In this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution (...) Tarski's Postulate 4 can be omitted, together with its versions 4' and 4". We also prove that the equivalence of postulates 4, 4' and 4" is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms. We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory. Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski. (shrink)

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. (shrink)

In this paper, we are concerned with the development of a general set theory using the single axiom version of Leśniewski’s mereology. The specification of mereology, and further of Tarski’s geometry of solids will rely on the Calculus of Inductive Constructions (CIC). In the first part, we provide a specification of Leśniewski’s mereology as a model for an atomless Boolean algebra using Clay’s ideas. In the second part, we interpret Leśniewski’s mereology in monadic second-order logic using names and (...) develop a full version of mereology referred to as CIC-based Monadic Mereology (λ-MM) allowing an expressive theory while involving only two axioms. In the third part, we propose a modeling of Tarski’s solid geometry relying on λ-MM. It is intended to serve as a basis for spatial reasoning. All parts have been proved using a translation in type theory. (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)

The paper presents the main idea of point-free theories of space based on Tarski's system of point-free geometry. First, the general idea of the so-called point-free ontology was discussed, as well as the epistemological and methodological reasons for its adoption. Next, Whitehead's method of extensive abstraction, which is the methodological basis for the construction of point-free theories of space, is presented, and the fundamental concepts of mereology are discussed. The main part of the paper is a discussion of Tarski’s (...) geometry of solids, its postulates and metatheoretical properties. The paper ends with a short description of the contribution of Polish researchers to the development of research on point-free theories of space. (shrink)

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a (...) “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

Even though the discovery of the regular polyhedra is attributed to the Pythagoreans, there is some fascinating evidence that they may have been known in prehistoric Scotland. In the Ashmolean Museum at Oxford University there are five rounded stones with regularly spaced bumps. The high points of each bump mark the vertices of each of the regular polyhedra. The stone balls also appear to demonstrate the duals of three of the regular polyhedra. For example, if the six faces of the (...) cube become points, they become the six vertices of the octahedron. If the eight faces of the octahedron become points, they become the eight vertices of a cube. Remarkably, groves in the Ashmolean stones indicate each of the duals, including the fact that the tetrahedron is its own dual. Instead of assuming that these ancient Scots were experts in solid geometry, we have hypothesized that they must have discovered all of this by sphere stacking, which will be demonstrate later in this article. (shrink)

In 1995 Slater argued both against Priest’s paraconsistent system LP (1979) and against paraconsistency in general, invoking the fundamental opposition relations ruling the classical logical square. Around 2002 Béziau constructed a double defence of paraconsistency (logical and philosophical), relying, in its philosophical part, on Sesmat’s (1951) and Blanche’s (1953) “logical hexagon”, a geometrical, conservative extension of the logical square, and proposing a new (tridimensional) “solid of opposition”, meant to shed new light on the point raised by Slater. By using n-opposition (...) theory (NOT) we analyse Beziau’s anti-Slater move and show both its right intuitions and its technical limits. Moreover, we suggest that Slater’s criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics. This last criticism has been addressed by Malinowski (1990) and Shramko and Wansing (2005), who developed a family of tenable logical counter-examples to it: trans-Suszkian systems are radically many-valued. This family of new logics has some strange logical features, essentially: each system has more than one consequence operator. We show that a new, deeper part of the aforementioned geometry of logical oppositions (NOT), the “logical poly-simplexes of dimension m”, generates new logical-geometrical structures, essentially many-valued, which could be a very natural (and intuitive) geometrical counterpart to the “strange”, new, non-Suszkian logics of Malinowski, Shramko and Wansing. By a similar move, the geometry of opposition therefore sheds light both on the foundations of paraconsistent logics and on those of many-valued logics. (shrink)

This article is devoted to the problem of ontological foundations of three-dimensional Euclidean geometry. Starting from Bertrand Russell’s intuitions concerning the sensual world we try to show that it is possible to build a foundation for pure geometry by means of the so called regions of space. It is not our intention to present mathematically developed theory, but rather demonstrate basic assumptions, tools and techniques that are used in construction of systems of point-free geometry and topology by (...) means of mereology and Whitehead-like connection structures. We list and briefly analyze axioms for mereological structures, as well as those for connection structures. We argue that mereology is a good tool to model so called spatial relations. We also try to justify our choice of axioms for connection relation. Finally, we briefly discuss two theories: Grzegorczyk’s point-free topology and Tarski’s geometry of solids. (shrink)

