Postulates, Theorems, and Proofs

Postulates and theorems are the building blocks for proof and deduction in any mathematical system, such as geometry, algebra, or trigonometry. By using postulates to prove theorems, which can then prove further theorems, mathematicians have built entire systems of mathematics.

Postulates, or axioms , are the most basic assumptions with which a reasonable person would agree. An example of an axiom is "parallel lines do not intersect." Postulates must be consistent, meaning that one may not contradict another. They are also independent, meaning not one of them can be proved by some combination of the others. There may also be a few undefined terms and definitions.

Postulates or axioms can then be used to prove propositions or statements, known as theorems. In doing so, mathematicians must strictly follow agreed-upon rules of argument known as the "logic" of the system. A theorem is not considered true unless it has been rigorously proved by valid arguments that have strictly followed this logic.

Deductive reasoning is a method by which mathematicians prove a theorem within the pre-defined system. Deduction begins by using some combination of the undefined terms, definitions, and postulates to prove a first theorem. Once that theorem has been proved by a valid argument, it may then be used to prove other theorems that follow it in the logical development of the system.

Euclid's Deductions

Perhaps the oldest and most famous deductive system, as well as a paradigm for later deductive systems, is found in a work called the Elements by the ancient Greek mathematician Euclid (c. 300 b.c.e.). The Elements is a massive thirteen-volume work that uses deduction to summarize most of the mathematics known in Euclid's time.

Euclid stated five postulates, equivalent to the following, from which to prove theorems that, in turn, proved other theorems. He thereby built his well-known system of geometry:

1. It is possible to draw a straight line from any point to any point.
2. It is possible to extend a finite straight line continuously in a straight line.
3. It is possible to draw a circle with any center and distance (radius).
4. All right angles are equal to one another.
5. Given a line and a point not on the line, there is exactly one line parallel to the given line.

Starting with these five postulates and some "common assumptions," Euclid proceeded rigorously to prove more than 450 propositions (theorems), including some of the most important theorems in mathematics. The Elements is one of the most influential treatises on mathematics ever written because of its unrelenting reliance on deductive proof. Its "postulate-theorem-proof" paradigm has reappeared in the works of some of the greatest mathematicians of all time.

Changing Postulates

What are considered "self-evident truths" may change from one generation to another. Until the nineteenth century, it was believed that the postulates of Euclidean geometry reflected reality as it existed in the physical world. However, by replacing Euclid's fifth postulate with another postulate—"Given a line and a point not on the line, there are at least two lines parallel to the given line"—the Russian mathematician Nikolai Ivanovich Lobachevski (1793–1856) produced a completely consistent geometry that models the space of Albert Einstein's theory of relativity. Thus the modern pure mathematician does not regard postulates as "true" or "false" but is only concerned with whether they are consistent and independent.

see also Consistency; Euclid and His Contributions; Proof.

Stephen Robinson

Bibliography

Moise, Edwin. Elementary Geometry from an Advanced Standpoint. Reading, MA: Addison-Wesley, 1963.

Narins, Brigham, ed. World of Mathematics. Detroit: Gale Group, 2001.