What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

GENERAL REFERENCES

D. J. Albers and G. L. Alexanderson (editors). Mathematical People. Boston: Birkhäuser, 1985.

D. J. Albers, G. L. Alexanderson, and Constance Reid (editors). More Mathematical People. New York: Academic Press, 1990.

B. Bollobás (editor). Littlewood’s Miscellany. Cambridge: Cambridge University Press, 1986.

J. L. Casti. Complexification. New York: HarperCollins, 1994.

J. Cohen and I. Stewart. The Collapse of Chaos. New York: Viking, 1993.

COMAP (editors). For All Practical Purposes. New York: Freeman, 1994.

P. J. Davis and R. Hersh. The Mathematical Experience. Boston: Birkhäuser, 1981.

———. Descartes’ Dream. Brighton: Harvester, 1986.

K. Devlin. All the Math That’s Fit to Print. Washington: Mathematical Association of America, 1994.

———. Mathematics: The New Golden Age. Harmondsworth: Penguin, 1988.

———. Mathematics, the Science of Patterns. New York: Scientific American Library, 1994.

I. Ekeland. The Broken Dice. Chicago: University of Chicago Press, 1993.

———. Mathematics and the Unexpected. Chicago: University of Chicago Press, 1988.

G. T. Gilbert, M. I. Krusemeyer, and L. C. Larson. The Wohascum County Problem Book. Dolciani Mathematical Expositions 14. Washington: Mathematical Association of America, 1993.

M. Golubitsky and M. J. Field. Symmetry in Chaos. Oxford: Oxford University Press, 1992.

M. Guillen. Bridges to Infinity. London: Rider, 1983.

R. Honsberger. Ingenuity in Mathematics. Washington: Mathematical Association of America, 1970.

———. Mathematical Gems I. Dolciani Mathematical Expositions 1. Washington: Mathematical Association of America, 1973.

———. Mathematical Gems II. Dolciani Mathematical Expositions 2. Washington: Mathematical Association of America, 1974.

———. Mathematical Gems III. Dolciani Mathematical Expositions 9. Washington: Mathematical Association of America, 1985.

K. Jacobs. Invitation to Mathematics. Princeton: Princeton University Press, 1992.

M. Kline. Mathematical Thought from Ancient to Modern Times. Oxford: Oxford University Press, 1972.

E. Maor. e: The Story of a Number. Princeton: Princeton University Press, 1994.

J. R. Newman (editor). The World of Mathematics 4 volumes. New York: Simon and Schuster, 1956.

I. Peterson. Islands of Truth. New York: Freeman, 1990.

———. The Mathematical Tourist. New York: Freeman, 1988.

C. Reid. Courant: In Goettingen and New York. New York: SpringerVerlag, 1976.

D. Ruelle. Chance and Chaos. Princeton: Princeton University Press, 1991.

M. Schroeder. Chaos, Fractals, Power Laws. New York: Freeman, 1991.

I. Stewart. Concepts of Modern Mathematics. New York: Dover, 1995.

———. Does God Play Dice?. Oxford: Blackwell, 1989.

———. From Here To Infinity. Oxford: Oxford University Press, 1996.

———. Nature’s Numbers. New York: Basic Books, 1995.

———. The Problems of Mathematics. Oxford: Oxford University Press, 1992.

I. Stewart and M. Golubitsky. Fearful Symmetry. Oxford: Blackwell, 1992.

M. Sved. Journey Into Geometries. Washington: Mathematical Association of America, 1991.

CHAPTER IX

§1. A Formula For Primes

M. Davis, Y. Matijasevic, and J. Robinson. “Hilbert’s Tenth Problem. Diophantine Equations: Positive Aspects of a Negative Solution.” In Proceedings of Symposia in Pure Mathematics 28: Mathematical Developments Arising from Hilbert Problems. Washington: American Mathematical Society, 1976, pp. 323-378.

M. Davis and R. Hersh. “Hilbert’s Tenth Problem.” Scientific American 229, no. 5 (1973): 84–91.

K. Devlin. Mathematics: The New Golden Age. Harmondsworth: Penguin, 1988.

J. P. Jones, D. Sato, H. Wada, and D. Wiens. “Diophantine Representations of the Set of Prime Numbers.” American Mathematical Monthly 83 (1976):449–464.

I. Stewart. Concepts of Modern Mathematics. New York: Dover, 1995.

§2. The Goldbach Conjecture and Twin Primes

K. Devlin. Mathematics: The New Golden Age. Harmondsworth: Penguin, 1988.

W. Yuan. Goldbach Conjecture. Singapore: World Scientific, 1984.

§3. Fermat’s Last Theorem

E. T. Bell. The Last Problem. Washington: Mathematical Association of America, 1990.

D. Cox. “Introduction to Fermat’s Last Theorem.” American Mathematical Monthly 101 (1994):3–14.

K. Devlin. Mathematics: The New Golden Age. Harmondsworth: Penguin, 1988.

I. Katz. “Fame by Numbers.” The Guardian Weekend, April 8 1995, 34–42.

P. Ribenboim. Thirteen Lectures on Fermat’s Last Theorem. New York: Springer-Verlag, 1979.

K. Rubin and A. Silverberg. “A Report on Wiles’ Cambridge Lectures.” Bulletin American Mathematical Society 31 (1994):15–38.

I. Stewart. “Fermat’s Last Time Trip.” Scientific American 269, no. 5 (1993):85–88.

———. From Here to Infinity. Oxford: Oxford University Press, 1996.

———. The Problems of Mathematics. Oxford: Oxford University Press, 1996.

