Theorems stated in one must enrich Tarski's theory with a new postulateĪsserting that the universe of discourse of the geometry of solids coincides withĪrbitrary mereological sums of balls, i.e., with solids. Was sketched by Tarski in his short paper. In this paper we give probably an exhaustive analysis of the geometry of solids which In the concluding section, I rely on the constructive side of Cohen’s criticisms to reconsider the philosophical aspects of Helmholtz’s theory and draw a few comparisons with contemporary measurement theory. Cohen’s criticism of some of Helmholtz’s assumptions notwithstanding, my emphasis is on some unexpected affinities between these two approaches. The main sections deal with Helmholtz’s and Cohen’s approaches to the foundations of measurement. The first part provides a brief introduction to this debate in its connection with the earlier discussion on geometrical axioms and the concept of space. This paper deals with Cohen’s view of magnitudes and measurement and with his – less known – review of Helmholtz’s paper “Zählen und Messen, erkenntnistheoretisch betrachtet” (1887), which contains one of the first attempts to formulate a theory of measurement in the modern sense. examination of scientific works and scientists’ epistemological views. His relation to Hermann von Helmholtz, who played a major role in the same debate, is an illuminating example of how some of the leading ideas of Marburg neo-Kantianism, although motivated independently of scientific debates, naturally led to the. It is well known that Hermann Cohen was one of the first philosophers who engaged in the debate about non-Euclidean geometries and the concept of space. Elementary geometry has no set-theoretical basis, because of which its formalization does not provide for variables of higher orders and no symbols are available to represent or denote geometrical figures and classes of geometrical figures. In the formalization of elementary geometry, only points are treated as individuals and are represented by first-order variables. As non-logical constants any predicates can be chosen denoting certain relations among points in terms of which all geometrical notions are known to be definable. The logical constants of the theory include the sentential connectives, the quantifiers, and the two special binary predicates. Elementary geometry is formalized within elementary logic that is essentially first-order predicate calculus. It focuses on the conception of elementary geometry, which can be described as the part of Euclidean geometry that can be formulated and established without the help of any set-theoretical devices. This chapter describes the significance of notions and methods of modern logic and metamathematics for the study of the foundations of geometry. Finally, having explicitly defined 'point' and 'line', we will derive the characteristic Paral-lel's Postulate (Playfair axiom) from regions-based axioms, and point the way toward deriving key Euclidean metrical properties. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit "extremal clause" (to the effect that "these are the only ways of generating regions"), and we have no axiom of induction other than ordinary numerical (mathematical) induction. Then we generalize this to arbitrary orientations, and then establishing an isomorphism between the space and the usual point-based R × R. We first derive the Archimedean property for a class of readily postulated orientations of certain special regions, "gener-alized quadrilaterals" (intended as parallelograms), by which we cover the entire space. The goal is to recover the classical contin-uum on a point-free basis. We extend the work presented in to a regions-based, two-dimensional, Euclidean theory. Finally, in Section 5, the adequacy of atomistic mereology as a framework for a formal reconstruction of Euclid’s system of geometry is discussed. An hypothesis is set forth why modern treatments of geometry abandon Euclid’s Axiom 5. 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. In Euclid’s system of geometry, megethology takes over the role played by the theory of congruence in modern accounts of geometry. Section 3.3 explains Euclid’s “megethology”, i.e., his theory of magnitudes. Section 3.2, then, develops the theories of incidence and order (of points on a line) using a blend of mereology and convex geometry. 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. The present article provides a mereological analysis of Euclid’s planar geometry as presented in the first two books of his Elements.
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