On Arithmetization

4. Conclusion There are a few respects which Cantor’s, Dedekind’s, and Kronecker’s arithmetization programmes share, in spite of all their manifest incompatibility with respect to finitist or constructivist requirements. First, all three authors considered mathematics as a science with a clearly defined domain of objects: as mentioned before, Kronecker viewed mathematics as a natural science;121 Dedekind considered his analysis of continuity via cuts as expressing the essence of this concept; Cantor seems to have considered even his transfinite numbers as something that he discovered, rather than invented.122 For all three authors arithmetization reduced the irrational numbers to the rational — or all the way to natural — numbers whose existence was taken to be evident. Second, they all executed this reduction to elementary given objects in away that they considered naturally adequate for the problem at hand: for Kronecker, this meant indeterminates and congruences à la Gauss, for Dedekind grouping together sets of primary objects was just as naturally adequate a procedure as the consideration of series of rational numbers was to Cantor. The overall image that this suggests of the movement of arithmetization in the 1870s and 1880s is therefore that of a novel theory of objects that had formerly been understood in terms of extrinsic notions (magnitudes), this novel theory being founded on an independently accepted basis (the natural numbers), and proceeding with ingredients or methods deemed to be acceptable. From this point of view, arithmetization, in spite of all its novelty, appears not as an expression of modernity — indeed, as far as new objects were created, they were not purely formal, nor were they objectivized tools, but regularly formed from existing integers — but as a new type of solid building, erected on a traditional base in a controlled and supposedly innocuous and stable construction. Our periodization has allowed us to isolate a transitional phase of arithmetization where Gaussian influence is detectible at least in two of the major authors. This influence operated via diverging fundamental positions (Kronecker’s constructivism vs. Dedekind’s completed infinites), but always in the direction of a novel but object-oriented rewriting of analysis. Gauss’s after-effect ended with the onset of purely set-theoretic, axiomatic or logicist approaches, i.e., at the same time as the Göttingen nostrified image of arithmetization took shape.