Proof

A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition. A mathematical statement that has been proven is called a theorem.

According to Hardy (1999, pp. 15-16), "all physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.... [This opinion], with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a mathematician ought to have some reply."

To prove Hardy's assertion, Feynman is reported to have commented, "A great deal more is known than has been proved" (Derbyshire 2004, p. 291).

There is some debate among mathematicians as to just what constitutes a proof. The four-color theorem is an example of this debate, since its "proof" relies on an exhaustive computer testing of many individual cases which cannot be verified "by hand." While many mathematicians regard computer-assisted proofs as valid, some purists do not. There are several computer systems currently under development for automated theorem proving, among them, THOREM.

A page of proof-related humor is maintained by Chalmers.

REFERENCES

Aigner, M. and Ziegler, G. M. Proofs from the Book. New York: Springer-Verlag, 1999.

Allenby, R. Numbers and Proofs. Oxford, England: Oxford University Press, 1997.

Benson, D. C. The Moment of Proof: Mathematical Epiphanies. Oxford, England: Oxford University Press, 1999.

Chalmers, D. "Philosophical Humor." http://consc.net/phil-humor.html.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Epstein, D. and Levy, S. "Experimentation and Proof in Mathematics." Not. Amer. Math. Soc. 42, 670-674, 1995.

Garnier, R. and Taylor, J. 100% Mathematical Proof. New York: Wiley, 1996.

Greenwald, S. J. "Simpsons Math." http://www.cs.appstate.edu/~sjg/simpsonsmath/blackboard.html.Hardy, G. H. "Mathematical Proof." Mind 38, 1-25, 1929.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Laczkovich, M. Conjecture and Proof. Washington, DC: Math. Assoc. Amer., 2001.

The Omega Group. "The OMEGA System." http://www.ags.uni-sb.de/~omega/soft/omega/.Pólya, G. How to Solve It: A New Aspect of Mathematical Method, 2nd ed. Princeton, NJ: Princeton University Press, 1988.

Pólya, G. Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, 2 vols. in One. New York: Wiley, 1981.

Pólya, G. Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics. Princeton, NJ: Princeton University Press, 1990.

Pólya, G. Mathematics and Plausible Reasoning, Vol. 2: Patterns of Plausible Inference. Princeton, NJ: Princeton University Press, 1990.

Krantz, S. G. Techniques of Problem Solving. Providence, RI: Amer. Math. Soc., 1997.

Research Institute for Symbolic Computation. "Overview of Theorema." http://www.risc.uni-linz.ac.at/research/theorema/description/.Solow, D. How to Read and Do Proofs: An Introduction to Mathematical Thought Process, 2nd ed. New York: Wiley, 1990.

Vakil, R. A Mathematical Mosaic: Patterns and Problem Solving. Washington, DC: Math. Assoc. Amer., 1997.

Wickelgren, W. A. How to Solve Mathematical Problems: Elements of a Theory of Problems and Problem Solving. New York: Dover, 1995.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1157, 2002.