DEMONSTRATION, AND NECESSARY TRUTHS. 287 



out 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 sym- 

 bols, the evidence fails. 



There is another circumstance, which, still more than that 

 which we have now mentioned, gives plausibility to the notion 

 that the propositions of arithmetic and algebra are merely 

 verbal. That is, that when considered as propositions respect- 

 ing Things, they all have the appearance of being identical 

 propositions. The assertion, Two and one are equal to three, 

 considered as an assertion respecting objects, as for instance 

 " Two pebbles and one pebble are equal to three pebbles," 

 does not affirm equality between two collections of pebbles, 

 but absolute identity. It affirms that if we put one pebble to 

 two pebbles, those very pebbles are three. The objects, there- 

 fore, being the very same, and the mere assertion that " ob- 

 jects 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 bear 

 examination. The expression " two pebbles and one pebble," 

 and the expression, " three pebbles," stand indeed for the 

 same aggregation of objects, but they by no means stand for 

 the same physical fact. They are names of the same objects, 

 but of those objects in two different states : though they de- 

 note the same things, their connotation is different. Three 

 pebbles in two separate parcels, and three pebbles in one 

 parcel, do not make the same impression on our senses ; and 

 the assertion that the very same pebbles may by an alteration 

 of place and arrangement be made to produce either the one 

 set of sensations or the other, though a very familiar proposi- 

 tion, is not an identical one. It is a truth known to us by 

 early and constant experience : an inductive truth ; and such 

 truths are the foundation of the science of Number. The 



