This paper provides a detailed study of David Hilbert’s axiomatization of the theory of plane area, in the classical monograph Foundation of Geometry. 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. (shrink)

We show that the first-order theory of a large class of plane geometries and the first-order theory of their groups of motions, understood both as groups with a unary predicate singling out line-reflections, and as groups acting on sets, are mutually inter-pretable.

We proved in the first part [1] that plane geometry over Pythagorean fields is axiomatizable by quantifier-free axioms in a language with three individual constants, one binary and three ternary operation symbols. In this paper we prove that two of these operation symbols are superfluous.

In this paper we examine the contributions of the Italian geometrical school to the Foundations of Projective Geometry. Starting from De Paolis' work we discuss some papers by Segre, Peano, Veronese, Fano and Pieri. In particular we try to show how a totally abstract and general point of view was clearly adopted by the Italian scholars many years before the publication of Hilbert's Grundlagen.We are particularly interested in the interrelations between the Italian and the German schools (mainly the (...) influence of Staudt's and Klein's works). We try also to understand the reason of the steady decline of the Italian school during the twentieth century. (shrink)

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as “natural”. We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry – i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3–34 years), and 25 (...) participants from the Amazon (age 5–67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in “sense”). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans’ knowledge in Euclidean geometry could possibly be grounded. (shrink)

In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The (...) article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids. (shrink)

The paper [Tarski: Les fondements de la géométrie des corps, Annales de la Société Polonaise de Mathématiques, pp. 29—34, 1929] is in many ways remarkable. We address three historico-philosophical issues that force themselves upon the reader. First we argue that in this paper Tarski did not live up to his own methodological ideals, but displayed instead a much more pragmatic approach. Second we show that Leśniewski's philosophy and systems do not play the significant role that one may be tempted to (...) assign to them at first glance. Especially the role of background logic must be at least partially allocated to Russell's systems of Principia mathematica. This analysis leads us, third, to a threefold distinction of the technical ways in which the domain of discourse comes to be embodied in a theory. Having all of this in place, we discuss why we have to reject the argument in [Gruszczyński and Pietruszczak: Full development of Tarski's Geometry of Solids, The Bulletin of Symbolic Logic, vol. 4, no. 4, pp. 481—540] according to which Tarski has made a certain mistake. (shrink)

§30. Significance of Desargues's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 CHAPTER VI. PASCAL'S THEOREM. §31. ...

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

We provide a quantifier-free axiom system for plane hyperbolic geometry in a language containing only absolute geometrically meaningful ternary operations (in the sense that they have the same interpretation in Euclidean geometry as well). Each axiom contains at most 4 variables. It is known that there is no axiom system for plane hyperbolic consisting of only prenex 3-variable axioms. Changing one of the axioms, one obtains an axiom system for plane Euclidean geometry, expressed in (...) the same language, all of whose axioms are also at most 4-variable universal sentences. We also provide an axiom system for plane hyperbolic geometry in Tarski's language L B which might be the simplest possible one in that language. (shrink)

A Paper on the Foundations of Projective Geometry - (Read before the Aristotelian Society, Dec. 13, 1897) is an unchanged, high-quality reprint of the original edition of 1898. Hansebooks is editor of the literature on different topic areas such as research and science, travel and expeditions, cooking and nutrition, medicine, and other genres. As a publisher we focus on the preservation of historical literature. Many works of historical writers and scientists are available today as antiques only. Hansebooks newly (...) publishes these books and contributes to the preservation of literature which has become rare and historical knowledge for the future. (shrink)

The Foundations of Geometry was first published in 1897, and is based on Russell's Cambridge dissertation as well as lectures given during a journey through the USA. This is the first reprint, complete with a new introduction by John Slater. It provides both an insight into the foundations of Russell's philosophical thinking and an introduction to the philosophy of mathematics and logic. As such it will be an invaluable resource not only for students of philosophy, but also (...) for those interested in Russell's philosophical development. Foundations of Geometry consists of four chapters which explore the various concepts of geometry and their philosophical implications, including a historical overview of the development of geometrical theory. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

ArgumentAccording to Hermann von Helmholtz, free mobility of bodies seemed to be an essential condition of geometry. This free mobility can be interpreted either as matter of fact, as a convention, or as a precondition making measurements in geometry possible. Since Henri Poincaré defined conventions as principles guided by experience, the question arises in which sense experiential data can serve as the basis for the constitution of geometry. Helmholtz considered muscular activity to be the basis on which (...) the form of space could be construed. Yet, due to the problem of illusion inherent in the subject’s self-assessment of muscular activity, this solution yielded new difficulties, in that if the manifold is abstracted from rigid bodies which serve as empirical justification of the geometrical notion of space, then illusionary bodies will produce fictive manifolds. The present article is meant to disentangle these difficulties. (shrink)

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid’s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid’s plane geometry (EPG). Euclid’s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based (...) unless diagrams are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: ( i ) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; ( ii ) EPG objects inherit some properties and relations from these diagrams. (shrink)

We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ∀∃∀, but that there is an axiom system, all of whose axioms are ∀∃∀∃ sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields.