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the (...) aforementioned areas. (shrink)

This paper begins with an assessment of the differences between two states of Jacopo Caraglio's engraved Diogenes after Parmigianino, and between each of those states and Parmigianino's preparatory drawing of the composition. What follows is an attempt to trace both the textual sources and the creative development of this unusual iconographie subject, culminating in a hypothesis about the chronological sequence of the earliest prints of Parmigianino's Diogenes. It is argued that, originally, the artist devised the composition in collaboration with a (...) learned advisor. His own interest in geometry then led him, incongruously, to introduce a geometric solid—an error corrected in Ugo da Carpi's chiaroscuro woodcut version. (shrink)

Abū al-Jūd Muḥammad ibn al-Layth is one of the mathematicians of the 10th century who contributed most to the novel chapter on the geometric construction of the problems of solids and super-solids, and also to another chapter on solving cubic and bi-quadratic equations with the aid of conics. His works, which were significant in terms of the results they contained, are moreover important with regard to the new relations they established between algebra and geometry. Good fortune transmitted (...) to us his correspondences with the mathematician and astronomer al-Bīrūnī. The questions they debated, and the answers they yielded, all offer us multiple in vivo perspectives on the research that was undertaken in that period. The reader would find in this article a critical edition and French translation of this correspondence, with historical and mathematical commentaries. Résumé Abū al-Jūd Muḥammad ibn al-Layth est l’un des mathématiciens du x e siècle qui ont le plus contribué au nouveau chapitre sur les constructions géométriques des problèmes solides et sur-solides, ainsi qu’à un autre chapitre, sur la solution des équations cubiques et biquadratiques à l’aide des coniques. Ses travaux, importants pour les résultats qu’ils renferment, le sont aussi par les nouveaux rapports qu’ils instaurent entre l’algèbre et la géométrie. La bonne fortune nous a transmis sa correspondance avec le mathématicien et astronome al-Bīrūnī. Les questions qui y sont débattues, les réponses apportées, nous offrent, in vivo, plusieurs perspectives sur la recherche qui était alors menée. Le lecteur trouve dans cet article l’édition critique de cette correspondance, sa traduction française ainsi qu’un commentaire historique et mathématique. (shrink)

I offer a new interpretation of Aristotle's philosophy of geometry, which he presents in greatest detail in Metaphysics M 3. On my interpretation, Aristotle holds that the points, lines, planes, and solids of geometry belong to the sensible realm, but not in a straightforward way. Rather, by considering Aristotle's second attempt to solve Zeno's Runner Paradox in Book VIII of the Physics , I explain how such objects exist in the sensibles in a special way. I conclude (...) by considering the passages that lead Jonathan Lear to his fictionalist reading of Met . M3,1 and I argue that Aristotle is here describing useful heuristics for the teaching of geometry; he is not pronouncing on the meaning of mathematical talk. (shrink)

Descartes holds that the tell-tale sign of a solid proof is that its entailments appear clearly and distinctly. Yet, since there is a limit to what a subject can consciously fathom at any given moment, a mnemonic shortcoming threatens to render complex geometrical reasoning impossible. Thus, what enables us to recall earlier proofs, according to Descartes, is God’s benevolence: He is too good to pull a deceptive switch on us. Accordingly, Descartes concludes that geometry and belief in God must (...) go hand in hand. However, I argue that, while theism adds a layer of psychological reassurance, the mind-independent reality of God would ensure the preservation of past demonstrations for atheists as well. (shrink)

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo (...) E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Kim. (shrink)

