i68 



REASONING. 



danger of error, every algebraical for- 

 mula in the books. The proposition, 

 2 (a + b) = 2 a + 2 6, is a truth co-ex- 

 tensive with all nature. Since then 

 algebraical truths are true of all things 

 whatever, and not, like those of 

 geometry, true of lines only or of 

 angles only, it is no wonder that the 

 symbols should not excite in our 

 minds ideas of any things in particu- 

 lar. When we demonstrate the forty- 

 seventh proposition of Euclid, it is 

 not necessary that the words should 

 raise in us an image of all right- 

 angled triangles, but only of some 

 one right-angled triangle ; so in 

 algebra we need not, under the 

 symbol a, picture to ourselves all 

 things whatever, but only some one 

 thing ; why not, then, the letter 

 itself ? The mere written characters, 

 a, 6, X, y, z, serve as well for repre- 

 sentatives of Things in general, as 

 any more complex and apparently 

 more concrete conception. That we 

 are conscious of them, however, in 

 their character of things, and not of 

 mere signs, is evident from the fact 

 that our whole process of reasoning is 

 carried on by predicating of them the 

 properties of things. In resolving an 

 algebraic equation, by what rules do 

 we proceed? By applying at each 

 step to a, b, and x, the proposition 

 that equals added to equals make 

 equals ; that equals taken from equals 

 leave equals ; and other propositions 

 founded on these two. These are not 

 properties of language, or of signs as 

 such, but of magnitudes, which is as 

 much as to say, of all things. The 

 inferences, therefore, which are suc- 

 cessively drawn, are inferences con- 

 cerning things, not symbols ; though 

 as any Things whatever will serve the 

 turn, there is no necessity for keeping 

 the idea of the Thing at all distinct, 

 and consequently the process of 

 thought may, in this case, be allowed 

 without danger to do what all pro- 

 cesses of thought, when they have 

 been performed often, will do if per- 

 mitted, namely, to become entirely 

 mechanical. Hence the general lan- 



guage 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 pro- 

 bative 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 evi- 

 dence 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 respecting Things, they 

 all have the appearance of beingidenti- 

 cal propositions. The assertion, Two 

 and one is 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 " objeqts are them- 

 selves " being insignificant, it seems 

 but natural to consider the proposition 

 Two and one is 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 c?enote 

 the same things, their coTinotation is 

 different. Three pebbles in two sepa- 

 rate parcels, and three pebbles in one 

 parcel, do not make the same impres- 

 sion on our senses ; and the assertion 

 that the very same pebbles may by 



