David Hilbert and the foundations of the theory of plane area

Archive for History of Exact Sciences https://doi.org/10.1007/s00407-021-00278-z David Hilbert and the foundations of the theory of plane area Eduardo N. Giovannini1,2 Received: 11 November 2020 © The Author(s) 2021 Abstract This paper provides a detailed study of David Hilbert’s axiomatization of the theory of plane area, in the classical monograph Foundation of Geometry (1899). On the one hand, we offer a precise contextualization of this theory by considering it against its nineteenth-century geometrical background. Specifically, we examine some crucial steps in the emergence of the modern theory of geometrical equivalence. On the other hand, we analyze from a more conceptual perspective the significance of Hilbert’s theory of area for the foundational program pursued in Foundations. We argue that this theory played a fundamental role in the general attempt to provide a new independent basis for Euclidean geometry. Furthermore, we contend that our examination proves relevant for understanding the requirement of “purity of the method” in the tradition of modern synthetic geometry. Keywords Hilbert · Axiomatic geometry · Polygonal area · De Zolt’s postulate · Purity of the method 1 Introduction Chapter IV of David Hilbert’s classical Foundations of Geometry, first published in 1899, develops the theory of plane polygonal area. This section of the influential monograph is usually praised not only for its unprecedented level of rigor, being the first modern axiomatization of this central part of elementary geometry, but also for the many innovative and original results contained therein. Among these, one can mention the systematic study of different relations of geometrical equivalence, the Communicated by Jeremy Gray. B Eduardo N. Giovannini 1 Department of Philosophy, University of Vienna, Universitätsstraße 7, 1010 Vienna, Austria 2 CONICET and Universidad Nacional del Litoral, Bv. Pellegrini 2750, S3000 Santa Fe, Argentina 123 E. N. Giovannini construction of the theory of area independently of continuity assumptions (viz. the axiom of Archimedes), as well as a sophisticated but elementary proof of the central geometrical proposition known as De Zolt’s postulate. Notwithstanding, despite the importance of this chapter, it has received less attention from historians and philosophers of mathematics than other sections of Foundations. 1 This paper aims to fill this gap in the specialized literature by offering the first detailed historical discussion of Hilbert’s axiomatic investigations into the theory of plane area. We will undertake this task by closely examining the development of this theory in Foundations. In addition, Hilbert’s notes for lecture courses on the foundations of mathematics will also be taken into account. These important sources offer a unique landscape to elaborate a more accurate historical account of his work.2 Hilbert’s theory of plane area will be investigated with an eye to two interpretative points. The first point concerns the historical background of Hilbert’s investigations. We will argue that, to a significant extent, his axiomatization of the theory of area was the culmination of a rich and intense foundational debate, which took place during the second half of the nineteenth century. This debate was triggered by the emergence of the modern theory of geometrical equivalence, which investigates criteria for the equality of area of polygonal figures on the basis of its decomposition and composition into polygonal components, respectively congruent. The main issue in these discussions concerned the role and logical status of a geometrical proposition known as “De Zolt’s postulate.” This central proposition states that if a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent (i.e., equal in area) to the given polygon. In discussing methodological and epistemological issues related to this new “geometrical axiom,” geometers involved in this debate delivered novel insights for the modern synthetic reconstruction of Euclidean geometry. The contextualization of Hilbert’s investigations within this specific geometrical background yields a better historical assessment of his contributions in Foundations and sheds new light on a central episode in the emergence of modern axiomatic geometry. In particular, a welcome offshoot of the present investigation is a better historical appraisal of the contributions of important nineteenth-century geometers, such as Friedrich Schur, to the foundations of modern geometry. The second interpretative issue relates to the historical and conceptual significance of Hilbert’s theory of plane area for the general axiomatic program pursued in Foundations. As is well known, this program aimed at providing a new independent basis for elementary Euclidean geometry, by removing the dependence on continuity and (implicit) numerical assumptions from the classical theories of proportion and plane area. In this regard, a key technical innovation was the construction of a purely geometrical calculus of segments, which allowed the derivation of the (abstract) algebraic structure of an ordered field from the axioms for the Euclidean plane. In this paper, we will argue that the problem of obtaining an adequate proof of the so-called De 1 Hilbert’s theory of plane area in Foundations has been analyzed recently by Baldwin (2018a, b) and Baldwin and Mueller (2019). These articles offer excellent expositions of the central ideas and results achieved by Hilbert. Nevertheless, the nineteenth-century geometrical background upon which Hilbert developed his theory is not taken into particular consideration. 2 Hilbert’s notes for lecture courses on the foundations of geometry, corresponding to the period 1891–1902, have been published in Hallett and Majer (2004). 123 David Hilbert and the foundations of the theory of area Zolt’s postulate was, for Hilbert, a central issue in the modern axiomatic development of the theory of plane area. More specifically, we will contend that a significant challenge was to deliver a rigorous proof of this proposition that was not only strictly geometrical—in the sense of avoiding numerical considerations—but also independent of the Archimedean axiom. The paper consists of two thematic parts. The first part provides a historical examination of the development of the theory of plane area in the second half of the nineteenth century, which set the stage for Hilbert’s axiomatic investigations. A central aspect of this geometrical background was a clear distinction between a “synthetic” and a “metrical” approach to the study of polygonal areas. While the former was identified with the theory of geometrical equivalence, the latter consisted in the (now standard) method of measuring the area of polygonal figures by means of (positive) real numbers. To put these geometrical developments into proper context, Sect. 2 presents a (...truncated)