DEMONSTRATION AND NECESSARY TRUTHS. 



169 



an alteration of place and arrange- 

 ment be made to produce either the 

 one set of sensations or the other, 

 though a very familiar proposition, is 

 not an identical one. It is a truth 

 known to us by early and constant 

 experience — an inductive truth ; and 

 such truths are the foundation of the 

 science of Numbers. The fundamental 

 truths of that science all rest on the 

 evidence of sense ; they are proved 

 by showing to our eyes and our fingers 

 that any given number of objects, 

 ten balls, for example, may by separa- 

 tion and rearrangement exhibit to 

 our senses all the different sets of 

 numbers the sum of which is equal to 

 ten. All the improved methods of 

 teaching arithmetic to children pro- 

 ceed on a knowledge of this fact. 

 All who wish to carry the child's 

 mind along with them in learning 

 arithmetic ; all who wish to teach 

 numbers, and not mere ciphers — now 

 teach it through the evidence of the 

 fsenses, in the manner we have de- 

 scribed. 



We may, if we please, call the pro- 

 position, "Three is two and one," a 

 definition of the number three, and 

 assert that arithmetic, as it has been 

 asserted that geometry, is a science 

 founded on definitions. But they are 

 definitions in the geometrical sense, 

 not the logical ; asserting not the 

 meaning of a term only, but along 

 with it an observed matter of fact. 

 The proposition, " A circle is a figure 

 bounded by a line which has all its 

 points equally distant from a point 

 within it," is called the definition of 

 a circle ; but the proposition from 

 which so many consequences follow, 

 and which is really a first principle 

 in geometry, is, that figures answering 

 to this description exist. And thus 

 we may call " Three is two and one " 

 a definition of three ; but the calcula- 

 tions which depend on that proposi- 

 tion do not follow from the definition 

 itself, but from an arithmetical theorem 

 presupposed in it, namely, that collec- 

 tions of objects exist, which while they 

 impress the senses thus, **^<', may be 



separated into two parts, thus, ©o e. 

 This proposition being granted, we 

 term all such parcels Threes, after 

 which the enunciation of the above- 

 mentioned physical fact will serve 

 also for a definition of the word 

 Three. 



The Science of Numbers is thus no 

 exception to the conclusion we previ- 

 ously arrived at, that the processes 

 even of deductive sciences are alto- 

 gether inductive, and that their first 

 principles are generalisations from ex- 

 perience. It remains to be examined 

 whether this science resembles geome- 

 try 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, 

 being true in no other sense than that 

 those propositions legitimately follow 

 from the hypothesis of the truth of 

 premises 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 pro- 

 positions respecting unequals may be 

 proved from these by a reductio ad 

 ahsurdum. 



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, without the hy- 

 pothetical assumption of unqualified 

 truth where an approximation to it is 

 all that exists. The conclusions, there- 

 fore, it will naturally be inferred, are 

 exactly true, and the science of 

 numbers is an exception to other 

 demonstrative sciences in this, that 