Tim Maudlin sets out a completely new method for describing the geometrical structure of spaces, and thus a better mathematical tool for describing and understanding space-time. He presents a historical review of the development of geometry and topology, and then his original Theory of Linear Structures.

In 1906, Frigyes Riesz introduced a preliminary version of the notion of a topological space. He called it a mathematical continuum. This development can be traced back to the end of 1904 when, genuinely interested in taking up Hilbert’s foundations of geometry from 1902, Riesz aimed to extend Hilbert’s notion of a two-dimensional manifold to the three-dimensional case. Starting with the plane as an abstract point-set, Hilbert had postulated the existence of a system of neighbourhoods, thereby introducing (...) the notion of an accumulation point for the point-sets of the plane. Inspired by Hilbert’s technical approach, as well as by recent developments in analysis and point-set topology in France, Riesz defined the concept of a mathematical continuum as an abstract set provided with a notion of an accumulation point. In addition, he developed further elementary concepts in abstract point-set topology. Taking an abstract topological approach, he formulated a concept of three-dimensional continuous space that resembles the modern concept of a three-dimensional topological manifold. In 1908, Riesz presented his concept of mathematical continuum at the International Congress of Mathematicians in Rome. His lecture immediately won the attention of people interested in carrying on his research. They promoted his ideas, thus assuring their gradual reception by several future founders of general topology. In this way, Riesz’s work contributed significantly to the emergence of this discipline. (shrink)

The main goal of part 1 is to challenge the widely held view that Poincaré orders the sciences in a hierarchy of dependence, such that all others presuppose arithmetic. Commentators have suggested that the intuition that grounds the use of induction in arithmetic also underlies the conception of a continuum, that the consistency of geometrical axioms must be proved through arithmetical induction, and that arithmetical induction licenses the supposition that certain operations form a group. I criticize each of these readings. (...) More fully, I argue that the justification Poincaré offers for the use of the group notion in geometry appears to extend to set-theoretic notions that would suffice to put arithmetic on a logical foundation, thus undermining his own case for the necessity of intuition in arithmetic. In part 2, I offer an interpretation of intuition’s role on which it justifies the use of group-theoretic, but not set-theoretic, notions. (shrink)

Part 1 of this article exposed a tension between Poincaré’s views of arithmetic and geometry and argued that it could not be resolved by taking geometry to depend on arithmetic. Part 2 aims to resolve the tension by supposing not merely that intuition’s role is to justify induction on the natural numbers but rather that it also functions to acquaint us with the unity of orders and structures and show practices to fit or harmonize with experience. I argue (...) that in this manner, intuition serves the epistemological function of warranting generalizations and justifying practices. In particular, it justifies the application of group-theoretic notions in geometry but not the use of set-theoretic notions in arithmetic. (shrink)

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view (...) that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at (...) Göttingen. Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)

In this paper we provide quantifier-free, constructive axiomatizations for 2-dimensional absolute, Euclidean, and hyperbolic geometry. The main novelty consists in the first-order languages in which the axiom systems are formulated.

We provide a universal axiom system for plane hyperbolic geometry in a firstorder language with two sorts of individual variables, ‘points’ and ‘lines’ , containing three individual constants, A0, A1, A2, standing for three non-collinear points, two binary operation symbols, φ and ι, with φ = l to be interpreted as ‘[MATHEMATICAL SCRIPT SMALL L] is the line joining A and B’ , and ι = P to be interpreted as [MATHEMATICAL SCRIPT SMALL L]P is the point of (...) intersection of g and h , and two binary operation symbols, π1 and —2, with πi = g to be interpreted as ‘g is one of the two limiting paralle lines from P to [MATHEMATICAL SCRIPT SMALL L]. (shrink)

We present the basic concepts of space and time, the Galilean and pseudo-Euclidean geometry. We use an elementary geometric framework of affine spaces and groups of affine transformations to illustrate the natural relationship between classical mechanics and theory of relativity, which is quite often hidden, despite its fundamental importance. We have emphasized a passage from the group of Galilean motions to the group of Poincaré transformations of a plane. In particular, a 1-parametric family of natural deformations of the (...) Poincaré group is described. We also visualized the underlying groups of Galilean, Euclidean, and pseudo-Euclidean rotations within the special linear group. (shrink)

In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a (...) reconstruction of this New Science that meets modern standards and to examine possible problems surrounding Frege’s original proposal. The paper is organized as follows: the first two sections summarize the main points of the Frege–Hilbert controversy and discuss some issues surrounding the problem of independence proofs. Section 3 contains an informal presentation of Frege’s proposal. In section 4 a more detailed reconstruction of Frege’s New Science is set out while section 5 examines what is left out. The concluding section is devoted to a discussion of Frege’s strategy and its significance from a broader perspective. (shrink)