164 BEASONIVG. 



inductions from experience : the simplest and easiest cases of generali- 

 zation from the facts furnished to us by our senses or by our internal 

 consciousness. 



While the axioms of demonstrative sciences thus appeared to be 

 experimental truths, the definitions, as they are incorrectly called, qf 

 those sciences, were found by us to be generalizations from experience 

 which are not even, accurately speaking, truths ; being propositions in 

 which, while we assert of some kind of object, some j^roperty or prop- 

 erties which observation shows to belong to it, we at the same time 

 deny that it possesses any other properties, although in truth other 

 properties do in every individual instance accompany, and in most or 

 even in all instances, modify the property thus exclusively predicated. 

 The denial, therefore, is a mere fiction, or 'supposition, made for the 

 purpose of excluding the consideration of those modifying circum- 

 stances, when their influence is of too trifling amount to be worth con- 

 sidering, or adjourning it, when important, to a more convenient 

 moment. 



From these considerations it would appear that Deductive or De- 

 monstrative Sciences are all, without exception. Inductive Sciences : 

 that their evidence is that of experience, but that they are also, in virtue 

 of the peculiar character of one indispensable portion of the general 

 formulae according to which their inductions are made. Hypothetical 

 Sciences. Their conclusions are only true upon certain suppositions, 

 which are, or ought to be, approximations to the truth, but are seldom, 

 if ever, exactly true ; and to this hypothetical character is to be ascribed 

 the peculiar certainty, which is supposed to be inherent in demon- 

 stration. 



What we have now asserted, however, cannot be received as univer- 

 sally true of Deductive or Demonstrative Sciences vmtil 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 d priori, but experimental truths, or that their pe- 

 cuHm- certainty is o\\nng to their being not absolute but only conditional 

 truths. This, therefore, is a case which mei'its 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 d 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 fi-om 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 lan- 

 guage, 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 agi-eed 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 terminology, by which equivalent expressions are substi- 

 tuted one for another ; a series of translations of the same fact, fi-om 



