188 REASONING. 



"What we have now asserted, however, can not be received as universally 

 true of Deductive or Demonstrative Sciences, until verified by being ap- 

 plied to the most remarkable of all those sciences, that of Numbers ; the 

 theory of the Calculus ; Arithmetic and Algebra. It is harder to believe 

 of the doctrines of this science than of any other, either that they are not 

 truths a priori, but experimental truths, or that their peculiar certainty is 

 owing to their being not absolute but only conditional truths. This, there- 

 fore, is a case which merits examination apart ; and the more so, because 

 on this subject we have a double set of doctrines to contend with ; that of 

 the a priori philosophers on one side ; "and on the other, a theory the most 

 opposite to theirs, which was at one time very generally received, and is 

 still far from being altogether exploded, among metaphysicians. 



§ 2. This theory attempts to solve the difficulty apparently inherent in 

 the case, by representing the propositions of the science of numbers as 

 merely verbal, and its processes as simple transformations of language, sub- 

 stitutions of one expression for another. The proposition, Two and one is 

 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 equiva- 

 lent 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 terminology, by which equiva- 

 lent 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 de- 

 monstrate a new geometrical theorem by algebra), they have not explain- 

 ed ; 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 Avhich render the theory in question very plausible, 

 and have not unnaturally made those sciences the stronghold of Nominal- 

 ism. 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 advances in philosophy to be- 

 \lieve it: men fly to so paradoxical a belief to avoid, as they think, some 

 leven 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 reconcilable with the nature of the Science of Numbers. For we 

 do not carry any ideas along with us when we use the S3anbols of arithme- 

 tic or of algebra. In a geometrical demonstration we have a mental dia- 

 gram, if not one on paper; AB, AC, are present to our imagination as lines, 

 intersecting other lines, forming an angle with one another, and the like ; 

 but not so a and h. These may represent lines or any other magnitudes, 

 but those magnitudes are never thought of ; nothing is realized in our im- 

 agination but a and h. The ideas which, on the particular occasion, they 

 happen to represent, are banished from the mind during every intermediate 

 part of the process, between the beginning, when the premises are trans- 

 lated from things into signs, and the end, when the conclusion is translated 

 back from signs into things. Nothing, then, being in the reasoner's mind 

 but the symbols, what can seem more inadmissible than to contend that the 

 reasoning process has to do with any thing more ? We seem to have come 

 to one of Bacon's Prerogative Instances ; an experimentum criicis on the 

 nature of reasoning itself. 



