138 WHAT IS SCIENCE? 



between weight and volume is a relation between numerals, 

 the numerals that are used to represent those properties. 

 The physical relation in the second case is not between 

 numbers at all, but between weight and volume which 

 are properties quite different from numbers ; it appears 

 very similar to that between numbers only because we 

 use numerals, originally invented to represent numbers, 

 to represent other properties. The relation stated by a 

 numerical law is a relation between numerals, and only 

 between numerals, though the idea that there may be 

 such a relation has been suggested to us by the study of 

 the physical property, number. 



If we understand this, we shall see what a very remark- 

 able thing it is that there should be numerical laws at all, 

 and shall see why the idea of such a law arose compara- 

 tively late in the history of science. For even when we 

 know the relations between numbers, there is no reason 

 to believe that there must be any relations of the same 

 kind between the numerals which are used to represent, 

 not only numbers, but also other properties. Until we 

 actually tried, there was no reason to think that it must 

 be possible to find at all numerical laws, stating numerical 

 relations such as those of division and multiplication. 

 The fact that there are such relations is a new fact, and 

 ought to be surprising. As has been said so often, it 

 does frequently turn out that suggestions made simply 

 by our habits of mind are actually true ; and it is because 

 they are so often true that science is interesting. But 

 every time they are true there is reason for wonder and 

 astonishment. 



And there is a further consequence yet more deserving 

 of our attention at present If we realize that the 

 numerical relations in numerical laws, though suggested 

 by relations between numbers, are not those relations, 

 we shall be prepared to find also numerical relations which 

 are not even suggested by relations between numbers, 



