From Magnitudes to Geometry and Back: De Zolt's Postulate

Abstract

A crucial trend of nineteenth-century mathematics was the search for pure foundations of specific mathematical domains by avoiding the obscure concept of magnitude. In this paper, we examine this trend by considering the “fundamental theorem” of the theory of plane area: “If a polygon is decomposed into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon.” This proposition, known as De Zolt's postulate, was conceived as a strictly geometrical expression of the general principle of magnitudes “the whole is greater than the part.” On the one hand, we illustrate this striving for purity in the foundations of geometry by analysing David Hilbert's classical proof of De Zolt's postulate. On the other hand, we connect this geometrical problem with the first axiomatizations of the concept of magnitude by the end of the nineteenth century. In particular, we argue that a recent result in the logical analysis of the concept of magnitude casts new light on Hilbert's proof. We also outline an alternative development of a theory of magnitude that includes a proof of De Zolt's postulate in an abstract setting.