t66 



REASONlNa 



to axioms, we found that, considered 

 as experimental truths, they rest on 

 superabundant and obviona evidence. 

 We inquired whether, since this is the 

 case, it be imperative to suppose any 

 other evidence of those ti-uths than 

 experimental evidence, any other ori- 

 gin for our belief of them than an 

 experimental origin. We decided 

 that the burden of proof lies with 

 those who maintain the affirmative, 

 and we examined, at considerable 

 length, such arguments as they have 

 produced. The examination having 

 led to the rejection of those argu- 

 ments, we have thought ourselves 

 warranted in concluding that axioms 

 are but a class, the most universal 

 class, of inductions from experience ; 

 the simplest and easiest cases of gene- 

 ralisation from the facts furnished to 

 us by our senses or by our internal 

 consciousness. 



While the axioms of demonstrative 

 sciences thus appeared to be experi- 

 mental truths, the definitions, as 

 they are incorrectly called, in those 

 sciences, were found by us to be 

 generalisations from experience which 

 are not even, accurately speaking, 

 truths ; being propositions in which, 

 while we assert of some kind of ob- 

 ject some property or properties which 

 observation shows to belong to it, we 

 at the same time deny that it pos- 

 sesses any other properties, though in 

 truth other properties do in every 

 individual instance accompany, and 

 in almost all instances modify, the 

 property thus exclusively predicated. 

 The denial, therefore, is a mere fic- 

 tion or supposition, made for the 

 purpose of excluding the considera- 

 tion of those modifying circumstances, 

 when their influence is of too trifling 

 amount to be worth considering, or 

 adjourning it, when important, to a 

 more convenient moment. 



From these considerations it would 

 appear that Deductive or Demonstra- 

 tive Sciences are all, without excep- 

 tion, 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 accord- 

 ing to which their inductions are made. 

 Hypothetical Sciences. Their conclu- 

 sions are only true on certain suppo- 

 sitions, which are, or ought to be, 

 approximations to the truth, but are 

 seldom, if ever, exactly true; and to this 

 h3rpothetical character is to be ascribed 

 the peculiar certainty which is sup- 

 posed to be inherent in demonstration. 

 What we have now asserted, how- 

 ever, cannot be received as universally 

 true of Deductive or Demonstrative 

 Sciences, until verified by being 

 applied 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 d 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 exa- 

 mination apart ; and the more so, 

 because on this subject we have a 

 double set of doctrines to contend 

 with ; that of the d 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 proposi- 

 tions 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 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 

 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 



