268 SPECULATIVE SCIENCE 



impossible, except as regards a small and constantly lessening 

 fringe or margin of the whole part. 



III. 



The mathematical relations between numbers and magnitudes 

 require separate consideration. 



In geometry they involve the comparison of dimensions, and 

 dimensions are, in the sense explained above, things. But where 

 mathematics deal with the relations between symbols, whether 

 algebraic or numerical, they do not compare things. They 

 essentially substitute a symbol for the thing itself, and use the 

 symbols to facilitate not measurement but ratiocination con- 

 cerning measurements. Number is not an object, a condition, 

 a property, an attribute, or even an abstraction. Numbers ex- 

 press the result of measurement; they cannot themselves be 

 measured. The relation between them is stated when they are 

 stated. The very conception of number involves the assumption 

 that two things can be identical, or may for a special purpose 

 be regarded as identical. This is neither the result of abstrac- 

 tion nor reasoning ; it is merely a short way of saying that 

 any four things which we choose to regard as identical, may be 

 counted in two ways, described as regards number in tw6 

 different phrases . 



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