§4. The Continuum Hypothesis

P. Bernays. Axiomatic Set Theory. New York: Dover, 1991.

P. J. Cohen and R. Hersh. “Non-Cantorian Set Theory.” In Mathematics in the Modern World, edited by M. Kline. San Francisco: Freeman, 1979.

K. Devlin. Mathematics: The New Golden Age. Harmondsworth: Penguin, 1988.

W. S. Hatcher. The Logical Foundations of Mathematics. Oxford: Pergamon Press, 1982.

S. Lavine. Understanding the Infinite. Cambridge: Harvard University Press, 1994.

I. Stewart. “A Subway Named Turing.” Scientific American 271, no. 3 (1994):90–92.

R. L. Vaught. Set Theory: An Introduction. Boston: Birkhäuser, 1985.

§5. Set-Theoretic Notation

I. Stewart. Concepts of Modern Mathematics. New York: Dover, 1995.

R. L. Vaught. Set Theory: An Introduction. Boston: Birkhäuser, 1985.

§6. The Four Color Theorem

K. Appel and W. Haken. “The Four-Color Problem.” In Mathematics Today, edited by L. A. Steen. New York: Springer, 1978.

———. “The Four-Color Proof Suffices.” The Mathematical Intelligencers 8, no. 1 (1986):10–20.

K. Devlin. Mathematics: The New Golden Age. Harmondsworth: Penguin, 1988.

G. Ringel. Map Color Theorem. New York: Springer, 1974.

T. L. Saaty. “Remarks on the Four Color Problem: The Kempe Catastrophe.” Mathematics Magazine 40 (1967):31–36.

I. Stewart. From Here to Infinity. Oxford: Oxford University Press, 1996.

———. The Problems of Mathematics. Oxford: Oxford University Press, 1992.

———. “The Rise and Fall of the Lunar M-pire.” Scientific American 268, no. 4 (1993):90–91.

§7. Hausdorff Dimension and Fractals

M. F. Barnsley. Fractals Everywhere. Boston: Academic Press, 1993.

B. B. Mandelbrot. The Fractal Geometry of Nature. New York: Freeman 1982.

H. O. Peitgen, H. Jürgens, and D. Saupe. Chaos and Fractals. New York: Springer-Verlag, 1992.

I. Stewart. From Here to Infinity. Oxford: Oxford University Press, 1996.

———. The Problems of Mathematics. Oxford: Oxford University Press, 1992.

§8. Knots

C. W. Ashley. The Ashley Book of Knots. London: Faber and Faber, 1947.

P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu. “A new Polynomial Invariant of Knots and Links.” Bulletin of the American Mathematical Society 12 (1985):239–246.

V. F. R. Jones. “A Polynomial Invariant for Knots via von Neumann Algebras.” Bulletin of the American Mathematical Society 12 (1985):103–111.

V. F. R. Jones. “Knot Theory and Statistical Mechanics.” Scientific American 263, no. 5 (1990):52–57.

W. B. R. Lickorish and K. C. Millett. “The New Polynomial Invariants of Knots and Links.” Mathematics Magazine 61 (1988):3–23.

C. Livingston. Knot Theory. Carus Mathematical Monographs 24. Washington: Mathematical Association of America, 1993.

I. Stewart. From Here to Infinity. Oxford: Oxford University Press, 1996.

———. “Knots, Links, and Videotape.” Scientific American 270, no. 1 (1994):136–138.

———. The Problems of Mathematics. Oxford: Oxford University Press, 1992.

§9. A Problem in Mechanics

T. Poston. “Au Courant with Differential Equations.” Manifold 18 (Spring 1976):6–9.

I. Stewart, Game, Set, and Math. Oxford: Blackwell, 1989.

§10. Steiner’s Problem

M. W. Bern and R. L. Graham. “The Shortest-Network Problem.” Scientific American 260, no. 1 (1989):66–71.

E. N. Gilbert and H. O. Pollak. “Steiner Minimal Trees.” SIAM Journal of Applied Mathematics 16 (1968):1–29.

Z. A. Melzak. Companion to Concrete Mathematics. New York: Wiley, 1973.

I. Stewart. “Trees, Telephones, and Tiles.” New Scientist 1795 (1991): 26–29.

P. Winter. “Steiner Problems in Networks: A Survey.” Networks 17 (1987):129–167.

§11. Soap Films and Minimal Surfaces

F. J. Almgren Jr. “Minimal Surface Forms.” The Mathematical Intelligencer 4 no. 4 (1982):164–171.

———. Plateau’s Problem, and Introduction to Varifold Geometry. New York: Benjamin, 1966.

F. J. Almgren Jr. and J. E. Taylor. “The Geometry of Soap Films and Soap Bubbles.” Scientific American 235 no. 1 (1976):82–93.

C. Isenberg. The Science of Soap Films and Soap Bubbles. New York: Dover Publications, 1992.

§12. Nonstandard Analysis

J. W. Dauben. Abraham Robinson: The Creation of Nonstandard Analysis. Princeton: Princeton University Press, 1995.

A. E. Hurd and P. A. Loeb. An Introduction to Nonstandard Real Analysis. New York: Academic Press, 1985.

M. J. Keisler. Foundations of Infinitesimal Calculus. New York: Prindle, Weber, and Schmidt, 1976.

A. Robinson. Introduction to Model Theory and to the Metamathematics of Algebra. Amsterdam: North-Holland, 1963.

K. D. Stroyan and W. A. U. Luxemburg. Introduction to the Theory of Infinitesimals. New York: Academic Press, 1976.