QUANTITATIVE ESTIMATES OF SENSORY EVENTS 297 



arithmetic pertain to numbers and to nothing else : there is nothing inherently 

 numerical in the structure of the phenomenal world. We are, however, so 

 familiar with the description of phenomena in numerical terms (or their 

 formal mathematical equivalents) that the association has become instinctive, 

 and we are apt to imagine that we directly perceive the metrical aspects of 

 nature as inherent constituents of phenomena, existing in their own right, 

 so to speak, and merely observed by us. This induces us to overlook the 

 essentially arbitrary and man-made nature of the association. When this 

 is of an unfamiliar cha racte r, as, for instance, when we associate the arith- 

 metical operation X \/ — i with the physical events constituting a phase 

 change of n/z in alternating current problems and others of a like nature, 

 we recognise it at once as a mere device of the mathematician. In principle 

 it is no more artificial than the more familiar associations of events with 

 arithmetical concepts which underlie all metrical processes. 



The phenomenal world presents itself to us as a complex relation structure. 

 We need not here enter into metaphysical questions concerning the parts 

 played by our sense organs and by things external to us in determining the 

 kind of relations exhibited by phenomena. We will take the phenomenal 

 world as we find it : a structure of related events, which we find it con- 

 venient to describe and classify in terms of various concepts. 



We have discovered — and this discovery is the foundation stone of 

 physical science — that by employing a simple but ingenious device some 

 aspects of phenomena can be classified so that certain phenomenal relations 

 existing between members of any such class are ' similar ' to the relations 

 between members of the class of numbers on which the laws of arithmetic 

 are based. In arithmetic, these relations are implicit in the meaning 

 assigned by two important symbols, namely =, the symbol of numerical 

 equality, and +, the symbol of the operation of adding one number to 

 another. All other arithmetical operations, subtraction, multiplication or 

 division, involution or evolution, etc., ultimately derive their significance 

 from the operation of addition. 



In any class of phenomenal aspects of the kind we are considering we can 

 perceive various relations. Further, by performing experimental operations 

 on the things which exhibit the aspects in question we can change the actual 

 relations exhibited. But there is nothing inherently numerical in these 

 phenomenal relations : in order to establish a connection we must arbitrarily 

 associate some unique symmetrical transitive phenomenal relation from 

 among those which may have perceptual significance with the arithmetical 

 relation of equality ; and, further, we must associate some suitable experi- 

 mental operation with the arithmetical operation of addition. If, now, we 

 base our phenomenal classification entirely in terms of this symmetrical 

 relation and this experimental operation we obtain a phenomenal series 

 whose members are related to each other in a similar manner to the members 

 of the series of numbers. We must not confuse ' similarity ' as here used 

 with identity. Relations are themselves things which can be classified in 

 virtue of certain characteristics irrespective of the kind of things related by 

 them. A relation may be symmetrical or unsymmetrical, transitive or 

 intransitive, etc., and it is these properties of relations themselves, and not 

 any specific properties of the things related by them, which confer relational 

 similarity or dissimilarity on classes defined by relations. However, it is 

 not here necessary to discuss the theory of similar relations, or go into the 

 conditions which must be imposed on our selected criteria of ' equality ' 

 and ' addition ' in order that phenomenal and numerical relations may be 

 similar. The important point to be noted is simply that there is no a priori 

 connection between phenomenal structure and number, and that to make 



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