72 MENTAL EVOLUTION IN MAN. 



Take, for example, another t)'pe of abstract ideation, 

 and one which not only serves better than most to show 

 the importance of signs as substitutes for ideas, but also 

 best illustrates the extraordinary results to which such 

 symbolism may lead when carried out persistently. I refer 

 to mathematics. Of course, before the idea of number or of 

 relation can arise at all, the faculty of conception must have 

 made great advances ; but let us take this faculty at the 

 point where the artifice of substituting signs for ideas has 

 gone as far as to enable a mind to count by means of simple 

 notation. It would clearly be impossible to conduct the 

 least intricate trains of reasoning which invoke any ideas of 

 number or proportion, were we deprived of the power of 

 attaching particular signs to particular ideas of number. 

 We could not even tell whether a clock had struck eleven or 

 twelve, unless we were able to mark off each successive stroke 

 with some distinctive sign ; so that when it is said, as it often 

 is, that an animal cannot count, we must remember that 

 neither could a senior wrangler count if deprived of his 

 symbols. " Man begins by counting things, grouping them 

 visibly \i.e. by the Logic of Recepts]. He then learns to 

 count simply the numbers, in the absence of things, using 

 his fingers and toes for symbols. He then substitutes 

 abstract signs, and Arithmetic begins. From this he passes 

 to Algebra, the signs of which are not merely abstract but 

 general ; and now he calculates numerical relations, not 

 numbers. From this he passes to the higher calculus of 

 relations." 



And just as in mathematics the symbols that are employed 

 contain in an easily manipulated form enormous bodies of 

 meaning — possibly, indeed, the entire meaning of a long 

 calculation, — so in all other kinds of abstract ideation, the 

 symbols which we employ — whether in gesture, speech, or 

 writing — contain more or less condensed masses of significa- 

 tion. Or, to take another illustration, which, like the last 

 example, I quote from Lewes, " It is the same with the 

 development of commerce. Men begin by exchanging things. 



