Take, for example, another type 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 signification. 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. They pass to the exchange of values. First money, then notes or bills, is the symbol of value. Finally men simply debit and credit one another, so that immense transactions are effected by means of this equation of equations. The complicated processes of sowing, reaping, collecting, shipping, and delivering a quantity of wheat, are condensed into the entry of a few words in a ledger.”