This article discusses some difficulties of the theory of color propounded by Meinong in his Re-marks on the Color Solid and the Mixture Law of 1903. First, I argue that Meinong’s geometrical approach faces at least three sets of difficulties related to the following assumptions: colors pos-sess a “nature” that can be grasped through intuition; they are separated from each other by continua in color space; there are an infinite number of a priori relations between colors. Second, I confront the (...) geometrical approach with Wittgenstein’s grammatical approach, contending that the latter escapes these difficulties. (shrink)

We investigate the Fourth Spatial Dimension, also known as ‘hyperspace’, by researching the capabilities of the human senses from the perspective of art and technology. The geometric approach of the fourth spatial dimension is studied through mathematical logic and the properties of simple geometric hyper-solids are examined. Focusing on the different ways that scientists and artists approached the Hyperspatial cognitive perception, we propose new aesthetic approaches by researching the capabilities of the human senses/bio-sensors and the brain. We present an (...) interactive art installation for approaching the hyper-sound, with reference to the Varèse and Helmholtz description, but also in conjunction with gravitational waves, an interactive art application about visualization of Hyperspace resulting from our evolution of the Hinton Coloured Cubes Method and an artistic approach to connect Johan Van Manen’s intuitive ‘hypersphere’ with the Quantum Geometry of String Theory, proposing a new cosmological model. Through these methods, a reasoning is developed between the objective scientific thought and the freedom that Art offers generating a practical result: a series of innovative audio and visual stimuli created via innovative audio-visual digital technology. Hence, we merge computer science with sensory technology and artistic knowhow, to define the required knowledge base that today can be found in an emerging professional, the ‘technartist’. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:62. 'HUME ON SPACE AND GEOMETRY': ONE RESERVATION In so far as Rosemary Newman disagrees with any2 thing said in my 'Infinite Divisibility in Hume's Treatise ' - which seems, happily, not to be so very far - I hasten to report that I am now persuaded. Thus my suggested reason for refusing to allow that an impression of blackness could give rise to the idea of extension (...) was not, after all, 3 Hume's. And no doubt his conviction that whatever ^s capable of being divided in_ infinitum, must consist of an infinite number of parts (T26) was in part sustained by "the atomistic type of phenomenalism advanced in the first Part 4 of Book I of the Treatise". For to any objection that this nowadays so plainly unacceptable conviction had been accepted as obviously correct by such formidable predecessors as Berkeley and Hobbes, it could be replied that they too had their commitments to their own brands of atomism. In her further comments upon "Hume's phenomenalist atomism" Newman notices, as "a very curious claim", his statement that when - like the true philosopher he was! - he locked at a table:- My senses convey to me only the impressions of colour 'd points, dispos 'd in a certain manner (T34). So I was sorry that she seems to have missed the bold priority claim which, referring to the same passage, I made on Hume's behalf in that article: "Anyone familiar with the theories and the paintings of Seurat might also mischievously characterize the Hume of this Section as 'the Father of Pointillisme' "! But at the end of her paper, although I think only at the end, Newman begins to go very wrong indeed. She says: "Turning to the Enquiry there is little to indicate any shift in Hume's opinion of geometry as a body of synthetic but necessary knowledge." After remarking that Hume does not repeat "any of his earlier arguments" against infinite divisibility "nor those in support of a system of mathematical 63. points", she notices that, in the paragraph characterizing propositions stating only the relations of ideas, geometry "is classed as a science alongside arithmetic and algebra (E25) ".7 Next she quotes the end of that paragraph, and at once comments: "Flew thinks that in the Enquiry Hume is to be seen as 'restoring pure geometry to its place alongside Q the other two elements in the trinity',..." Newman does not, so far as I can see, offer any reason for dismissing this suggestion - apart, that is, from asserting that the paragraph from which she has just quoted "can be interpreted so as to remove any apparent inconsistency with the claim that the impossibility of conceiving the negation of a geometrical axiom has the same ground in the Enquiry as in the Treatise, namely, the invariant character of sensible space." She just tells us that "Where geometry is concerned I follow Atkinson's opinion in finding no reason to suppose that 9 Hume abandoned the Treatise stand." This I find, to say the least, surprising since the very sentence of Hume's Philosophy of Belief from which she quoted my contrary opinion does provide what I still consider to be a quite decisive reason. Since she does not discuss either my reason or the Hume passage to which I was referring, I can here do nothing else or better than repeat myself; adding only and by the way that Professor Atkinson's paper 'Hume on Mathematics' was published a year before that book:Besides restoring pure geometry to its place alongside the other two elements of the trinity, the Inquiry also sketches an account of applied mathematics. This is something the earlier book did not provide at all. 'Every part of mixed mathematics proceeds on the supposition that certain laws are established by nature in her operations, and abstract reasonings are employed, either to assist experience in the discovery of these laws or to determine their influence in particular instances... Thus it is a law of motion, discovered by experience, that the moment or force of any body in motion is in 64. the compound ratio or proportion of its solid contents and its velocity, and, consequently... (shrink)

