i2 4 SCIENTIFIC METHOD IN PHILOSOPHY 



with its object as a part of the physical world. Never- 

 theless, both these authors, and especially Mach, deserve 

 mention as having made serious contributions to the con- 

 sideration of our problem. 



When a point or an instant is defined as a class of 

 sensible qualities, the first impression produced is likely 

 to be one of wild and wilful paradox. Certain considera- 

 tions apply here, however, which will again be relevant 

 when we come to the definition of numbers. There is 

 a whole type of problems which can be solved by such 

 definitions, and almost always there will be at first an 

 effect of paradox. Given a set of objects any two of 

 which have a relation of the sort called " symmetrical and 

 transitive," it is almost certain that we shall come to 

 regard them as all having some common quality, or as 

 all having the same relation to some one object outside 

 the set. This kind of case is important, and I shall 

 therefore try to make it clear even at the cost of some 

 repetition of previous definitions. 



A relation is said to be " symmetrical ' when, if one 

 term has this relation to another, then the other also has 

 it to the one. Thus " brother or sister " is a " symmet- 

 rical ' relation : if one person is a brother or a sister of 

 another, then the other is a brother or sister of the one. 

 Simultaneity, again, is a symmetrical relation ; so is 

 equality in size. A relation is said to be " transitive " 

 when, if one term has this relation to another, and the 

 other to a third, then the one has it to the third. The 

 symmetrical relations mentioned just now are also tran- 

 sitive provided, in the case of " brother or sister," we 

 allow a person to be counted as his or her own brother 

 or sister, and provided, in the case of simultaneity, we 

 mean complete simultaneity, i.e. beginning and ending 

 together. 



