Gödel, Escher, Bach: An Eternal Golden Braid (138 page)

Davis, Martin. The Undecidable. Hewlett, N. Y.: Raven Press, 1965. An anthology of some of the most important papers in metamathematics from 1931 onwards (thus quite complementary to van Heijenoort's anthology). Included are a translation of Godel's 1931 paper, lecture notes from a course which Godel once gave on his results, and then papers by Church, Kleene, Rosser, Post, and Turing.

Davis, Martin, and Reuben Hersh. "Hilbert's Tenth Problem". Scientific American, November 1973, p. 84. How a famous problem in number theory was finally shown to be unsolvable, by a twenty-two-year old Russian.

** DeLong, Howard. A Profile of Mathematical Logic. Reading, Mass.: AddisonWesley, 1970. An extremely carefully written book about mathematical logic, with an exposition of Godel's Theorem and discussions of many philosophical questions. One of its strong features is its outstanding, fully annotated bibliography. A book which influenced me greatly.

Doblhofer, Ernst. Voices in Stone. New York: Macmillan, Collier Books, 1961. Paper back. A good hook on the decipherment of ancient scripts.

* Drevfus, Hubert. What Computers Can't Do: A Critique of Artificial Reason. New York:Harper & Roc-, 1972. A collection of many arguments against Artificial Intelligence from someone outside of the field. Interesting to try to refute. The Al community and Drevfus enjoy a relation of strong mutual antagonism. It is important to have people like Dreyfus around, even if you find them very irritating.

Edwards, Harold M. "Fermat's Last Theorem". Scientific American, October 1978, pp.

104-122. A complete discussion of this hardest of all mathematical nuts to crack, from its origins to the most modern results. Excellently illustrated.

* Ernst, Bruno. The Magic Mirror of M. C. Escher. New York: Random House, 1976.

Paperback. Escher as a human being, and the origins of his drawings, are discussed with devotion by a friend of mans' years. A "must" for any lover of Escher.

** Escher, Maurits C., et al. The World of M. C. Escher. New York: Harry N. Abrarns, 1972. Paperback. The most extensive collection of reproductions of Escher's works.

Escher comes about as close as one can to recursion in art, and captures the spirit of Godel's Theorem in some of his drawings amazingly well,

Feigenbaum, Edward, and Julian Feldman, eds. Computers and Thought. New York: McGraw-Hill, 1963. Although it is a little old now, this book is still an important collection of ideas about Artificial Intelligence. Included are articles on Gelernters Index

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geometry program, Samuel's checkers program, and others on pattern recognition, language understanding, philosophy, and so on.

Finsler, Paul. "Formal Proofs and Undecidability", Reprinted in van Heijenoorcs anthology From Frege to Godel (see below). A forerunner of Godel's paper, in which the existence of undecidable mathematical statements is suggested, though not rigorously demonstrated.

Fitzpatrick, P. J. "To Godel via Babel"- Mind 75 (1966): 332-350. An innovative exposition of Godel's proof which distinguishes between the relevant levels by using three different languages: English, French, and Latin'

* Gablik, Suzi. Magritte. Boston, Mass.: New York Graphic Society, 1976. Paperback.

An excellent book on Magritte and his works by someone who really understands their setting in a wide sense; has a good selection of reproductions.

* Gardner, Martin. Fads and Fallacies. New York: Dover Publications, 1952. Paperback.

Still probably the best of all the anti-occult books. Although probably not intended as a book on the philosophy of science, this book contains many lessons therein. Over and over, one faces the question, "What is evidence?" Gardner demonstrates how unearthing "the truth" requires art as much as science.

Gebstadter, Egbert B. Copper, Silver, Gold: an Indestructible Metallic Alloy. Perth: Acidic Books, 1979. A formidable hodge-podge, turgid and confused-yet remarkably similar to the present work. Professor Gebstadter's Shandean digressions include some excellent examples of indirect self-reference. Of particular interest is a reference in its well-annotated bibliography to an isomorphic, but imaginary, book.

** Godel, Kurt. On Formally Undecidable Propositions. New York: Basic Books, 1962.

A translation of Godel's 1931 paper, together with some discussion.

