190 REASONING. 



But when we look back to see from whence the probative force of the 

 process is derived, we find that at every single step, unless we suppose our- 

 selves to bo thinking and talking of the things, and not the mere symbols, 

 the evidence fails. 



There is another circumstance, which, still more than that which we have 

 now mentioned, gives plausibility to the notion that the propositions of 

 arithmetic and algebra are merely verbal. That is, that when considered 

 as propositions respecting Things, they all have the appearance of being 

 identical propositions. The assertion, Two and one is equal to three, con- 

 sidered as an assertion respecting objects, as for instance, " Two pebbles 

 and one pebble are equal to three pebbles," does not affirm equality be- 

 tween two collections of pebbles, but absolute identity. It affirms that if 

 we put one pebble to two pebbles, those very pebbles are three. The ob- 

 jects, therefore, being the very same, and the mere assertion that " objects 

 are themselves" being insignificant, it seems but natural to consider the 

 proposition, Two and one is equal to three, as asserting mere identity of 

 signification between the two names. 



This, however, though it looks so plausible, will not bear examination. 

 The expression " two pebbles and one pebble," and the expression " three 

 pebbles," stand indeed for the same aggregation of objects, but they by no 

 means stand for the same physical fact. They are names of the same ob- 

 jects, but of those objects in two different states ; though they deuoto the 

 same things, their connotation is different. Three pebbles in two separate 

 parcels, and three pebbles in one parcel, do not make the same impression 

 on our senses ; and the assertion that the very same pebbles may by an al- 

 teration of place and arrangement be made to produce either the one set of 

 sensations or the other, though a very famiUar proposition, is not an iden- 

 tical 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 Num- 

 ber. 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 separation and 

 re-arrangement exhibit to our senses all the different sets of numbers the 

 sums of which is equal to ten. All the improved methods of teaching 

 arithmetic to children proceed 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 ev- 

 idence of the senses, in the manner we have described. 



We may, if we please, call the proposition, " 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 mean- 

 ing 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 de- 

 scription exist. And thus we may call "Three is two and one "a defini- 

 tion of three; but the calculations which depend on that proposition do 

 not follow from the definition itself, but from an arithmetical theorem pre- 

 supposed in it, namely, that collections of objects exist, which while they 

 impress the senses thus, °o°, may be separated into two parts, thus, o o o- 

 Tjiis proposition being granted, we term all such parcels Threes, after 



