The Arithmetization of Classical Mathematics

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We have had occasion to refer to the notion of a "model" or an "interpreta­tion" of a math theory constructed with the help of another theory. This is not a recent idea, and it should undoubtedly be seen as a continually recurring mani­festation of a deep-lying feeling of the unity of the various "math sciences". If the traditional maxim "All things are numbers" of the early Pythagoreans is tak­en as authentic, it may be considered as the vestige of a first attempt to reduce the geometry and algebra of the time to arithmetic.

The increasingly widespread use of the notion of "model" was to permit the nineteenth-century mathematicians to achieve the unification of maths dreamed of by the Pythagoreans. At the beginning of the century, whole numbers and continuous quantities were as irreconcilable as they had been in antiquity; the real numbers were related to the notion of geometric magnitude (especially that of length), and the "models" of negative numbers and imaginary number! were constructed on this basis. Even the rational numbers were traditionally associated with the idea of "subdivision" of magnitude into equal parts. Only the integers remained apart, as "exclusive products of our intellect", as Gauss said in 1832.

The first attempts to bring together arithmetic and analysis were concerned with the rational numbers (positive and negative); they were taken up around 1860 by several authors, notably Grassman, Hankel, and Weierstrass. To Weierstrass is, apparently, due the idea of obtaining a "model" of the positive of negative rational numbers by considering classes of ordered pairs of natural integers. But the most important step still remained to be taken, namely, to construct a "model"  of the irrational numbers from within the theory of rational numbers.

By 1870 this had become an urgent problem because of the necessity, after the discovery of "pathological"  phenomena in analysis, to purge every trace of geometrical intuition and the vague notion of "magnitude" from the definition of real numbers. The problem was, in fact, solved at about this time almost simultaneously Cantor, Dedekind, Meray and Weierstrass, using quite different methods from one another.

From then on, the integers became the foundation of all classical maths. Furthermore,  the "models" founded on arithmetic acquired still greater importance with the extension of the axiomatic method and the conception of math objects as free creations of the human intellect. For there remained a limitation to the freedom claimed by Cantor, namely, the limitation raised by the question of "existence " which had already occupied the Greeks, and which arose now so much  more  urgently precisely because all appeal to intuitive representation was now abondoned.

We shall not discuss what a philosophico-math maelstrom was to be generated by notion of "existence" in the early years of the twentieth century. But in the  nineteenth century this stage had not yet been reached, and to prove the existence of a math object having a given set of properties meant simply, just as Euclid, "to construct" an object with the required properties. This was purpose of the arithmetical "models".

Once the real numbers had been "interpreted" in terms of integers, then the same was done for complex numbers and Euclidean geometry, thanks to analytic geometry, and likewise for all the new algebraic objects introduced since the beginning' of the century. Finally, in a discovery which achieved great fame Beltrami and Klein obtained Euclidean "models" of the non-Euclidean geometries of Lobachevsky and Riemann and thereby "arithmetized" (and completely justified) these theories which at first had aroused so much distrust.