ANNOTATED BIBLIOGRAPHY
- Asger Aaboe, Episodes from the Early History of Mathematics, New Mathematical Library, Mathematical Association of America, 1964. Describes the archaeological work on the Babylonian cuneiform tablets, presents examples of sexagesimal notation and of solutions of quadratic equations with very large coefficients. This is followed by chapters on Euclid, Archimedes, and Ptolemy. I present Euclid's axioms and postulates, emphasizing the 5th (parallel). I also use Archimedes' famous determination of the relative volumes of the sphere, cone, and pyramid. Archimedes' use of “unorthodox” methods (infinitesimals, centers of gravity) is very clearly explained by Aaboe.
- E. T. Bell, Men of Mathematics, Simon and Schuster, New York, 1937. This book is an unrivaled collection of novelizations of the lives of mathematicians. Students can enjoy Bell's narrative flair, while learning the lesson that not everything in print is gospel truth.
- Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics, 2nd ed., Cambridge University Press, Cambridge, 1983. This is a standard collection of articles in the philosophy of mathematics, overwhelmingly weighted toward the foundationist viewpoint.
- E. Bishop, Foundations of Constructive Analysis, McGraw-Hill, New York, 1967. Most of this book is technical mathematics. The first chapter is an eloquent statement of the intuitionist-constructivist viewpoint.
- E. Bishop, Aspects of constructivism, Lecture Notes, New Mexico State University, 1972.
- C. Boyer, History of the Calculus and Its Conceptual Development, New York, 1949. Dover reprint, 1959. This book could well merit a course in itself. The leading theme is the struggle to clarify the notions of infinitesimal and limit. Boyer, writing before Robinson's nonstandard analysis had been created, takes it for granted that the necessary and desirable resolution was the abandonment of infinitesimals.
- H. Curry, Some aspects of the problem of mathematical rigor, Bulletin of the American Mathematical Society 47 (1941) 221–241. Takes an extreme formalist stance.
- Tobias Dantzig, Number, The Language of Science, Doubleday Anchor Books, Garden City, NY, 1956. The story of number systems from natural to complex, written with the style of a poet and the depth of a philosopher.
- Philip J. Davis and Reuben Hersh, The Mathematical Experience, Birkhäuser, Boston, 1981. One of my favorite books.
- Philip J. Davis and Reuben Hersh, Descartes Dream, Harcourt Brace Jovanovich, Boston, 1986. Describes the impact of mathematics and computing on today's society.
- A. Dresden, Some philosophical aspects of mathematics, Bulletin of the American Mathematical Society 34 (1928) 438–452. In sympathy with Brouwer, but not as extreme.
- Howard Eves and Carroll V. Newsom, An Introduction to the Foundations and Fundamental Concepts of Mathematics, Holt Rinehart and Winston, New York, 1965. This is the closest thing to a textbook for the course. It is a historically presented account of the main ideas and results of mathematics from the Greeks to Hilbert and Gödei. Beautifully written, with many interesting problems which unfortunately are too hard for students who aren't mathematics or science majors. I use their chapters on Euclidean and on non-Euclidean geometry. I find their work on calculus and on 19th- and 20th-century mathematics needs to be supplemented.
- Gottlob Frege, The Foundations of Arithmetic, Harper and Brothers, Harper Torchbooks, New York, 1960. Logicism from the horse's mouth. Written in a lively, polemical, sarcastic style that keeps my students awake.
- K. Gödel, What is Cantor's continuum problem?, American Mathematical Monthly 54 (1947) 515–525. Takes a Platonist stand.
- Hans Hahn, The crisis in intuition, in J. R. Newman, ed., The World of Mathematics, Simon and Schuster, New York, 1956, pp. 1956–1976. Beautifully written, presents the “pathological” space-filling and nowhere-differentiable curves that undermined the authority of visual intuition in the 19th century.
- G. H. Hardy, Mathematical proof, Mind 30 (1929) 1–25. “Typical” philosophy of the working mathematician.
- Reuben Hersh, Some proposals for reviving the philosophy of mathematics, Advances in Mathematics 31 (1979) 31–56. My personal view on what's wrong and what to do about it.
- M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, Oxford, 1972. An amazing compendium, up through the early 20th century, carefully and reliably presented. By far the most complete and authoritative history of mathematics available today.
- Philip Kitcher, The Nature of Mathematical Knowledge, Oxford University Press, New York, 1983. The most ambitious of the attempts to ground the philosophy of mathematics in its history and in the actual practice of mathematicians today. The later chapters, on history, are probably more accessible to students than the earlier chapters, which are more technical in the philosophical sense.
- S. Korner, The Philosophy of Mathematics, Harper Torchbooks, New York, 1962. A useful compendium of the three foundationist dogmas (logicist, formalist, intuitionist) with critiques of each.
- Imre Lakatos, Proofs and Refutations, Cambridge University Press, Cambridge, 1976. A landmark in 20th century philosophy of mathematics. A veritable geyser (or avalanche) of wit and learning. The nature of mathematics is expressed by means of a classroom dialog on the Euler formula V - E + F = 2.
- Jean Piaget, Genetic Epistemology, Columbia University Press, New York, 1970. Piaget was involved in the philosophy of mathematics, not just its psychology and pedagogy. The philosophical importance of his ideas has yet to be recognized.
- J. Pierpont, Mathematical rigor, past and present, Bulletin of the American Mathematical Society 34 (1928) 23–53. Describes the crisis in foundations with tongue firmly in cheek.
- M. Polanyi, Personal Knowledge: Towards a Post-Critical Philosophy, University of Chicago Press, Chicago, 1960. This is mostly philosophy of science, not of mathematics, but it is a fascinating, exciting book that everyone should read. On mathematics, he offers the startling insight that in order to be accepted as part of mathematics, an idea or method has to be interesting.
- G. Pólya, How to Solve It, Princeton University Press, Princeton, 1945. Like Piaget, Pólya's work on teaching and learning had deep philosophical import, which has yet to be recognized. This book is one of the intellectual master works of the 20th century.
- Alfred Renyi, Dialogues on Mathematics, Holden Day, San Francisco, 1967. These dialogues can be read by anybody. Painlessly and accurately they answer, what is mathematics? What is applied mathematics? How is mathematics useful in science?
- Rudy Rucker, Infinity and the Mind, Birkhäuser, Boston, 1982. This is a fun book, a melange of logic, mathematics, and mysticism. A special bonus is Rucker's personal recollections of conversations with Gödei.
- Lynn Arthur Steen (ed.), Mathematics Today, Vintage Books (Random House), New York, 1980. A collection of articles widely varying in level, subject, and originality. I always use the article by Martin Davis (“What is a Computation?”) to teach Turing machines and Gödel's theorem.
- Thomas Tymoczko (ed.), New Directions in the Philosophy of Mathematics, Birkhäuser, Boston, 1986. An anthology of recent writings, most of which try to base philosophy on the actual practice and experience of mathematicians.
- L. A. White, The locus of mathematical reality, Philosophy of Science 14 (1947) 289–303. Reprinted in The World of Mathematics, J. R. Newman (ed.), Vol. 4, Simon and Schuster, New York, 1956, pp. 2348–2364. A beautiful presentation, by a famous anthropologist, of the view that mathematics' “locus of reality” is in human culture, the shared consciousness of communities.