In this paper I draw a clear and precise distinction between pure or mathematical geometry and applied or physical geometry. I make this distinction inside two contexts : one, the reflections about foundations of geometry due to the source of non-Euclidean geometry and, other one, the discussions by the logical positivists on general structure of empirical theories. In particular, such and like propose the logical positivists, I defend that pure geometry is a formal system (...) that doesn’t tell us anything about physical reality, whereas applied geometry is a theory about physical space that comes of to interpret a mathematical geometry. I support this thesis in some Einstein ’s ideas about the general theory of relativity. Finally, even though this picture of structure of physical geometry is relatively appropriate, I insist on the thesis of that main mistake of logical empiricist philosophy would be in making of this picture the dominant character of structure of scientific theories in general. (shrink)

En este artículo se traza una distinción clara y precisa entre geometría pura y geometría aplicada dentro del marco de las reflexiones sobre los fundamentos de la geometría promovidas por la aparición de geometrías no-euclidianas y en el contexto de las discusiones mantenidas por los empiristas lógicos sobre la estructura general de las teorías empíricas. De manera más particular, se defiende, tal y como proponen los empiristas lógicos, que una geometría pura es un sistema formal que no nos dice nada (...) sobre la realidad física, mientras que una geometría aplicada es una teoría empírica, una teoría física (una teoría del espacio) que resulta de dotar de significado a una geometría matemática. Para sostener esta tesis se recurre, en parte, a ciertas ideas expresadas por Einstein sobre la teoría general de la relatividad. Por último, si bien parece que esta imagen de la estructura de una geometría física es relativamente adecuada, se insiste en la tesis de que el error principal de los empiristas lógicos estaría entonces en pretender hacer de ella el carácter predominante de la estructura de las teorías científicas en general. (shrink)

The purpose of this work is to address what notion of geometrical object and geometrical figure we have in different kinds of geometry: practical, pure, and applied. Also, we address the relation between geometrical objects and figures when this is possible, which is the case of pure and applied geometry. In practical geometry it turns out that there is no conception of geometrical object.

With In Focus Sacred Geometry, learn the fascinating history behind this ancient tradition as well as how to decipher the geometrical symbols, formulas, and patterns based on mathematical patterns. People have searched for the meaning behind mathematical patterns for thousands of years. At its core, sacred geometry seeks to find the universal patterns that are found and applied to the objects surrounding us, such as the designs found in temples, churches, mosques, monuments, art, architecture, and nature. Learn (...) the fundamental principles behind: Interpreting the sacred symbolism and principles behind shapes Calculating numbers used in sacred geometry Applying the golden ratio and Fibonacci’s Sequence to objects Translating symbolic and geometrical letters and alphabets This accessible and beautifully designed guide to sacred geometry includes a frameable poster of the main sacred geometrical shapes and their unique, innate arrangements. The In Focus series applies a modern approach to teaching the classic body, mind, and spirit subjects. Authored by experts in their respective fields, these beginner's guides feature smartly designed visual material that clearly illustrates key topics within each subject. As a bonus, each book includes reference cards or a poster, held in an envelope inside the back cover, that give you a quick, go-to guide containing the most important information on the subject. (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)

Wittgenstein famously ended his Tractatus Logico-Philosophicus (Wittgenstein 1922) by writing: "Whereof one cannot speak, one must pass over in silence." (Wovon man nicht sprechen kann, darüber muss man schweigen.) In that earliest work, Wittgenstein gives no clue about whether this aphorism applied to animal minds, or whether he would have included philosophical discussions about animal minds as among those displaying "the most fundamental confusions (of which the whole of philosophy is full)" (1922, TLP 3.324), but given his later writings (...) on language and thought, it seems a plausible hypothesis. Years later he wrote in the Philosophical Investigations (1968, p. 223) another aphoristic statement: "If a lion could speak, we would not understand him." (Wenn der Löwe sprechen könnte, wir könnten ihn nicht verstehen.) Decades of philosophical discussion about what Wittgenstein meant by these remarks illustrate another point: when philosophers speak, we do not fully understand them. (shrink)

Euclidean geometry, statics, and classical mechanics, being in some sense the simplest physical theories based on a full-fledged mathematical apparatus, are well suited to a historico-philosophical analysis of the way in which a physical theory differs from a purely mathematical theory. Through a series of examples including Newton’s Principia and later forms of mechanics, we will identify the interpretive substructure that connects the mathematical apparatus of the theory to the world of experience. This substructure includes models of experiments, models (...) of measurement, and modular connections with partial theories. It evolves during the life of a theory as physicists learn how to apply it in various contexts. It should nevertheless be regarded as an integral part of a genuine physical theory since the theory would otherwise degenerate into pure mathematics. (shrink)

The paper presents the diagrammatic calculus CL, which combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. In its basic form, `CL' (= Cubus Logicus) organizes terms in the form of a square or cube. By applying the arrows of the square of opposition to CL, judgments and inferences can be displayed. Thus CL offers on the one hand an intuitive method to display ontologies and on the other hand a diagrammatic tool to check inferences. The paper focuses (...) mainly on the adaptation of the square of opposition in CL and offers an algebraic notation, which corresponds to the diagrammatic representation. (shrink)

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of (...) physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl. (shrink)

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of (...) physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl. (shrink)

