632 SCIENCE AND HYPOTHESIS 



the equality: (1) x + a = [x + (a !)] + !. We shall know what 

 x + a is when we know what x+(a 1) is, and as I have assumed 

 that to start with we know what x + 1 is, we can define suc- 

 cessively and " by recurrence " the operations x + 2, x + 3, etc. This 

 definition deserves a moment's attention; it is of a particular nature 

 which distinguishes it even at this stage from the purely logical defi- 

 nition; the equality (1), in fact, contains an infinite number of dis- 

 tinct definitions, each having only one meaning when we know the 

 meaning of its predecessor. 



Properties of Addition 



Associative. I say that a + (& + c) = (a + &)+ c; in fact the theo- 

 rem is true for c = 1. It may then be written a +(& + 1)= (a + &) 

 -f- 1 ; which, remembering the difference of notation, is nothing but the 

 equality (1) by which I have just defined addition. Assume the 

 theorem true for c y, I say that it will be true for c = y + 1. 

 Let (a + &)+y = a+(&+y), it follows that [(a + 6)+y] + l = 

 [a + (& + y)]+l; or by del (1) (a + &) + (y + l)=o+(& + y + 

 l)=a + [&+(y + l)L which shows by a series of purely analytical 

 deductions that the theorem is true for y + 1. Being true for c = 1, we 

 see that it is successively true for c = 2, c = 3, etc. 



Commutative. (1)1 say that a + 1 = 1 + a. The theorem is evi- 

 dently true for a = 1 ; we can verify by purely analytical reasoning that 

 if it is true for a = y it will be true for a = y + I. 1 Now, it is true for 

 a = 1, and therefore is true for a = 2, a = 3, and so on. This is what 

 is meant by saying that the proof is demonstrated " by recurrence." 



(2) I say that a + b & + a. The theorem has just been shown to 

 hold good for & = 1, and it may be verified analytically that if it is true 

 for b = , it will be true for b = ft + 1. The proposition is thus estab- 

 lished by recurrence. 



Definition of Multiplication 



We shall define multiplication by the equalities: (l)aXl = a. (2) 

 a X 6 = [a X (& 1)] -f a. Both of these include an infinite number 

 of definitions; having defined a X 1, it enables us to define in succession 

 a X 2, a X 3, and so on. 



Properties of Multiplication 



Distributive. I say that (a + &) X c = (a X c) + (6 X c) . We can 

 verify analytically that the theorem is true for c = 1 ; then if it is true 

 for c y, it will be true for c = y + 1. The proposition is then proved 

 by recurrence. 



i For ( y +i ) +1= ( i+ y ) + 1=1+ ( y + 1 ) . [TB.J 



