Page:Popular Science Monthly Volume 68.djvu/32

 28 and observing the numbers placed at its vertices, we may arrange in the same column the numbers of the points that appear together in the same line.

The array thus obtained is known as a 'triadic system' in the seven digits 0, 1, ··· 6.

If the undefined element, 'point' of our axioms is any one of the digits 0 ··· 6 and 'line' is a column, then the six axioms incompletely describe the triadic system. They describe it completely if we add:

I say they describe it completely because we have proved that the axioms are satisfied by seven points arranged as in the triadic system, while from 7 it follows that no other arrangement or number of points is in harmony with the axioms. There is only one kind of thing which satisfies all the axioms 1 ··· 7. In other terms, any two systems of objects (for example, the points of the diagram on p. 28, and the triadic system on p. 29) that satisfy axioms 1 ··· 8 are reciprocally in a one-to-one correspondence which preserves all the relations of the kind specified in the axioms.

This is what is meant by a categorical system of axioms. Thus in geometry, a categorical system is capable of distinguishing Euclidean space from all essentially different constructions of the mind—and this in spite of the fact that the fundamental elements of geometry are never defined in the ordinary sense of the term definition.

If we have before us a categorical system of axioms, every proposition which can be stated in terms of our fundamental (undefined) symbols either is or is not true of the system of objects satisfying the axioms. In this sense it either is a consequence of the axioms or is in contradiction with them. But if it is a consequence of the axioms, can it be derived from them by a syllogistic process? Perhaps not.