CONCEPTIONS AND METHODS OF MATHEMATICS 467 



It is convenient to have a term to designate a class of objects 

 associated with a class of relations between these objects. Such an 

 aggregate we will speak of as a mathematical system. If now we have 

 two different mathematical systems, and if a one-to-one correspond- 

 ence can be set up between the two classes of objects, and also 

 between the two classes of relations in such a way that whenever 

 a certain ordered set of objects of the first system satisfies a relation 

 of that system, the set consisting of the corresponding objects of the 

 second system satisfies the corresponding relation of that system, 

 and vice versa, then it is clear that the two systems are, from our 

 present point of view, mathematically equivalent, however different 

 the nature of the objects and relations may be in the two cases. 1 To 

 use a technical term, the two systems are simply isomorphic. 2 



It will be noticed that in the definition of mathematics just given 

 nothing is said as to the method by which we are to ascertain whether 

 or not a given relation holds between the objects of a given set. The 

 method used may be a purely empirical one, or it may be partly or 

 wholly deductive. Thus, to take a very simple case, suppose our class 

 of objects to consist of a large number of points in a plane and sup- 

 pose the only relation between them with which we are concerned 

 is that of collinearity. Then, if the points are given us by being 

 marked in ink on a piece of white paper, we can begin by taking three 

 pins, sticking them into the paper at three of the points; then, by 

 sighting along them, we can determine whether or not these points 

 are collinear. We can do the same with other groups of three 

 points, then with all groups of four points, etc. The same result 

 can be obtained with much less labor if we make use of certain 

 simple properties which the relation of collinearity satisfies, pro- 

 perties which are expressed by such propositions as: 



R(a, b, c) implies R(b, a, c), 



R(a, b, c, d) implies R(a, b, c), 



R(a, b, c) and R(a, b, d) together imply R(a, b, c, d), etc. 



By means of a small number of propositions of this sort it is easy 

 to show that no empirical observations as to the collinearity of 

 groups of more than three points need be made, and that it may 

 not be necessary to examine even all groups of three points. Having 



1 The point of view here brought out, including the term isomorphism, was 

 first developed in a special case, the theory of groups. 



2 Inasmuch as the relations in a mathematical system are themselves objects, 

 we may, if we choose, take our class of objects so as to include these relations as 

 well as what we called objects before, some of which, we may remark in passing, 

 may themselves be relations. Looked at from this point of view, we need one 

 additional relation which is now the only one which we explicitly call a relation. 

 If we denote this relation by inclosing the objects which satisfy it in parentheses, 

 then if the relation denoted before by R(a, b} is satisfied, we should now write 

 (R, a, 6), whereas we should not have (a, R, b) (S, R, a, b}, etc. Thus we see that 

 any mathematical system may be regarded as consisting of a class of objects and 

 a single relation between them. 



