330 REASONING. 



strative Sciences until verified by being applied to the 

 most remarkable of all those sciences, that of Num- 

 bers ; 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, therefore, 

 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 Mr. Whewell 

 and the a priori philosophers on one side ; and on the 

 other, a philosophical 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 ver- 

 bal, 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 philosophers, 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 equi- 

 valent 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 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 



