Theorem | Research & Encyclopedia Articles

Theorem

A theorem (the term is derived from the Greek theoreo, which means I look at) denotes either a proposition yet to be proven, or a proposition proven correct on the basis of accepted results from some area of mathematics. Since the time of the ancient Greeks, proven theorems have represented the foundation of mathematics. Perhaps the most famous of all theorems is the Pythagorean theorem.

Mathematicians develop new theorems by suggesting a proposition based on experience and observation which seems to be true. These original statements are only given the status of a theorem when they are proven correct by logical deduction. Consequently, many propositions exist which are believed to be correct, but are not theorems because they can not be proven using deductive reasoning alone.

Historical background

The concept of a theorem was first used by the ancient Greeks. To derive new theorems, Greek mathematicians used logical deduction from premises they believed to be self-evident truths. Since theorems were a direct result of deductive reasoning, which yields unquestionably true conclusions, they believed their theorems were undoubtedly true. The early mathematician and philosopher Thales (640-546 b.c.) suggested many early theorems, and is typically credited with beginning the tradition of a rigorous, logical proof before the general acceptance of a theorem. The first major collection of mathematical theorems was developed by Euclid around 300 b.c. in a book called The Elements.

The absolute truth of theorems was readily accepted up until the eighteenth century. At this time mathematicians, such as Karl Friedrich Gauss (1777-1855), began to realize that all of the theorems suggested by Euclid could be derived by using a set of different premises, and that a consistent non-Euclidean structure of theorems could be derived from Euclidean premises. It then became obvious that the starting premises used to develop theorems were not self-evident truths. They were in fact, conclusions based on experience and observation, and not necessarily true. In light of this evidence, theorems are no longer thought of as absolutely true. They are only described as correct or incorrect based on the initial assumptions.

Characteristics of a theorem

The initial premises on which all theorems are based are called axioms. An axiom, or postulate, is a basic fact which is not subject to formal proof. For example, the statement that there is an infinite number of even integers is a simple axiom. Another is that two points can be joined to form a line. When developing a theorem, mathematicians choose axioms which seem most reliable based on their experience. In this way, they can be certain that the theorems which are proved are as near to the truth as possible. However, absolute truth is not possible because axioms are not absolutely true.

To develop theorems, mathematicians also use definitions. Definitions state the meaning of lengthy concepts in a single word or phrase. In this way, when we talk about a figure made by the set of all points which are a certain distance from a central point, we can just use the word circle.