What are theorems, corollaries,and lemmas?
In mathematics and logic a theoremis a statement demonstrated to be true by accept-
ed mathematical operations and arguments. In general, a theorem is usually based on
some general principle that makes it part of a larger theory; it differs from an axiom in
that a proof is required for its acceptance. Some of the more well-known theorems are
named after their discoverers, such as the Pythagorean Theorem (involving right trian-
gles) and Fermat’s Last Theorem. It is interesting to note that American Richard Feyn-
man (1918–1988), one of the most brilliant physicists of the 20th century, stated that
any theorem, no matter how difficult to prove in the first place, is viewed as “trivial” by
mathematicians once it has been proved. Thus, according to Feynman, there are only
two types of mathematical objects: trivial ones and those that have not yet been proved.
A corollaryis a theorem that has been proved in only a few steps from an estab-
lished theorem, or one that follows as a direct consequence of another theorem or
axiom. Finally, a lemmais a theorem proved as a preliminary or intermediate step in
the proof of another, more basic theorem; or, it is a brief theorem used to prove a larg-
er theorem.
What are existence theorems?
An existence theorem is one that has a statement beginning with “There exist(s) ...,”
or, more generally, “for all x, y,... there exist(s)....” Existence theorems are presented
in several ways, including showing the exact formulas for the solution, describing in
their proofs’ iteration processes how to approach the problem, and by simply deducing
the solutions without showing any methods as to how it was determined.
Many mathematicians do not believe in existence theorems, stating that any theo-
rems in which entities cannot be constructed are worthless. Other mathematicians
cite the existence of such theorems—but prefer to use tried-and-true theorems that
offer specific proven methods.
What is a proof?
A proof is simply the process of showing a theorem to be correct, although the process
itself might not be simple. These mathematical arguments are often quite rigorous,
and they are used to demonstrate the truth of a given proposition. The result of the
proved statement is a theorem.
Interestingly enough, there are several computer systems now being developed to
automate proofs. But some mathematicians (mostly purists) do not believe these com-
puter-assisted proofs are valid; they believe that only humans can understand the
nuances and have the intuition needed to develop a theorem’s proof. One good exam-
ple is called the four-color theorem: Its proof relies on meticulous computer testing of
many separate cases, all of which can’t be verified by hand. 117