Page:Popular Science Monthly Volume 68.djvu/31

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We are now in a position to develop considerably more theory. By 6 and 4 and 1 there must be at least two lines which by 3 meet in a point A. Hence there must be four points at least, (B, C, D, E) which do not lie in the same line. For if D were in the line BC, by 2, the lines AB and AD would be the same, which is contrary to hypothesis.

A set of four points, such as A, B, C, D, of which no three are collinear, when taken together with the lines (called the sides) joining the six pairs of points, AB, BC, CD, DA, AC, BD, is called a complete quadrangle. In the diagram below, the vertices of a complete quadrangle are 0, 1, 4, 6. The three additional points 2, 3, 5, in which the sides of the quadrangle intersect, are called the diagonal points.

We have shown our axioms sufficient to establish the existence of a complete quadrangle; are they sufficient to prove the ordinary properties of such a figure? They are not. Axioms 1-6 do not decide whether the three diagonal points, 2, 3, 5, are or are not collinear. In the ordinary geometry, those points are non-collinear and form what is called the diagonal triangle. If, however, we suppose that they are collinear (one may assist one's imagination by means of the dotted line) then on rereading our six postulates they will all be found verified. In order to obtain the usual geometry it is necessary to assume as an axiom that the diagonal points of a complete quadrangle are non-collinear. What we have just done is a simple case of an 'independence proof.' We have proved that the proposition that the diagonal points of a complete quadrangle are not collinear, is independent of propositions 1,. . . 6, that is, it is not a logical consequence of them. Similarly, the non-Euclidean geometry is an independence proof for Euclid's axiom 12. The ideal of students of foundations of geometry is a system of axioms every one of which is independent of all the rest. To attain this ideal it is necessary to construct for each axiom an example in which it is untrue while all the rest are verified.

After seeing the bizarre construction that this process gives rise to, one is tempted to raise the question, how can we be sure that the complete system which we use applies uniquely to the space of our intuition or experience and not also to one of these mathematical dreams? In answering this question we define what is meant by a categorical system of axioms.

Returning to our complete quadrangle with collinear diagonal points