334 REASONING. 



case, be allowed without danger to do what all pro- 

 cesses of thought, when they have been performed 

 often, will do if permitted, namely, to become entirely 

 mechanical. Hence the general language of algebra 

 comes to be used familiarly without exciting ideas, as 

 all other general language is prone to do from mere 

 habit, though in no other case than this can it be done 

 with complete safety. But when we look back to see 

 from whence the probative force of the process is 

 derived, we find that at every single step, unless we 

 suppose ourselves to be thinking and talking of the 

 things, and not the mere symbols, the evidence fails. 



There is another circumstance, which,, still more 

 than that which we have now mentioned, gives plausi- 

 bility to the notion that the propositions of arithmetic 

 and algebra are merely verbal. This is, that when 

 considered as propositions respecting Things, they all 

 have the appearance of being identical propositions. 

 The assertion, Two and one are equal to three, consi- 

 dered as an assertion respecting objects, as for instance 

 * ' Two pebbles and one pebble are equal to three 

 pebbles," does not affirm equality between two collec- 

 tions of pebbles, but absolute identity. It affirms 

 that if we put one pebble to two pebbles, those very 

 pebbles are three. The objects, therefore, being the 

 very same, and the mere assertion that " objects are 

 themselves" being insignificant, it seems but natural 

 to consider the proposition, Two and one are equal to 

 three, as asserting mere identity of signification 

 between the two names. 



This, however, though it looks so plausible, will 

 not stand examination. The expression, "two peb- 

 bles and one pebble/' and the expression, " three 

 pebbles/ 7 stand indeed for the same aggregation of 

 objects, but they by no means stand for the same 



