Geometry

In this paper I present critical evaluations of Ayer and Putnam's views on the analyticity of geometry. By drawing on the historico-philosophical work of Michael Friedman on the relativized apriori; and Roberto Torretti on the foundations of geometry, I show how we can make sense of the assertion that pure geometry is analytic in Carnap's sense.

This short two-page essay provides irrefutable evidence that the Bible was encoded by its original redactors. Critical scholarship on the Bible needs to take ancient Jewish hermeneutical methods seriously.

This paper introduces some basic ideas and formalism of physics in non-commutative geometry. My goals are three-fold: first to introduce the basic formal and conceptual ideas of non-commutative geometry, and second to raise and address some philosophical questions about it. Third, more generally to illuminate the point that deriving spacetime from a more fundamental theory requires discovering new modes of `physically salient' derivation.

With the discovery of consistent non-Euclidean geometries, the a priori status of Euclidean proof was radically undermined. In response, philosophers proposed two revisionary interpretations of the practice: some argued that Euclidean proof is a purely formal system of deductive logic; others suggested that Euclidean reasoning is empirical, employing concepts derived from experience. I argue that both interpretations fail to capture the true nature of our geometrical thought. Euclidean proof is not a system of pure logic, but one in which our (...) grasp of the content of geometrical concepts plays a central role; moreover, our grasp of this content is a priori. (shrink)

Euclidean diagrammatic reasoning refers to the diagrammatic inferential practice that originated in the geometrical proofs of Euclid’s Elements. A seminal philosophical analysis of this practice by Manders (‘The Euclidean diagram’, 2008) has revealed that a systematic method of reasoning underlies the use of diagrams in Euclid’s proofs, leading in turn to a logical analysis aiming to capture this method formally via proof systems. The central premise of this paper is that our understanding of Euclidean diagrammatic reasoning can be fruitfully advanced (...) by confronting these logical and philosophical analyses with the field of cognitive science. Surprisingly, central aspects of the philosophical and logical analyses resonate in very natural ways with research topics in mathematical cognition, spatial cognition and the psychology of reasoning. The paper develops these connections, concentrating on four issues: (1) the cognitive origins of Euclidean diagrammatic reasoning, (2) the cognitive representations of spatial relations in Euclidean diagrams, (3) the nature of the cognitive processes and cognitive representations involved in Euclidean diagrammatic reasoning seen as a form of visuospatial relational reasoning and (4) the complexity of Euclidean diagrammatic reasoning for the human cognitive system. For each of these issues, our analysis generates concrete experiment proposals, opening thereby the way for further empirical investigations. The paper is thus a prolegomenon to a research program on Euclidean diagrammatic reasoning at the crossroads of logic, philosophy and cognitive science. (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)

From the exponential function of Euler’s equation to the geometry of a fundamental form, a calculation of the fine-structure constant and its relationship to the proton-electron mass ratio is given. Equations are found for the fundamental constants of the four forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. Symmetry principles are then associated with traditional physical measures.

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

This paper surveys John von Neumann's work on the mathematical foundations of quantum theories in the light of Hilbert's Sixth Problem concerning the geometrical axiomatization of physics. We argue that in von Neumann's view geometry was so tied to logic that he ultimately developed a logical interpretation of quantum probabilities. That motivated his abandonment of Hilbert space in favor of von Neumann algebras, specifically the type II1II1 factors, as the proper limit of quantum mechanics in infinite dimensions. Finally, we present (...) the reasons why his axiomatic program remained an “unsolved problem” in mathematical physics. A recent unpublished result by Huzimiro Araki, proving that no algebra with a tracial state defined on it, such as the type II1II1 factors, can support any (regular) representation of the canonical commutation relations, is also reviewed and its consequences for von Neumann's projects are discussed. (shrink)

Actes du colloque de Grenoble (8-9 octobre 2009), avec les contributions de Samuel Gessner (Lisbone), Eberhard Knobloch (Berlin), Jorge Galindo Díaz (Bogotá), Joël Sakarovitch (Paris) et Dominique Raynaud (Grenoble).

