The Concept of Number from Gauss to Kronecker

5. Conclusion The conquest of several mathematical areas by arithmetic, which we mentioned in the introduction, was thus achieved through two rival ways, both stemming from Gauss’s work. The first one, followed by Weierstrass, Dedekind and Cantor, borrowed the conception of number from Gauss. This conception entailed a development of mathematics through the extension of this concept of number and the introduction of new numbers. At the close of the conquest, which was to lead, in particular, to the arithmetization of analysis, “the captive analysis captured its savage victor,”70 and arithmetic was reduced to a simple province of analysis. The second way, followed by Kronecker who rejected this annexation of arithmetic by analysis, borrowed other concepts and methods, also from Gauss. These concepts, more related to operations than to objects, had to avoid the extension of the concept of number and the proliferation of new numbers. Thus Kronecker proposed this deployment of operations as an alternative to the development of the mathematical subject matter through conceptual extension. These two approaches — conceptual extension (and the consequent widening of the domain of objects) or operative deployment — appear more generally as distinct paths to mathematical progress. These two ways, usually coming one after the other, are combined in Gauss’s work. Thus this work stops the pendular swing from the concept (or the object) to the operation and vice versa.