Postulate

A postulate is an assumption, that is, a proposition or statement, that is assumed to be true without any proof. Postulates are the fundamental propositions used to prove other statements known as theorems. Once a theorem has been proven it is may be used in the proof of other theorems. In this way, an entire branch of mathematics can be built up from a few postulates. Postulate is synonymous with axiom, though sometimes axiom is taken to mean an assumption that applies to all branches of mathematics, in which case a postulate is taken to be an assumption specific to a given theory or branch of mathematics. Euclidean geometry provides a classic example. Euclid based his geometry on five postulates and five "common notions," of which the postulates are assumptions specific to geometry, and the "common no tions" are completely general axioms.

The five postulates of Euclid that pertain to geometry are specific assumptions about lines, angles, and other geometric concepts. They are:

1. Any two points describe a line.
2. A line is infinitely long.
3. A circle is uniquely defined by its center and a point on its circumference.
4. Right angles are all equal.
5. Given a point and a line not containing the point, there is one and only one parallel to the line through the point.

The five "common notions" of Euclid have application in every branch of mathematics, they are:

1. Two things that are equal to a third are equal to each other.
2. Equal things having equal things added to them remain equal.
3. Equal things having equal things subtracted from them have equal remainders.
4. Any two things that can be shown to coincide with each other are equal.
5. The whole is greater than any part.

On the basis of these ten assumptions, Euclid produced the Elements, a 13 volume treatise on geometry (published c. 300 B.C.) containing some 400 theorems, now referred to collectively as Euclidean geometry.

When developing a mathematical system through logical deductive reasoning any number of postulates may be assumed. Sometimes in the course of proving theorems based on these postulates a theorem turns out to be the equivalent of one of the postulates. Thus, mathematicians usually seek the minimum number of postulates on which to base their reasoning. It is interesting to note that, for centuries following publication of the Elements, mathematicians believed that Euclid's fifth postulate, sometimes called the parallel postulate, could logically be deduced from the first four. Not until the nineteenth century did mathematicians recognize that the five postulates did indeed result in a logically consistent geometry, and that replacement of the fifth postulate with different assumptions led to other consistent geometries.

Postulates figure prominently in the work of the Italian mathematician Guiseppe Peano (1858-1932), formalized the language of arithmetic by choosing three basic concepts: zero; number (meaning the non-negative integers); and the relationship "is the successor of." In addition, Peano assumed that the three concepts obeyed the five following axioms or postulates:

1. Zero is a number.
2. If b is a number, the successor of b is a number.
3. Zero is not the successor of a number.
4. Two numbers of which the successors are equal are themselves equal.
5. If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

Based on these five postulates, Peano was able to derive the fundamental laws of arithmetic. Known as the Peano axioms, these five postulates provided not only a formal foundation for arithmetic but for many of the constructions upon which algebra depends.

Indeed, during the nineteenth century, virtually every branch of mathematics was reduced to a set of postulates and resynthesized in logical deductive fashion. The result was to change the way mathematics is viewed. Prior to the nineteenth century mathematics had been seen solely as a means of describing the physical universe. By the end of the century, however, mathematics came to be viewed more as a means of deriving the logical consequences of a collections of axioms.

In the twentieth century, a number of important discoveries in the fields of mathematics and logic showed the limitation of proof from postulates, thereby invalidating Peano's axioms. The best known of these is Gödel's theorem, formulated in the 1930s by the Austrian mathematician Kurt Gödel (1906-1978). Gödel demonstrated that if a system contained Peano's postulates, or an equivalent, the system was either inconsistent (a statement and its opposite could be proved) or incomplete (there are true statements that cannot be derived from the postulates).