Letters exchanged by scientists are a crucial source by which to trace the process that accompanies their scientific evolution. In this paper -accomplished through a historical approach- I aim to throw new light on Leibniz's continuing interest in classical geometry and to stress the significance of his correspondence with the Italian mathematician Vitale Giordano.

It is shown that the automorphism group of a relation algebra ${\cal B}_P$ constructed from a projective geometry P is isomorphic to the collineation group of P. Also, the base automorphism group of a representation of ${\cal B}_P$ over an affine geometry D is isomorphic to the quotient of the collineation group of D by the dilatation subgroup. Consequently, the total number of inequivalent representations of ${\cal B}_P$ , for finite geometries P, is the sum of the numbers ${\mid Col(P)\mid\over (...) {\mid Col(D)\mid/\mid Dil(D)\mid}}$ where D ranges over a list of the non-isomorphic affine geometries having P as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations. (shrink)

Let ${\cal Y}$ be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its $1$-homology group has notorsion. Weak limits of graphs of smooth maps $u_k:B^n\rightarrow {\cal Y}$ with equibounded total variation give riseto equivalence classes of cartesian currents in $\mathop {\rm cart}\nolimits ^{1,1}$ for which we introduce a natural$BV$-energy.Assume moreover that the first homotopy group of ${\cal Y}$ iscommutative. In any dimension $n$ we prove that every element $T$ in $\mathop {\rm cart}\nolimits ^{1,1}$ can be approximatedweakly in the (...) sense of currents by a sequence of graphs of smoothmaps $u_k:B^n\rightarrow {\cal Y}$ with total variation converging to the$BV$-energy of $T$. As a consequence, we characterize the lowersemicontinuous envelope of functions of bounded variations from$B^n$ into ${\cal Y}$. (shrink)

Este ensaio introdutório faz uma breve apresentação do tratado de óptica atribuído a Euclides de Alexandria, inserindo-o no contexto das teorias sobre a visão formuladas pelas doutrinas filosóficas antigas. Ressalta-se o antagonismo entre a análise geométrica da visão, empreendida por Euclides, e as considerações filosóficas acerca dos processos físicos subjacentes à sensação visual. Pretende-se mostrar que o objeto da óptica euclidiana é a percepção visual daquilo que Aristóteles denomina "sensível comum". This introductory essay provides an abridged presentation of the optical (...) treatise attributed to Euclid of Alexandria, placing it in the context of theories about vision formulated by the ancient philosophical doctrines. I emphasize the antagonism between the geometric analysis of vision, undertaken by Euclid, and the philosophical considerations about the physical processes underlying visual sensation. In addition, I aim to show that the object of Euclidean optics is the visual perception of what Aristotle calls "common sensible". (shrink)

The definition of a point in geometry is primordial in order to understand the different elements of this branch of mathematics ( line, surface, solids…). This paper aims at shedding fresh light on the concept to demonstrate that it is related to another one named, here, the Primary Geometric Object; both concepts concur to understand the multiplicity of geometries and to provide hints as concerns a new understanding of some concepts in physics such as time, energy, mass….

This textbook for the second year undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering most of the usual linear algebra topics. -/- Geometric algebra and its extension to geometric calculus simplify, unify, and generalize vast areas of mathematics that involve geometric ideas. Geometric algebra is an extension of linear algebra. The treatment of many linear algebra topics is enhanced by geometric algebra, for example, determinants and orthogonal transformations. And geometric algebra does much (...) more, as it incorporates the complex, quaternion, and exterior algebras, among others. (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)

The perfect compass, used by al-Qūhī, al-Sijzī and his successors for the continuous drawing of conic sections, reappeared after a long eclipse in the works of Renaissance mathematicians like Francesco Barozzi in Venice. The resurgence of this instrument seems to have depended on its interest to solve new optico-perspective problems. Having reviewed the various instruments designed for the drawing of conic sections, the article is focused on the sole conic compass. Theoretical and empirical applications are detailed. Contrarily to the common (...) thesis of an independant discovery, various elements suggest a direct descent between the birkār al-tāmm of the Arabic mathematical tradition and the Italian conic compass. Then we present the most probable transmission hypothesis involving: 1° Ibn Yūnus and his disciples of Mosul, 2° Sultan Malik al-Kāmil in Damas, 3° Master Theodore and Frederick II at the court of Sicily, 4° Andalò di Negro and Robert of Anjou in Naples, 5° Lorenzo della Volpaia, Vinci, Sangallo and Michelangelo in Florence, 6° Ausonio, Contarini, Thiene and Francesco Barozzi in Venice. (shrink)

