DEMONSTRATION, AND NECESSARY TRUTHS. 191 



which the enunciation of the above-mentioned physical fact will serve also 

 for a definition of the word Three. 



The Science of Number is thus no exception to the conclusion we pre- 

 viously arrived at, that the processes even of deductive sciences are alto- 

 gether inductive, and that their first principles are generalizations from ex- 

 perience. It remains to be examined whether this science resembles geom- 

 etry in the further circumstance, that some of its inductions 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, be- 

 ing true in no other sense than that those propositions legitimately follow 

 from the hypothesis of the truth of premises which are avowedly mere ap- 

 proximations 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, 

 etc., which may be called the definitions of the various numbers, in the im- 

 proper 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 sufiicient ; for the corresponding pi'opositions re- 

 specting unequals may be proved from these by a reductio ad absurdum. 



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

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

 seem, exactly true, Avithout the 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 categorical 

 certainty which is predicable of its demonstrations is independent 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 assumption which 

 may be false. The condition is, that 1 = 1; that all the numbers are num- 

 bers of the same or of equal units. Let this be doubtful, and not one of 

 the propositions of arithmetic will hold true. How can we know that one 

 oound and one pound make two pounds, if one of the pounds may be troy, 

 md the other avoirdupois ? They may not make two pounds of either, or 

 if any weight. How can we know that a forty-horse power is always equal 

 :o itself, unless we assume that all horses are of equal strength ? It is cer- 

 ,ain that 1 is always equal in number to 1 ; and where the mere number of 

 )bjects, or of the parts of an object, without supposing them to be equiv- 

 ilent in any other respect, is all that is material, the conclusions of arith- 

 netic, so far as they go to that alone, are true without mixture of hypoth- 

 'sis. There are such cases in statistics ; as, for instance, an inquiry into 

 he amount of the population of any country. It is indifferent to that in- 

 [uiry whether they are grown people or children, strong or weak, tall or 

 : hort ; the only thing we want to ascertain is their number. But when- 

 ' ver, from equality or inequality of number, equality or inequality in any 

 • ither respect is to be inferred, arithmetic carried into such inquiries be- 

 < omes as hypothetical a science as geometry. All units must be assumed 

 :o be equal in that other respect; and this is never accurately true, for one 

 : ctual pound weight is not exactly equal to another, nor one measured mile's 

 1 3ngth to another ; a nicer balance, or more accurate measuring instruments, 

 ' /ould alwavs detect some difference. 



