Selections from R.L. Wilder, Introduction to the Foundations of Mathematics
I. The Axiomatic Method
Since the axiomatic method as it is now understood and practiced by mathematicians is the result of a long evolution in human thought, we shall precede our discussion of it by a brief description of some older uses of the term axiom. The modern usage of the term represents a high degree of maturity, and a better understanding of it may be achieved by some acquaintance with the course of its evolution.
1. Evolution of the Method
If the reader has at hand a copy of an elementary plane geometry, of a type frequently used in high schools, he may find two groupings of fundamental assumptions, one entitled “Axioms,” the other entitled “Postulates.” The intent of this grouping may be explained by such accompanying remarks as: “An axiom is a self-evident truth.” “A postulate is a geometrical fact so simple and obvious that its validity may be assumed.” The “axioms” themselves may contain such statements as: “The whole is greater than any of its parts.” “The whole is the sum of its parts.” “Things equal to the same thing are equal to one another.” “Equals added to equals yield equals.” It will be noted that such geometric terms as “point” or “line” do not occur in these statements; in some sense the axioms are intended to transcend geometry — to be “universal truths.” In contrast, the “postulates” probably contain such statements as: “Through two distinct points one and only one straight line can be drawn.” “A line can be extended indefinitely.” “If L is a line and P is a point not on L, then through P there can be drawn one and only one line parallel to L.” (Some so-called “definitions” of terms usually precede these statements.)
This grouping into “axioms” and “postulates” has its roots in antiquity. Thus we find in Aristotle (384–321 B.C.) the following viewpoint: †
“Every demonstrative science must start from indemonstrable principles; otherwise, the steps of demonstration would be endless. Of these indemonstrable principles some are (a) common to all sciences, others are (b) particular, or peculiar to the particular science; (a) the common principles are the axioms, most commonly illustrated by the axiom that, if equals be subtracted from equals, the remainders are equal. In (b) we have first the genus or subject-matter, the existence of which must be assumed.”
† As summarized by T.L. Heath, [The Thirteen Books of Euclid's Elements, I, 119, Cambridge University Press, Cambridge, UK, 1908]. The reader is referred to this book for citations from Aristotle, Proclus, et al.
Wilder, Raymond L., Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY, 1952.