René Descartes, Discourse on Method, Optics, Geometry, and Meteorology. Trans., with an Introduction, by Paul J. Olscamp. Indianapolis: The Bobbs-Merrill Co., 1965. Pp. xxxvi + 361. = The Library of Liberal Arts, 211. Paper, $2.25. -/- From the notice in Journal of the History of Philosophy 5 (1967), 311: "In the introduction, Professor Olscamp calls attention to the fact that Descartes intended the other three pieces in this volume to serve as examples of the method set forth in the Discourse. (...) I t is only just that Descartes' intentions should now be fulfilled in fairly clear, if occasionally awkward English versions. Seeing these pieces brought together in this way also sheds light on Descartes' philosophical opinions, frequently distorted by over-simplified distinctions like that between empiricist and rationalist. 0lscamp succeeds in showing that Descartes is better thought of as a forerunner to modern-day hypothetico-deductive theories of scientific explanation. Perhaps it is what we should have realized all along in one so preoccupied with methodology. --A. R. Louch". (shrink)

We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...) brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept. (shrink)

This paper applies the ideas presented in "Time, Euclidean Geometry and Relativity" ID 1290 , to a specific problem in temporal measurement. It is shown that, under very natural assumptions, that if there is a minimum time interval T in ones collection of clocks, it is impossible to measure an interval of time 1/2T save by the accidental construction of a clock which pulses in that interval. This situation is contrasted to that for length, in which either the Euclidean Algorithm (...) or a ruler and compass construction can be used to construct a lengh 1/2L from a length Lo. (shrink)

A detailed axiomatisation of diagrams (in affine geometry) is presented, which supports typing of geometric objects, calculation of geometric quantities and automated proof of theorems.

In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, (...) and we show that the two approaches are equivalent. (shrink)

Based on our research regarding the relationship between physics and mathematics in HPS, and recently on Geneva Edition of Newton's Philosophiae Naturalis Principia Mathematica (1739–42) by Thomas Le Seur (1703–70) and François Jacquier (1711–88), in this paper we present some aspects of such Edition: a combination of editorial features and scientific aims. The proof of Proposition XLIII is presented and commented as a case study.

Thomas Reid (1710–1796) presented a two-dimensional geometry of the visual field in his Inquiry into the human mind (1764), whose axioms are different from those of Euclidean plane geometry. Reid’s ‘geometry of visibles’ is the same as the geometry of the surface of the sphere, described without reference to points and lines outside the surface itself. Interpreters of Reid seem to be divided in evaluating the significance of his geometry of visibles in the history of the discovery of non-Euclidean geometries. (...) The question will be reexamined with particular attention given to his unpublished manuscripts. These include comments on Saccheri’s work and Reid’s repeated attempts to derive Euclid’s parallel postulate from the axioms of incidence. (shrink)

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

A simplistic image of twentieth century French philosophy sees Merleau-Ponty’s death in 1961 as the line that divides two irreconcilable moments in its history: existentialism and phenomenology, on the one hand, and structuralism on the other. The structuralist generation claimed to recapture the dimension of objectivity and impersonality, which the previous generation was supposedly incapable of. As a matter of fact, in 1962, Derrida’s edition of Husserl’s The Origin of Geometry was taken to be a turning point that announced the (...) structuralist revolution by introducing a reflection on the historicity and the materiality of impersonal idealities. And yet, the 1998 publication of Merleau-Ponty’s notes from his Collège de France lecture course on the same topic make manifest that he was already taking phenomenology in another direction. His 1959 reading of The Origin of Geometry shows how unexpectedly close the early Derrida is to the late Merleau-Ponty. By identifying the tension between archaeology and teleology as the basic problem of Husserlian phenomenology, Merleau-Ponty and Derrida each disclose the fundamental importance of history and of writing. Comparing the two readings in their specific context not only brings about a more complex picture of the intellectual debates of the time, but also shows how, with Merleau-Ponty’s interpretation of The Origin of Geometry, Derrida’s “différance” predates itself and receives another genealogy. (shrink)

