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REASONING. 



rally received, and is still far from being altogether exploded, 

 among metaphysicians. 



2. This theory attempts to solve the difficulty appa- 

 rently inherent in the case, by representing the propositions 

 of the science of numbers as merely verbal, and its processes 

 as simple transformations of language, substitutions of one 

 expression for another. The proposition, Two and one are 

 equal to three, according to these writers, is not a truth, is 

 not the assertion of a really existing fact, but a definition of 

 the word three ; a statement that mankind have agreed to use 

 the name three as a sign exactly equivalent to two and one ; 

 to call by the former name whatever is called by the other 

 more clumsy phrase. According to this doctrine, the longest 

 process in algebra is but a succession of changes in termi- 

 nology, by which equivalent expressions are substituted one 

 for another ; a series of translations of the same fact, from 

 one into another language ; though how, after such a series 

 of translations, the fact itself comes out changed (as when 

 we demonstrate a new geometrical theorem by algebra,) they 

 have not explained ; and it is a difficulty which is fatal to 

 their theory. 



It must be acknowledged that there are peculiarities in the 

 processes of arithmetic and algebra which render the theory 

 in question very plausible, and have not unnaturally made 

 those sciences the stronghold of Nominalism. The doctrine 

 that we can discover facts, detect the hidden processes of 

 nature, by an artful manipulation of language, is so contrary 

 to common sense, that a person must have made some ad- 

 vances in philosophy to believe it : men fly to so paradoxical 

 a belief to avoid, as they think, some even greater difficulty, 

 which the vulgar do not see. What has led many to believe 

 that reasoning is a mere verbal process, is, that no other 

 theory seemed reconcileable with the nature of the Science of 

 Numbers. For we do not carry any ideas along with us when 

 we use the symbols of arithmetic or of algebra. In a geome- 

 trical demonstration we have a mental diagram, if not one on 

 paper ; AB, AC, are present to our imagination as lines, in- 



