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, TH
OREM
.
A page of proof-related humor is maintained by Chalmers.
SEE ALSO: Conjecture,
Constructive Proof,
Deep Theorem,
Dissection
Proof,
Existence Problem,
Existence
Theorem,
Experimental Mathematics,
Hypothesis,
Lemma,
Nonconstructive
Proof,
Paradox,
Proof
without Words,
Proposition,
Q.E.D.,
Reductio ad Absurdum,
Rigorous,
Theorem,
Trivial,
Two-Column
Proof
REFERENCES:
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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.
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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.
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Princeton University Press, 1988.
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New York: Wiley, 1981.
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NJ: Princeton University Press, 1990.
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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.
Referenced on Wolfram|Alpha:
Proof
CITE THIS AS:
Weisstein, Eric W. "Proof." From MathWorld--A
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Are (2,-1) and (4,2) linearly independent?