We classify two of Bertrand Russell's theories of events within the point-free ontology. The first of such approaches was presented informally by Russell in ‘The World of Physics and the World of Sense’ (Lecture IV in Our Knowledge of the External World of 1914). Based on this theory, Russell sketched ways to construct instants as collections of events. This paper formalizes Russell's approach from 1914. We will also show that in such a reconstructed theory, we obtain all axioms of Russell's (...) second theory from 1936 and all axioms of Thomason's theory of events from 1989. Russell's work certainly influenced the works of Stanisław Leśniewski, his student Alfred Tarski, and Czesław Lejewski – prominent members of the Lvov-Warsaw School (LWS). We see our work in the tradition of the research of Leśniewski and Tarski. Building on the technical tools developed in this environment and in the spirit of the traditional research of the LWS, we engage here, in particular, with two classic works by Russell on fundamental ontology. (shrink)

According to the conventionalist doctrine of space elaborated by the French philosopher-scientist Henri Poincaré in the 1890s, the geometry of physical space is a matter of definition, not of fact. Poincaré’s Hertz-inspired view of the role of hypothesis in science guided his interpretation of the theory of relativity (1905), which he found to be in violation of the axiom of free mobility of invariable solids. In a quixotic effort to save the Euclidean geometry that relied on this (...) axiom, Poincaré extended the purview of his doctrine of space to cover both space and time. The centerpiece of this new doctrine is what he called the ‘‘principle of physical relativity,’’ which holds the laws of mechanics to be covariant with respect to a certain group of transformations. For Poincaré, the invariance group of classical mechanics defined physical space and time (Galilei spacetime), but he admitted that one could also define physical space and time in virtue of the invariance group of relativistic mechanics (Minkowski spacetime). Either way, physical space and time are the result of a convention. (shrink)

According to the conventionalist doctrine of space elaborated by the French philosopher-scientist Henri Poincaré in the 1890s, the geometry of physical space is a matter of definition, not of fact. Poincaré's Hertz-inspired view of the role of hypothesis in science guided his interpretation of the theory of relativity (1905), which he found to be in violation of the axiom of free mobility of invariable solids. In a quixotic effort to save the Euclidean geometry that relied on this (...) axiom, Poincaré extended the purview of his doctrine of space to cover both space and time. The centerpiece of this new doctrine is what he called the "principle of physical relativity," which holds the laws of mechanics to be covariant with respect to a certain group of transformations. For Poincaré, the invariance group of classical mechanics defined physical space and time (Galilei spacetime), but he admitted that one could also define physical space and time in virtue of the invariance group of relativistic mechanics (Minkowski spacetime). Either way, physical space and time are the result of a convention. (shrink)

This paper argues that Socrates’ second definition of figure in Plato’s Meno (76a5–7) is deliberately insufficient: It states only a necessary condition for something’s being a figure, not a condition that is necessary as well as sufficient. For although it is true that every figure (in plane geometry) is (or corresponds to) a limit of a solid, not every limit of solid is a figure, i.e. not if the solid has a curved surface. It is argued that this mistake (...) is one Meno was meant to detect, since one of the three concepts he has been asked to agree to beforehand (75e1–76a3), namely the concept ‘plane’, has not been used in the definition. If this concept is put to work, we get a proper definition of figure, i. e. figure is the plane limit of a solid. Meno’s failure shows that he did not grasp the import of Socrates’ first definition of figure (at 75b9–1), and hence not what this example was meant to show him on a definition generally. Socrates’ first account of figure did indeed state a proper definition of figure, taking the definiens to be a necessary as well as a sufficient condition for the definiendum, as shown by the two words standing in for quantifiers, ‘only’ and ‘always’. Some general lessons about the importance of the dramatic elements in a Platonic dialogue are drawn from this interpretation. (shrink)