. "Uber Formal Unentscheidbare Satze der Principia Mathematica and Verwandter Systeme, I." Monatshefte fur Mathematik and Physik, 38 (1931), 173-198. Godel's 1931 paper.

Golf man, Ering. Frame Analysis. New York: Harper & Row, Colophon Books, 1974.

Paperback. A long documentation of the definition of "systems" in human communication, and how in art and advertising and feporting and the theatre, the borderline between "the system" and "the world" is perceived and exploited and violated.

Goldstein, Ira. and Seymour Papert. "Artificial Intelligence, Language, and the Study of Knowledge". Cognitive Science 1 (January l9/ 7): 84-123. A survey article concerned with the past and future of Al. The authors see three periods so far: "Classic",

"Romantic,., and "Modern".

Guod, I. I. "Human and Machine Logic". British Journal for the Philosophy of Science In (196/): 144. One of the most interesting attempts to refute Lucas, having to do with whether the repeated application of the diagonal method is itself a mechanizable operation.

. "Godel's Theorem is a Red Herring". British Journal for the Philosophy of Science 19

(1969): 357. In which Good maintains that Lucas' argument has nothing to do with Godel's Theorem, and that Lucas should in fact have entitled his article "Minds, Machines, and Transfinite Counting". The Good-Lucas repartee is fascinating.

Goodman, Nelson. Fact, Fiction, and Forecast. 3rd ed. Indianapolis: Bobbs-Merrill, 1973.

Paperback. A discussion of contrary-to-fact conditionals and inductive logic, including Index

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Goodman's famous problem-words "bleen" and "grue". Bears very much on the question of how humans perceive the world, and therefore interesting especially from the Al perspective.

Press, 1976. A short hook about bees, apes, and other animals, and whether or not they are "conscious"-and particularh whether or not it is legitimate to use the word

"consciousness" in scientific explanations of animal behavior.

deGroot, Adriaan. Thought and Choice in Chess. The Hague: Mouton, 1965. A thorough study in cognitive psychology, reporting on experiments that have a classical simplicity and elegance.

Gunderson, Keith. Mentality and Machines. New York: Doubleday, Anchor Books, 1971.

Paperback. A very anti-Al person tells why. Sometimes hilarious.

Hanawalt, Philip C., and Robert H. Haynes, eds. The Chemical Basis of Life. San Francisco: W. H. Freeman, 1973. Paperback. An excellent collection of reprints from the Scientilic American. One of the best ways to get a feeling for what molecular biology is about.

Hardy G. H. and E. M. Wright. An Introduction to the Theory of Numbers, 4th ed. New York: Oxford University Press, 1960. The classic book on number theory. Chock-full of information about those mysterious entities. the whole numbers.

Harmon, Leon. "The Recognition of Faces". Scientific American, November 1973, p. 70.

Explorations concerning how we represent faces in our memories, and how much information is needed in what form for us to be able to recognize a face. One of the most fascinating of pattern recognition problems.

van Heijenoort, Jean. From Frege to Godel: A Source Book in Mathematical Logic.

Cambridge, Mass.: Harvard University Press, 1977. Paperback. A collection of epoch-making articles on mathematical logic, all leading up to Godel's climactic revelation, which is the final paper in the book.

Henri, Adrian. Total Art: Environments, Happenings, and Performances. New York: Praeger, 1974. Paperback. In which it is shown how meaning has degenerated so far m modern art that the absence of meaning becomes profoundly meaningful (whatever that means).

* Goodstein, R. L. Development of Mathematical Logic. New York: Springer Verlag, 1971. A concise survey of mathematical logic, including much material not easily found elsewhere. An enjoyable book, and useful as a reference.

Gordon, Cyrus. Forgotten Scripts. New York: Basic Books, 1968. A short and nicely written account of the decipherment of ancient hieroglyphics, cuneiform, and other scripts.

