168 REASONING, 



from experience. It remains to be examined whether this science 

 resembles geometry in the further circumstance, that some of its induc- 

 tions are not exactly true ; and that the peculiar certainty ascribed to 

 it, on account of which its propositions are called Necessary Truths, is 

 fictitious and hypothetical, being true in no other sense than that those 

 propositions necessarily follow from the hypothesis of the truth of prem- 

 isses which are avowedly mere approximations to truth. 



§ 3. The inductions of arithmetic are of two sorts : first, those which 

 we have just expounded, such as One and one are two, Two and one 

 are three, &c., which may be called the definitions of the various 

 numbers, in the improper or geometrical sense of the word Definition ; 

 and secondly, the two following axioms : The sums of equals are equal, 

 The differences of equals are equal. These two are sufficient ; for the 

 corresponding propositions respecting unequals may be proved from 

 these, by the process well known to mathematicians under the name of 

 reductio ad ahsurdum. 



These axioms, and likewise the so-called definitions, are, as already 

 shown, results of induction ; true of all objects whatever, and, as it may 

 seem, exactly true, without any hypothetical assumption of unqualified 

 truth where an approximation to it is all that exists. The conclusions, 

 therefore, it will naturally be inferred, are exactly true, and the science 

 of number is an exception to other demonstrative sciences in this, that 

 the absolute certainty which is predicable of its demonstrations is inde- 

 pendent of all hypothesis. 



On more accurate investigation, however, it will be found that, even 

 in this case, there is one hypothetical element in the ratiocination. In 

 all propositions concerning numbers, a condition is implied, without 

 which none of them would be true ; and that condition is an assump- 

 tion which maybe false. The condition is, that 1 = 1; that all the 

 numbers are numbers of the same or of equal units. liet this be doubt- 

 ful, and not one of the propositions of arithmetic will hold true. How 

 can we know that one pound and one pound make two pounds, if one 

 of the pounds may be troy, and the other avoirdupois ] They may not 

 make two pounds of either, or of any weight. How can we know that 

 a forty-horse power is always equal to itself, unless we assume that all 

 horses are of equal strength % It is certain that 1 is always equal in 

 number to 1 ; and where the inere number of objects, or of the parts 

 of an object, without supposing them to be equivalent in any other 

 respect, is all that is material, the conclusions of aiithmetic, so far as 

 they go to that alone, are true without mixture of hypothesis. There 

 are a few such cases ; as, for instance, an inquiry into the amount of 

 population of any country. It is indifferent to that inquiry whether 

 they are grown people or children, strong or weak, tall or short ; the 

 only thing we want to ascertain is their number. Bvit whenever, from 

 equality or inequality of number, equality or inequality in any other 

 respect is to be infen-ed, arithmetic carried into such inquiries becomes 

 as hypothetical a science as geometry. All units must be assumed to 

 be equal in that other respect ; and this is never precisely time, for 

 one pound weight is not exactly equal to another, nor one mile's length 

 to another; a nicer balance, or more accurate measuring instruments, 

 would always detect some diflference. 



What is commonly called mathematical certainty, therefore, which 