The occasion for this collection of four essays—by Vlastos, Ostwald, Callahan, and Solmsen—was Plato’s 2400th birthday in 1974. We note at once the book’s most disappointing and inexplicable flaw: it includes no example of North’s own fine classical scholarship and luminous understanding of the spirit of Platonic thought. Ostwald’s essay, ranging over much of the Platonic corpus, tills the well-plowed field of Plato’s contribution to the nomos-physis problem. His thesis is clear and familiar: Plato introduces new objects, eide, into nature, (...) the knowledge of which objects founds both the techne [[sic]] of the true king and justice. The science of such knowledge is the bulwark against Thrasymachus’s, Callicles', and Glaucon’s arguments concerning ‘natural law'. Ostwald seems unwilling, though, to grapple with the practical details of this techne [[sic]] in the political sphere. Rep. 546b-e and Polit. 310d ff. are programmatic, not substantive, and supply more puzzlement than answer. Callahan takes no smaller a topic than dialectic, myth, and history in Plato. His most interesting claim is that myth provides a setting for dialectic; each dialogue, he argues, is itself a myth, providing a locus wherein mythical topics, historical personages, and records of dialectical examinations can themselves be examined. Solmsen’s quite suggestive essay asks why Plato should investigate the physical world, which is intrinsically less than fully knowable, as well as being an insufficient source of guidance for man on issues of justice, moderation, and conduct. Though pointing out that physical bodies and their science must be of lesser rank than dialectic, Solmsen reminds us that not only did bodies provide a starting-point for mathematical demonstrations, and the incentive for solid geometry, but also that the planetary motions provided a taxis, an order, which was an image of an ordered cosmos, itself an image of an ordered eidetic realm. (shrink)

An account of how the mathematical sciences turn the mind away from becoming and towards being. There are four main conclusions. 1. The study of numbers, when treated independently of the other sciences, uses a particular conception of the nature of numbers to detach the mind from the influence of perceptible objects. 2. The study of ratios and proportions, explicitly the core of Plato's harmonics, is fundamental also to plane and solid geometry and astronomy. 3. Ratios and proportion form (...) the basis for the synoptic vision of all the mathematical sciences. This vision is an essential preparation for dialectic because it makes mathematicians become aware of the limitations of the mathematics they have been engaged in but it also shows them how they can begin to overcome those limitations. 4. The main purpose of the study of numbers, treated finally in relation to the other sciences, is to calculate ratios. (shrink)

Spherical geometry studies the sphere not simply as a solid object in itself, but chiefly as the spatial context of the elements which interact on it in a complex three-dimensional arrangement. This compels to establish graphical conventions appropriate for rendering on the same plane—the plane of the diagram itself—the spatial arrangement of the objects under consideration. We will investigate such “graphical choices” made in the Theodosius’ Spherics from antiquity to the Renaissance. Rather than undertaking a minute analysis of every (...) particular element or single variant, we will try to uncover the more general message each author attempted to convey through his particular graphical choices. From this analysis, it emerges that the different kinds of representation are not the result of merely formal requirements but mirror substantial geometrical requirements expressing different ways of interpreting the sphere and testify to different ways of reasoning about the elements that interact on it. (shrink)

Name der Zeitschrift: Logic and Logical Philosophy Jahrgang: 22 Heft: 3 Seiten: 255-293.