Griffin, Donald. The Question of Animal Awareness. New York: Rockefeller University von Foerster, Heinz and James W. Beauchamp, eds. Music by Computers. New York:

**

John Wiles', 1969. This book contains not only a set of articles about various types of computer-produced music, but also a set of four small phonograph records so you can actually hear (and judge) the pieces described. Among the pieces is Max Mathews'

mixture of "Johnny Comes Marching Home" and "The British Grenadiers".

Fraenkel, Abraham, Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory, 2nd ed. Atlantic Highlands, N. J.: Humanities Press, 1973. A fairly nontechnical Index

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discussion of set theory, logic, limitative Theorems and undecidable statements.

Included is a long treatment of intuitionism.

*

Frev, Peter W. Chess Skill in Man and Machine. New York: Springer Verlag, 1977.

Anexcellent survey of contemporary ideas in computer chess: why programs work, why they don't work, retrospects and prospects.

Friedman, Daniel P. The Little Lisper. Palo Alto, Calif.: Science Research Associates, 1974. Paperback. An easily digested introduction to recursive thinking in LISP. You'll eat it up!

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Hoare, C. A. R, and D. C. S. Allison. "Incomputability". Computing Surveys 4, no. 3 September 1972). A smoothly presented exposition of why the halting problem is unsolvable. Proves this fundamental theorem: ' Any language containing conditionals and recursive function definitions which is powerful enough to program its own interpreter cannot be used to program its own 'terminates' function."

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Hofstadter, Douglas R. "Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields". Physical Review B, 14, no. 6 (15 September 1976). The author's Ph.D. work, presented as a paper. Details the origin of "Gplot", the recursive graph shown in Figure 34.

Hook, Sidney. ed. Dimensions of Mind. New York: Macmillan, Collier Books, 1961.

Paperback. A collection of articles on the mind-body problem and the mind-computer problem. Some rather strong-minded entries here.

• Hornev, Karen. Self-Analysis. New York: W. W. Norton, 1942. Paperback. A fascinating description of how the levels of the self must tangle to grapple with problems of self-definition of any individual in this complex world. Humane and insightful.

Hubbard, John I. The Biological Basis of Mental Activity. Reading, Mass.: AddisonWesley, 1975. Paperback. Just one more book about the brain, with one special virtue. however: it contains man long lists of questions for the reader to ponder, and references to articles which treat those questions.

• Jackson. Philip C. Introduction to Artificial Intelligence. New York, Petrocelli Charter.

1975. A recent book, describing, with some exuberance, the ideas of Al. There are a huge number of vaguely suggested ideas floating around this book, and for that reason it is very stimulating just to page through it. Has a giant bibliography, which is another reason to recommend it.

Jacobs, Robert L. Understanding Harmony. New York: Oxford University Press, 1958.

Paperback. A straightforward book on harmony, which can lead one to ask many questions about why it is that conventional Western harmony has such a grip on our brains.

Jaki, Stanley L. Brain, Mind, and Computers. South Bend. Ind.: Gateway Editions, 1969.

Paperback. A polemic book whose every page exudes contempt for the computational paradigm for understanding the mind. Nonetheless it is interesting to ponder the points he brings up.

• Jauch, J. M. Are Quanta Real? Bloomington, Ind.: Indiana University Press, 1973. A delightful little book of dialogues, using three characters borrowed from Galileo, put in a modern setting. Not only are questions of quantum mechanics discussed, but also issues of pattern recognition, simplicity, brain processes, and philosophy of science enter. Most enjoyable and provocative.

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• Jeffrey, Richard. Formal Logic: Its Scope and Limits. New York: McGraw-Hill, 1967.

An easy-to-read elementary textbook whose last chapter is on Godel's and Church's Theorems. This book has quite a different approach from many logic texts, which makes it stand out.

• Jensen, Hans. Sign, Symbol, and Script. New York: G. P. Putnam's, 1969. A-or perhaps the-top-notch book on symbolic writing systems the world over, both of now and long ago. There is much beauty and mystery in this book-for instance, the undeciphered script of Easter Island.

Kalmar, Laszlo. "An Argument Against the Plausibility of Church's Thesis". In A.

Heyting, ed. Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam, 1957, North-Holland, 1959. An interesting article by perhaps the bestknown disbeliever in the Church-Turing Thesis.