Klaus Robering

The present article provides a mereological analysis of Euclid’s planar geometry as presented in the first two books of his Elements. As a standard of comparison, a brief survey of the basic concepts of planar geometry formulated in a set-theoretic framework is given in Section 2. Section 3.2, then, develops the theories of incidence and order using a blend of mereology and convex geometry. Section 3.3 explains Euclid’s “megethology”, i.e., his theory of magnitudes. In Euclid’s system of geometry, megethology takes over the role played by the theory of congruence in modern accounts of geometry. Mereology and megethology are connected by Euclid’s Axiom 5: “The whole is greater than the part.” Section 4 compares Euclid’s theory of polygonal area, based on his “Whole-Greater-Than-Part” principle, to the account provided by Hilbert in his Grundlagen der Geometrie. An hypothesis is set forth why modern treatments of geometry abandon Euclid’s Axiom 5. Finally, in Section 5, the adequacy of atomistic mereology as a framework for a formal reconstruction of Euclid’s system of geometry is discussed.

Following ideas elaborated by Hering in his celebrated analysis of color, the psychologist and gestalt theorist Otto Selz developed in the 1930s a theory of “natural space”, i.e., space as it is conceived by us. Selz’s thesis is that the geometric laws of natural space describe how the points of this space are related to each other by directions which are ordered in the same way as the points on a sphere. At the end of one of his articles, Selz tries to derive within his framework two of Hilbert’s axioms for Euclidean geometry. Such derivations would, according to Selz, disclose the psychological origin and meaning of the geometric axioms and would thus contribute to a clarification of their epistemological status. In the present article Selz’s theory is explained and analyzed, his basic principles are amended, and implicit assumptions are made explicit. It is shown that the resulting system is one of ordered affine geometry in which all of Hilbert’s axioms except the axioms of congruence and those of continuity are derivable. The primary aim of the present paper is to make explicit the basic principles behind Selz’s geometry of natural space. The question of the logical independence of these principle is not investigated.

In the 1930s Otto Selz developed a novel approach to the psychology of perception which he called “synthetic psychology of wholes”. This “synthetic psychology” is based on a phenomenological description of the structural relationships between elementary items building up integral wholes. The present article deals with Selz’s account of spatial cognition within this general framework. Selz Zeitschrift für Psychologie, 114, 351–362 argues that his approach to spatial cognition delivers answers to the long-discussed question of the epistemological status of the laws of geometry. More specifically he tries to derive the Euclidean axioms from the structural laws valid for phenomenal space. After a brief description of the discussion of the status of geometry in the 1920s/1930, the present article explains Selz’s understanding of “phenomenology”. Section 4 then deals with Selz’s attempt to derive the Euclidean laws from the structural phenomenological laws of space. Selz’s attempted derivation suffers from some formal shortcomings, which however can be repaired. The question arises, though, whether the necessary improvements do not rely upon more intricate geometric intuitions and thus render Selz’s attempt to base geometry upon the phenomenology of spatial cognition circular.

The present article first places Mach’s consideration about space and geometry into the context of the discussion of these issues in the nineteenth and early twentieth century and then proposes three interpretations of Mach’s thesis, put forward in chapter XXI of his Knowledge and Error, that the problem of measuring the volumes of material bodies is the origin of geometry. According to the first of these interpretations, Mach’s thesis is an assertion about the historical origin of the science of geometry. Alternatively, one may understand Mach as suggesting that our geometric theorizing is best understood by relating it to our handling of material bodies and our interest in their volumes. Finally, Mach’s thesis may be conceived as asserting that the most appropriate form of geometry would be a metric geometry of the volumes of solids. The article concludes with a discussion of objections raised by Brentano against both Mach’s thesis of the priority of the measurement of volumes and his conception of bodies as complexes as sensations.