28 POPULAR SCIENCE MONTHLY 



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 • • • 6 and ' line ' is a column, then the six axioms incompletely 

 describe the triadic system. They describe it completely if we add: 



7. There are not more than three points on a line. 



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 proposi- 

 tion 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. 