Neste artigo, tentamos mostrar que um problema comum aos capítulos XVI e XX das Observações filosóficas é o da aplicabilidade dos conceitos e «proposições» da geometria à realidade física e perceptiva (visual, em particular), e que o modo pelo qual Wittgenstein aborda esse problema nessa obra difere radicalmente, a despeito de aparentes similitudes, daquele que caracteriza as teorias semânticas do a priori visual em termos de estipulações (notadamente, o de Carnap em 1922). O esclarecimento do estatuto dos enunciados sobre os (...) objetos do espaço visual como regras de sintaxe e a abordagem do problema da aplicação de nossos conceitos geométricos à realidade perceptiva em termos de condições práticas de emprego permite a Wittgenstein dissolver, não sem alguma ambiguidade no vocabulário empregado, as falsas aparências ontológicas suscitadas pelas teorias que postulam a existência de um universo de discurso intermediário entre os corpos físicos e os números. (shrink)

Abstract: Space is from two kinds of energy in standing waves; (1) energy with mass which is finite energy and (2) energy without mass which is infinite energy. Given light speed is equal to frequency times wavelength C = f λ then photon half waves are twice light speed on contraction before reversal expansion at light speed. Light speed is a constant relative to mass in Special Relativity but photon half waves are twice light speed on contraction from the fundamental (...) frequency. Infinity is half wave photon energy on contraction at twice light speed without mass-time. Standing half wave photons are oscillating as medium (ether) at twice light speed where energy is infinite; consequently, energy on reversal expansion creates mass and time in the electromagnetic field. (shrink)

Abstract: Space is from two kinds of energy in standing waves; (1) energy with mass which is finite energy and (2) energy without mass which is infinite energy. Given light speed is equal to frequency times wavelength C = f λ then photon half waves are twice light speed on contraction before reversal expansion at light speed. Light speed is a constant relative to mass in Special Relativity but photon half waves are twice light speed on contraction from the fundamental (...) frequency. Infinity is half wave photon energy on contraction at twice light speed without mass-time. Standing half wave photons are oscillating as medium (ether) at twice light speed where energy is infinite; consequently, energy on reversal expansion creates mass and time in the electromagnetic field. (shrink)

Thomas Reid's Geometry of Visibles, according to which the geometrical properties of an object's perspectival appearance equal the geometrical properties of its projection on the inside of a sphere with the eye in its centre allows for two different interpretations. It may (1) be understood as a theory about phenomenal visual space – i.e. an account of how things appear to human observers from a certain point of view – or it may (2) be seen as a mathematical model of (...) viewpoint-relative but mind-independent relational properties of objects. This paper makes a systematic and a historical claim. I shall argue, first, that given certain features of the human visual system phenomenal visual space differs in several aspects from Reidean visual space. Secondly, I suggest that, since Reid was aware of some of these empirical facts, we should interpret Reid as endorsing the second interpretation of the Geometry of Visibles. (shrink)

L'essor de la perspective linéaire a suscité de nombreuses polémiques tout au long du Quattrocento et du Cinquecento, opposant les partisans d'une géométrisation artificialiste de la vision à ceux qui vantaient les qualités du dessin d'après nature ou invoquaient des arguments de nature physiologique. Ces débats peuvent être retracés à partir des quatre alternatives qui en constituent le noyau dur : champ de vision restreint vs. large ; immobilité vs. mobilité oculaire ; tableau plan vs. curviligne ; vision monoculaire vs. (...) binoculaire. En retenant les premiers termes de ces quatre alternatives, l'histoire de la perspective a rejeté de nombreux systèmes hétérodoxes. Du point de vue de la mathématisation, l'intérêt de ces débats tient à ce qu'ils ont succédé, et non précédé, l'adoption du dispositif perspectif comme intersection de la pyramide visuelle. L'histoire de la perspective linéaire offre ainsi un authentique cas de justification a posteriori des fondements ou, pour ainsi dire, de mathématisation à rebours. (shrink)

