What Is Theorem?
Theorems is what mathematics consists of, isn't it? We may learn the etymology of the word from The Words of Mathematics by S. Schwartzman:
theorem (noun), theory (noun), theoretical (adjective): from Greek theorema, from theorein "to look at," of unknown origin. A related borrowing is Theater, since you go to a theater to look at a play. A theorem was originally a sight or the act of seeing. Something that is looked at for any moment of time becomes an object of study. In mathematics, after studying a situation or a class of objects, a person hopes to make speculations and then prove them, so theorem came to mean the proof of a speculation that has been arrived at by looking at something. |
A theorem is a proven mathematical statement, although, as an exception, some statements (notably Fermat's Last Theorem, or FLT) have been traditionally called theorems even before their proofs have been found. To prove a statement means to derive it from axioms and other theorems by means of logic rules, like modus ponens. A proof is needed to establish a mathematical statement. A single counterexample suffices to refute such a statement. (Say, in a right triangle with hypotenuse c and legs a and b, the inradius
There is certainly an ambiguity mathematicians live with. Some rules (like the Law of Excluded Middle) and some axioms (like the Axiom of Choice) are not universally accepted by all mathematicians.
Commonly, auxiliary theorems of a lesser significance are called lemmas. If there is a need to emphasize an importance of a theorem in proving another theorem, the latter is called a corollary from the former, especially when the proof at hand is short.
A theorem is a theorem in a certain theory which is a collection of whatever is derivable from a selection of axioms. Whether a particular statement is a theorem or not depends on such a selection of axioms. What in one theory is an axiom in another may be a theorem, and vice versa.
References
- E. J. Borowski & J. M. Borwein, The Harper Collins Dictionary of Mathematics, Harper Perennial, 1991
- J. Daintith, R. D. Nelson (eds), The Penguin Dictionary of Mathematics, Penguin Books, 1989
- J. Mason, Thinking Mathematically, Addison-Wesley, 1985
- G. Polya, How To Solve It, Princeton University Press, 2nd ed, 1957
- S. Schwartzman, The Words of Mathematics, MAA, 1994
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