The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. (...) Specifically, we want to see in which ways geometers’ preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been influenced by reflections on duality. (shrink)

Dans un discours prononcé à Zurich vers la fin des années 1940, Hermann Weyl a examiné l'épistémologie dialectique de Ferdinand Gonseth et l'a considérée comme trop strictement limitée aux aspects de changement historique. Son expérience de la philosophie diaclectique post-kantienne, en particulier la dérivation du concept de l'espace et de la matière chez Johann Gottlieb Fichte, avait constitué une base dialectique solide pour ses propres études de 1918 en une géométrie purement infinitésimale et la théorie antérieure d'un champ de matière (...) géométriquement unifié (en prolongement du programme Mie-Hilbert). Bien que Weyl se fût alors éloigné des aspects spéculatifs de la philosophie de sa jeunesse et eût montré, en particulier, moins d'enthousiasme qu'auparavant pour Fichte, il nourrissait encore de nombreux doutes sur la base culturelle de la science mathématique moderne et sur son rôle dans la culture matérielle du modernisme de pointe. Pour Weyl, la « réflexion » philosophique était une nécessité culturelle ; il se tourna alors vers lexistentialisme de Karl Jaspers et de Martin Heidegger pour trouver des raisons plus profondes, dans un mouvement comparable à son orientation vers la philosophie de Fichte après la Première Guerre mondiale. Le débat de la fin des années 1940 peut être compris comme une sorte de supplément postérieur à la Seconde Guerre mondiale par rapport à ses commentaires plus connus sur les mathématiques et les sciences naturelles publiés au milieu des années 1920. (shrink)

This largely expository lecture deals with aspects of traditional solid geometry suitable for applications in logic courses. Polygons are plane or two-dimensional; the simplest are triangles. Polyhedra [or polyhedrons] are solid or three-dimensional; the simplest are tetrahedra [or triangular pyramids, made of four triangles]. -/- A regular polygon has equal sides and equal angles. A polyhedron having congruent faces and congruent [polyhedral] angles is not called regular, as some might expect; rather they are said to be subregular—a word coined (...) for this lecture. To repeat, a subregular polyhedron has congruent faces and congruent [polyhedral] angles. A subregular polyhedron whose faces are all regular polygons is regular—using standard terminology. -/- Geometers before Euclid showed that there are “essentially” only five regular polyhedra: every regular polyhedron is a tetrahedron (4 faces), a hexahedron or cube (6 faces), an octahedron (8 faces), a dodecahedron (12 faces), or an icosahedron (20 faces). -/- The first question is whether there are subregular polyhedra that are not regular. For example, are there tetrahedra having congruent angles and congruent triangular faces but whose faces are not equilateral triangles? -/- Another question is the classification of subregular polyhedra if they exist. For example, considering the fact that the regular tetrahedra all have equilateral triangles as faces, we ask which triangles other than equilaterals are faces of subregular tetrahedra. Similarly, considering the fact that the regular hexahedra all have squares as faces, we ask which quadrangles other than squares are faces of subregular hexahedra. -/- After introductory remarks that include historical and philosophical points, we concentrate on tetrahedra. A triangle that is congruent to each of the four faces of a tetrahedron is called a generator of the tetrahedron. The main result proved is that every acute triangle is a generator of a subregular tetrahedron. The proof includes an algorithm –implementable with scissors and paper –that constructs from any given acute triangle a subregular tetrahedron whose faces are congruent to the given triangle. -/- Algorithm: Given any acute triangle. Construct a similar triangle whose sides are double the sides of the given triangle. Draw the three lines connecting the three midpoints of the sides (making four triangles congruent to the given triangle—a central triangle surrounded by three peripheral triangles). Make three “hinges” along the lines connecting the midpoints. “Fold” the peripheral triangles together (into a tetrahedron). [LIGHTLY EDITED VERSION OF PRINTED ABSTRACT] Acknowledgements: William Lawvere, Colin McLarty, Irvin Miller, Frango Nabrasa, Lawrence Spector, Roberto Torretti, and Richard Vesley. -/- . (shrink)

