The Axiomatization of Arithmetic

Following the arithmetization of classical maths, it was natural, in this line of development, that attention should next be directed toward the foundations of arithmetic itself, and, in fact, this is what happened around 1880. Apparently, before the nineteenth century, no one had attempted to define addition and multiplication in any other way than by a direct appeal to intuition.

Leibnitz alone, faithful to his principles, pointed out explicitly that "truths" as "obvious" as 2 + 2 = 4 are no less capable of proof, if one reflects on the definitions of the numbers which appear there in, and he did not by any means regard the commutativity of addition and multiplication as self-evident. As examples of non-commutative operations, he cited subtraction, division, and exponentiation. At one time he even attempted to introduce such operations into his logical calculus. He did not, however, push his reflections on this subject any further, and in the middle of the nineteenth century progress has still not been made in this direction.

Even Weierstrass, whose lectures contributed greatly to the spread of "arith-metizing" point of view, did not realize the necessity of a logical clarification of the theory of integers. The first steps in this direction are, apparently, due to Grassman who, in 1861 gave a definition of addition and multiplication of inte­gers and proved their fundamental properties (commutativity, associativity, dis-tributivity), using nothing but the operation x —>x + 1 and the principle of induction.

The latter had been clearly conceived and used for the first time in the seven­teenth century by B.Pascal though more or less explicit applications of it are to be found in the maths of antiquity - and it was in current use by mathematicians from the second half of the seventeenth century.

But it was not until 1888 that Dedekind enunciated a complete system of axioms for arithmetic (reproduced three years later by Peano and usually known under his name), which contained, in particular, a precise formulation of the principle of induction (which Grassman had used without enunciating it explicitly).

With this axiomatization it seemed that the definitive foundations of maths had been attained. But, in fact, at the very moment when the axioms of arithmetic were being clearly formulated, arithmetic, itself was being dethroned from this role of primordial science in the eyes of many mathematicians (beginning with Dedekind and Peano), in favour of the most recent of math theories, namely, the theory of sets; and the controversies which were to surround the notion of integers cannot be isolated from the great "crisis of foundations" of the years 1900-1930.