Theory of Parallels — Preliminary Theorems (1–15)

Summary

Mathematical terms cannot be defined ex nihilo. The words that one uses in any given definition require further definitions of their own; these secondary definitions necessitate tertiary definitions; these in turn require still others. To escape infinite regress, geometers must leave a handful of socalled primitive terms undefined. These primitive terms represent the basic building blocks from which the first defined terms may be constructed. From there, one may build upward indefinitely; all subsequent development will be grounded upon the primitive terms, and circular definitions will be avoided.

Only in the late 19th-century was such clarity achieved in the foundations of geometry. Euclid never identifies his primitive terms and several of his early definitions founder in ambiguity. His vague definition of a straight line, “a line which lies evenly with the points on itself” is useless from a logical standpoint: since Euclid does not tell us what “lying evenly” means, we have no way of deciding whether a given curve is straight or not. Euclid has given us a description rather than a genuine definition of a line, and as such, he has given us something that is worthless in a strict logical development of geometry.

Mathematics encompasses more than logic, however. The very fact that Euclid attempts to describe a line has philosophical significance. It suggests that, for Euclid, straight lines are “out there”, capable of description. It implicitly asserts that straight lines exist independently of the mathematicians who study them. For one who accepts this Platonic concept of geometry, the logical gaps in The Elements are so superficial as to scarcely merit mention.