In his Republic Plato considers grasping the unity of mathematics as the ultimate goal of the mathematical studies in which the future philosopher-rulers must engage before they turn to philosophy. How the unity of mathematics is supposed to be understood is not explained, however. This book argues that Plato conceives of the unity of mathematics in terms of the mutually benefiting links between its branches, just as he conceives of the unity of the state outlined in the Republic in terms (...) of the common benefit for all citizens. Evidence for this view is provided by a discussion of his conception of astronomy as a propedeutic to philosophy, which can be best understood as hinting at a historically possible link between fourth-century-BC astronomy and solid geometry. The monograph also includes a detailed discussion of two well-known stories about Plato: not only he motivated Greek mathematicians to solve a difficult problem in solid geometry with his interpretation of a Delphic oracle given to the inhabitants of the island of Delos but he also posed the question which led to the development of the astronomical theory of homocentric spheres. It is argued that these stories are best understood as fictional episodes in Plato's life, constructed from passages in his works. (shrink)

According to the conventionalist doctrine of space elaborated by the French philosopher-scientist Henri Poincaré in the 1890s, the geometry of physical space is a matter of definition, not of fact. Poincaré’s Hertz-inspired view of the role of hypothesis in science guided his interpretation of the theory of relativity (1905), which he found to be in violation of the axiom of free mobility of invariable solids. In a quixotic effort to save the Euclidean geometry that relied on this (...) axiom, Poincaré extended the purview of his doctrine of space to cover both space and time. The centerpiece of this new doctrine is what he called the ‘‘principle of physical relativity,’’ which holds the laws of mechanics to be covariant with respect to a certain group of transformations. For Poincaré, the invariance group of classical mechanics defined physical space and time (Galilei spacetime), but he admitted that one could also define physical space and time in virtue of the invariance group of relativistic mechanics (Minkowski spacetime). Either way, physical space and time are the result of a convention. (shrink)

The evidence for a kinematics of the mind is confounded by uncontrolled properties of pictures. Effects of illumination and of picture-plane geometry may underlie some evidence given for a process of mental rotation. Pictured rotation is confounded by picture similarity, gauged by gray -level correlations. An example is given involving the depicted rotation of Shepard-Metzler solids in depth. [Hecht; Kubovy & Epstein; Shepard; Todorovic].

The geometry of the state space of a finite-dimensional quantum mechanical system, with particular reference to four dimensions, is studied. Many novel features, not evident in the two-dimensional space of a single spin, are found. Although the state space is a convex set, it is not a ball, and its boundary contains mixed states in addition to the pure states, which form a low-dimensional submanifold. The appropriate language to describe the role of the observer is that of flag manifolds.

This book explores and compares the reflections on space and quantity found in the works of five philosophers: Spinoza, Leibniz, Bergson, Whitehead, and Deleuze. What unites these philosophers is a series of metaphysical concerns rooted in 17th-century rationalism and embraced in 20th-century philosophies of process and difference. At the heart of these concerns is the need for a comprehensive metaphysical account of the diversity and individuality of things. This demand leads to a shared critique of Cartesian and Newtonian conceptions of (...) space. The most problematic aspect of those notions of space is homogeneity. In essence, uniform space fails to explain the differences between locations, thus violating the principle of sufficient reason. Cartesian and Newtonian theories of space thereby fail to meet the metaphysical requirement for explaining diversity and individuality. The traditional concept of quantity faces similar issues. Motivated by these problems, these five philosophers develop an alternative conception of space and quantity. By examining these theories, the book sheds new light on an unexplored relation between rationalism and 20th-century Continental philosophy. A Geometry of Sufficient Reason will appeal to scholars and graduate students working in Continental philosophy, history of philosophy, metaphysics, and the history and philosophy of science. (shrink)

There is a venerable position in the philosophy of space and time that holds that the geometry of spacetime is conventional, provided one is willing to postulate a “universal force field.” Here we ask a more focused question, inspired by this literature: in the context of our best classical theories of space and time, if one understands “force” in the standard way, can one accommodate different geometries by postulating a new force field? We argue that the answer depends on (...) one’s theory. In Newtonian gravitation the answer is yes; in relativity theory, it is no. (shrink)

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. (shrink)

