DEMONSTRATION AND NECESSARY TRUTHS. 



147 



cal phenomena with which those 

 sciences are conversant have been 

 found to answer to determinable 

 varieties in the quantity of some 

 circumstance or other ; or at least to 

 varieties of form or position for which 

 corresponding equations of quantity 

 had already been, or were susceptible 

 of being, discovered by geometers. 



In these various transformations, 

 the propositions of the science of 

 number do but fulfil the function 

 proper to all propositions forming a 

 train of reasoning, viz. that of enabling 

 us to arrive in an indirect method, by 

 marks of marks, at such of the pro- 

 perties of objects as we cannot directly 

 ascertain (or not so conveniently) by 

 experiment. We travel from a given 

 visible or tangible fact through the 

 truths of numbers to the facts sought. 

 The given fact is a mark that a cer- 

 tain relation subsists between the 

 quantities of some of the elements 

 concerned ; while the fact sought pre- 

 supposes a certain relation between 

 the quantities of some other elements. 

 Now, if these last quantities are de- 

 pendent in some known manner upon 

 the former, or mce versd, we can 

 argue from the numerical relation 

 between the one set of quantities to 

 determine that which subsists between 

 the other set ; the theorems of the 

 calculus affording the intermediate 

 links. And thus one of the two 

 physical facts becomes a mark of the 

 other, by being a mark of a mark of 

 a mark of it. 



CHAPTER V. 



or DEMONSTRATION AND NECESSARY 

 TRUTHS. 



§ I. If, as laid down in the two 

 preceding chapters, the foundation of 

 all sciences, even deductive or demon- 

 strative sciences, is Induction ; if every 

 step in the ratiocinations even of geo- 

 metry is an act of induction ; and if 

 a train of reasoning is but bringing 

 many inductions to b«ar upon the 



same subject of inquiry, and drawing 

 a case within one induction by means 

 of another ; wherein lies the pecu- 

 liar certainty always ascribed to the 

 sciences which are entirely, or almost 

 entirely, deductive ? Why are they 

 called the Exact Sciences? Why 

 are mathematical certainty, and the 

 evidence of demonstration, common 

 phrases to express the very highest 

 degree of assurance attainable by 

 reason ? Why are mathematics by 

 almost all phi^-isophers, and (by some) 

 even those branches of natural philo- 

 sophy which, through the medium of 

 mathematics, have been converted in- 

 to deductive sciences, considered to be 

 independent of the evidence of experi- 

 ence and observation, and character- 

 ised as systems of Necessary Truth ? 



The answer I conceive to be, that 

 this character of necessity ascribed 

 to the truths of mathematics, and even 

 (with some reservations to be hereafter 

 made) the peculiar certainty attributed 

 to them, IS an illusion ; in order to 

 sustain which, it is necessary to sup- 

 pose that those truths relate to, and 

 express the properties of purely ima- 

 ginary objects. It is acknowledged 

 that the conclusions of geometry are 

 deduced, partly at least, from the so- 

 called Definitions, and that those de- 

 finitions are assumed to be correct 

 representations, as far as they go, trf 

 the objects with which geometry is 

 conversant. Now we have pointed 

 out that, from a definition as such, no 

 proposition, unless it be one concern- 

 ing the meaning of a word, can ever 

 follow ; and that what apparently 

 follows from a definition, follows in 

 reality from an implied assumption 

 that there exists a real thing con- 

 formable thereto. This assumption 

 in the case of the definitions of geo- 

 metry, is not strictly true : there 

 exist no real things exactly conform- 

 able to the definitions. There exist 

 no points without magnitude ; no 

 lines without breadth, nor perfectly 

 straight ; no circles with all their 

 radii exactly equal, nor squares with 

 all their angles perfectly right. It 



