to mind, machines that replicate themselves, and games that modify their own rules. Examples are: self-reference, self-description, self-documentation, self-contradiction, self-questioning, self-response, self-justification, self-refutation, self-parody, self-doubt, self-definition, self-creation, self-replication, self-modification, self-amendment, self-limitation, self-extension, self-application, self-scheduling, self-watching, and on and on. When realized in different ways, this abstraction becomes a concrete phenomenon. A very general name for this is reflexivity. The twists dealt with here are those whereby a system (sentence, picture, language, organism, society, government, mathematical structure, computer program, etc.) twists back on itself and closes a loop. The title of this section conveys the image of problematical twistiness.

Metamagical Themas by Douglas R. Hofstadter