Picture geometry is often regarded as an area of technical knowledge that accompanies or provides useful information for basic research on visual culture and almost never as a methodological one. Despite the historical and conceptual connections between mathe­matics and the visual, even a basic geometric competence is by no means a common of image and visual culture researchers. At the same time, the overwhelming majority of this kind of work belong to the field of technical knowledge, the history of (...) mathemat­ics, or positivism based theories; in other words, they do not address the most important issues for humanities research. Rare non-technical works in this area do not find due recognition and dissemination among a wider community of specialists. The article offers the main result of the research, the subject of which was pictorial depth. Pictorial depth is the version of the specific experience of distance with which the image is traditionally associated. The problem of distance is a classical area of philosophical reflection on the image. The article presents an attempt to geometrically express one of its versions, namely pictorial and visual depth. At the same time, the important dimension of inacces­sibility and heterogeneity is preserved, that is, the experience of depth is not reduced to visual data or the literal geometry of a flat picture. By constructing a geometrically object that can be considered depth, it is possible to extract intuitively non-obvious properties that would not be available through direct analysis of experience and other methods of naked reflection. The article presents only some of the findings obtained during the study. (shrink)

Fix an algebraically closed field of characteristic zero and let G be its geometry of transcendence degree one extensions. Let X be a set of points of G. We show that X extends to a projective subgeometry of G exactly if the partial derivatives of the polynomials inducing dependence on its elements satisfy certain separability conditions. This analysis produces a concrete representation of the coordinatizing fields of maximal projective subgeometries of G.

The political integration of the European Union is fragile for many reasons, not least the reassertion of nationalism. That said, if we examine specific practices and infrastructures, a more complicated story emerges. We juxtapose the political fragility of the EU in relation to the ongoing formation of data infrastructures in official statistics that take part in postnational enactments of Europe’s populations and territories. We develop this argument by analyzing transformations in how European populations are enacted through new technological infrastructures that (...) seek to integrate national census data in “cubes” of cross-tabulated social topics and spatial “grids” of maps. In doing so, these infrastructures give meaning to what “is” Europe in ways that are both old and new. Through standardization and harmonization of social and geographical spaces, “old” geometries of organizing and mapping populations are deployed along with “new” topological arrangements that mix and fold categories of population. Furthermore, we consider how grids and cubes are generative of methodological topologies by closing the distances or differences between methods and making their data equivalent. By paying attention to these practices and infrastructures, we examine how they enable reconfiguring what is known and imagined as Europe and how it is governed. (shrink)

Photonic Crystals is the first book to address one of the newest and most exciting developments in physics--the discovery of photonic band-gap materials and their use in controlling the propagation of light. Recent discoveries show that many of the properties of an electron in a semiconductor crystal can apply to a particle of light in a photonic crystal. This has vast implications for physicists, materials scientists, and electrical engineers and suggests such possible developments as an entirely optical computer. Combining cutting-edge (...) research with the basic theoretical concepts behind photonic crystals, the authors present to undergraduates and researchers a concise, readable, and comprehensive text on these novel materials and their applications. The first chapters develop the theoretical tools of photonic crystals in a broad, intuitive fashion, starting from nothing more than Maxwell's equations and Fourier analysis, and include analogies to traditional solid-state physics and quantum theory. There follows an investigation of the unique phenomena that take place within photonic crystals, at defect sites, and at surfaces and interfaces. The authors offer a new treatment of the traditional multilayer film (a one-dimensional photonic crystal), which allows for the extension to higher dimensions and more complex geometries. After exploring the capabilities of photonic crystals to guide and localize light, the authors demonstrate how these notions can be put to work. (shrink)

In 2003, Peratt demonstrated that rock art images worldwide bear a remarkable similarity to high-energy plasma discharge formations. In later papers, Peratt located the plasma discharge column in which all of these would have occurred at the Earth’s South Pole. This article accepts the relation between the rock art images and the plasma formations, but concludes that the geometry of the reconstruction is incompatible with the global occurrence of the rock art images. As a corollary, the finer details of (...) the reconstructed column must also be called into question. In particular, the reconstruction of the top cusp, the two upper plasmoids, and the filamentary sheath in a single column at the South Pole cannot be reliably deduced from the data as presented by Peratt. All evidence points to a worldwide distribution of the phenomena. (shrink)

This note extends earlier results on geometrical interpretations of the logic KR to prove some additional results, including a simple undecidability proof for the four-variable fragment of KR.