The Principles of Mathematics (1903)

§ 116

It is easy to deduce from the above definitions the usual connection of addition and multiplication, which may be thus stated. If k be a class of b mutually exclusive classes, each of which contains a terms, then the logical sum of k contains a×b terms[80]. It is also easy to obtain the definition of ab, and to prove the associative and distributive laws, and the formal laws for powers, such as abac = ab+c. But it is to be observed that exponentiation is not to be regarded as a new independent operation, since it is merely an application of multiplication. It is true that exponentiation can be independently defined, as is done by Cantor[81], but there is no advantage in so doing. Moreover exponentiation unavoidably introduces ordinal notions, since ab is not in general equal to ba. For this reason we cannot define the result of an infinite number of exponentiations. Powers, therefore, are to be regarded simply as abbreviations for products in which all the numbers multiplied together are equal.(§ 116 ¶ 1)

From the data, which we now posess, all those propositions which hold equally of finite and infinite numbers can be deduced. The next step, therefore, is to consider the distinction between the finite and the infinite.(§ 116 ¶ 2)

§ 116 n. 1. See Whitehead, loc. cit..