CONCEPTIONS AND METHODS OF MATHEMATICS 465 



just cited, a given point may or may not lie on a given line; a given 

 line may or may not touch a given circle; three or more points may 

 or may not be collinear, etc. This example shows how in a single 

 mathematical problem a large number of relations may be involved, 

 relations some of which connect two objects, others three, etc. 

 Moreover these relations may connect like or they may connect 

 unlike objects; and finally the order in which the objects are taken 

 is not by any means immaterial in general, as is shown by the relation 

 between three points which states that the third is collinear with and 

 lies between the first two. 



But even this is not all; for, besides these objects and relations 

 of various kinds, we often have operations by which objects can be 

 combined to yield another object, as, for instance, addition or multi- 

 plication of numbers. Here the objects combined and the resulting 

 object are all of the same kind, but this is by no means necessary. 

 We may, for instance, consider the operation of combining two 

 points and getting the perpendicular bisector of the line connecting 

 them; or we may combine a point and a line and get the perpen- 

 dicular dropped from the point on the line. 



These few examples show how diverse the relations and operations, 

 as well as the objects of mathematics, seem at first sight to be. Out 

 of this apparent diversity it is not difficult to obtain a very great 

 uniformity by simply restating the facts in a little different language. 

 We shall find it convenient to indicate that the objects a, b, c, . . . , 

 taken in the order named, satisfy a relation R by simply writing 

 R(a. b, c, . . . ), where it should be understood that among the 

 objects a, b. c, . . . the same object may occur a number of times. 

 On the other hand, if two objects a and b are combined to yield 

 a third object c, we may write a o b = c, 1 where the symbol o is 

 characteristic of the special operation with which we are concerned. 



Let us first notice that the equation aob=c denotes merely 

 that the three objects a, b, c bear a certain relation to one another, 

 say R(a, b, c). In other words the idea of an operation or law of 

 combination between the objects we deal with, however convenient 

 and useful it may be as a matter of notation, is essentially merely 

 a way of expressing the fact that the objects combined bear a certain 

 relation to the object resulting from their combination. Accordingly, 

 in a purely abstract discussion like the present, where questions of 

 practical convenience are not involved, we need not consider such 

 rules of combination. 2 



1 I speak here merely of dyadic operations, i. e., of operations by which 

 two objects are combined to yield a third, these being by far the most import- 

 ant as well as the simplest. What is said, however, obviously applies to opera- 

 tions by which any number of objects are combined. 



2 Even from the point of view of the technical mathematician it may some- 

 times be desirable to adopt the point of view of a relation rather than that of an 

 operation. This is seen, for instance, in laying down a system of postulates for the 