In the Quattrocento and Cinquecento the rise of linear perspective caused many polemics which opposed the supporters of an artificial geometrisation of sight to those who were praising the qualities of the drawing according to nature, or were invoking some arguments on a physiological basis. ese debates can be grouped according to the four alternatives that form their central concerns: restricted vs. broad field of vision; ocular immobility vs. mobility; curvilinear vs. planar picture; monocular vs. binocular vision. By retaining the (...) first terms of these four alternatives, the history of perspective eliminated many heterodox constructions. From the viewpoint of mathematisation the interest of these debates is that they succeeded, rather than preceded, the adoption of a perspective system defined by the intersection of the visual pyramid. Thus the history of linear perspective constitutes a genuine case of a posteriori justification, or, put differently, it gives us a case of upside down mathematisation. (shrink)

H. Tietze has proved algebraically that the geometry of uniquely determined ruler and compass constructions coincides with the geometry of ruler and set square constructions. We provide a new proof of this result via new universal axiom systems for Euclidean planes of characteristic ≠ 2 in languages containing only operation symbols.

Twentieth-century developments in logic and mathematics have led many people to view Euclid’s proofs as inherently informal, especially due to the use of diagrams in proofs. In _Euclid and His Twentieth-Century Rivals_, Nathaniel Miller discusses the history of diagrams in Euclidean Geometry, develops a formal system for working with them, and concludes that they can indeed be used rigorously. Miller also introduces a diagrammatic computer proof system, based on this formal system. This volume will be of interest to mathematicians, computer (...) scientists, and anyone interested in the use of diagrams in geometry. (shrink)

John Playfair , professor of mathematics and natural philosophy at the University of Edinburgh, is a relatively obscure figure today, best known as the popularizer of James Hutton's theory of geology. However, Playfair was also involved in mathematics for most of his active career, with his most widely distributed publication, Elements of Geometry , shaping the mathematics education for at least thirteen thousand British students during the nineteenth century. This study focuses on the mathematical context surrounding Elements of Geometry. Specifically, (...) after recounting the background of the text, the paper explores the ways in which Playfair's presentation of elementary geometry reflected three understandings of the terms ‘analysis’ and ‘synthesis’, which were intrinsic components of mathematical culture at the turn of the nineteenth century. In one sense, the words denoted differing styles of mathematical practice in Great Britain and in France. In a second sense, the terms evoked contemporary appeals to ancient methods of proof. Finally, ‘analysis’ and ‘synthesis’ were understood in reference to separate approaches to mathematics education. Playfair's appeals to these understandings help reveal how he viewed himself as a mathematician. Overall, then, this study enriches the standard portrait of a professor who straddled the eighteenth and nineteenth centuries. (shrink)

The aim of the paper is to analyse the geometrical aspects of a series of modern paintings and to show the parallel between them and the development of modern geometry. It starts with El Greco, offering a geometrical explanation of his painting the figures in a prolonged manner. Further the analogy between the impressionist way of creating space and the geometrical idea of Cayley to use projective space as a basis for non-Euclidean geometry is reconstructed. Next the paper describes the (...) parallel between the creation of space in the paintings of Cézanne and Picasso and the concept of space in algebraic topology. In conclusion the paper deals with the analogy between Kandinski's abstract paintings and the set-theoretical foundations of geometry. (shrink)

The aim of this paper is a philosophical generalisation of the results, which we obtained through the analysis of the development of synthetic geometry. I our papers Náčrt analytickej teórie subjektu and Topológia versus teória množín we proposed a method of analysis of the development of geometry based on Wittgensteinś Picture theory of meaning from the Tractatus. It turned out, that the concept of the form of language can be effectively used to characterise the changes, which occured in the course (...) of development of geometry. In the present paper we would like to generalize our approach and outline a universal approach to epistemology, which we call formal epistemology. (shrink)