In the central chapters of Leviathan, Hobbes offers a demonstration of the "true doctrine of the laws of nature," which is identified with the "science of virtue andvice" and the "true moral philosophy." In his deduction of the laws of nature, Hobbes attempts to mimic the science of geometry, which he says is the "only science God had hitherto bestowed on mankind. "In this paper, I discuss some of the problems associated with Hobbes's application of the method of (...) class='Hi'>geometry to civil philosophy. After locating the root of these problems in Hobbes's in ability to recognize the distinction between formal and applied sciences, I discuss a possible solution. According to this solution, Hobbes's "science of morality" is considered to be a formal science that is applied to the world by an act of human creation. (shrink)

Summary In his Theoremata de lumine, et umbre (1521), Francesco Maurolyco (1494–1575) discussed, inter alia, the problem of the pinhole camera. Maurolyco outlined a framework based on Euclidean geometry in which he applied the rectilinear propagation of light to the casting of shadow on a screen behind a pinhole. We limit our discussion to the problem of how the image behind an aperture is formed, and follow the way Maurolyco combined theory with instrument to solve the problem of (...) the projection of light through small apertures. We show that Maurolyco not only reformed the classical sources which, he thought, were no longer the authoritative code of textual knowledge, but also established with the dioptra a novel linkage of method, theory, and instrument. He thereby demonstrated the importance of optics to the science of astronomy. (shrink)

In this paper, we will make explicit the relationship that exists between geometric objects and geometric figures in planar Euclidean geometry. That will enable us to determine basic features regarding the role of geometric figures and diagrams when used in the context of pure and applied planar Euclidean geometry, arising due to this relationship. By taking into account pure geometry, as developed in Euclid’s Elements, and practical geometry, we will establish a relation between geometric objects (...) and figures. Geometric objects are defined in terms of idealizations of the corresponding figures of practical geometry. We name the relationship between them as a relation of idealization. This relation, existing between objects and figures, is what enables figures to have a role in pure and applied geometry. That is, we can use a figure or diagram as a representation of geometric objects or composite geometric objects because the relation of idealization corresponds to a resemblance-like relationship between objects and figures. Moving beyond pure geometry, we will defend that there are two other ‘layers’ of representation at play in applied geometry. To show that, we will consider Euclid’s Optics. (shrink)

§1. Introduction. With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever one-off. Still others regarded the argument as magical and asked whether such sorcery could unlock (...) the secrets of a wide coterie of number theoretic problems.In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields. (shrink)

The literary and critical discourse about characters and characterization in Anglophone drama and fiction since the Renaissance shows a persistent but underrecognized presence of three idioms and vocabularies, two highly developed and one nascent, that either derive from the rhetoric of mathematics in classical antiquity or participate in its modern afterlife. Those discourses—which this article studies in detail—are, first, an explicitly Theophrastan one, in which taxonomies of character are constructed; second, an explicitly Euclidean one, in which characterization is discussed and (...) accomplished in relatively conventional or commonsensical geometric terms; and third, a non- or post-Euclidean one, in which characterization is discussed and inchoately accomplished in the terms of the “new geometries” that emerged during the nineteenth century. What taxonomy in the Aristotelian mode and geometry in the Euclidean mode have in common is that when their vocabularies are applied to characterization, they delimit characters in terms of established categories—whether of ideal shapes or statistically probable types—while discounting whatever features may be unique to individuals. An obliviousness to these three discourses can limit seriously what can be said about, and what can be said on behalf of, the literary and critical texts framed in their terms and also, most importantly, what can be said about the nature of literary characters. (shrink)

Spatial perception and spatial representation are not less central to experimental psychology than to visual art. Geometry allows their description and formalization. Therefore, geometrical language can be considered as a kind of generative grammar, which is embedded in the human perceptual experience of space. The paper outlines the suggestion that Euclidean geometry, along with most perspective geometries, even when applied to geometrical problem solving, have phenomenal bases, since they emerge from direct experience of the world, and not (...) necessarily from higher order cognitive processes. (shrink)

Historians of modern philosophy have been paying increasing attention to contemporaneous scientific developments. Isaac Newton's Principia is of course crucial to any discussion of the influence of scientific advances on the philosophical currents of the modern period, and two philosophers who have been linked especially closely to Newton are John Locke and Immanuel Kant. My dissertation aims to shed new light on the ties each shared with Newtonian science by treating Newton, Locke, and Kant simultaneously. I adopt Newton's philosophy of (...) geometry as the starting point of investigation, for here I believe we have a constructive means by which to assess Locke and Kant's relationship to Newton, In particular, I defend the thesis that the justification Newton, Locke and Kant offer for applying geometrical principles to nature is central to understanding their respective ties to a Newtonian science characterized by the intermingling of mathematics and experiment, Although little is said by Locke in regard to a mathematical approach to nature, I hope to show that his interpretation of the origins of our geometrical ideas has a close affinity to Newton's own characterization of geometry, leading us to reexamine the extent of Locke's 'Newtonianism.' Kant famously attempts to bridge the gap between geometry and the empirical world by establishing space as a "pure form of intuition." My discussion of Kant's Newtonian approach to nature centers on the imagination, and I argue that the mediating work completed by this faculty in geometrical construction and experience in general is equally important to understanding Kant's application of geometry to the empirical realm, In the end, I hope my treatment of the strategies employed by Newton, Locke, and Kant to account for a mathematical-experimental method of natural philosophy sheds further light on the importance of Newton to the progress of modern philosophy. (shrink)

Cartesian method both organizes and impoverishes the domains to which Descartes applies it. It adjusts geometry so that it can be better integrated with algebra, and yet deflects a full-scale investigation of curves. It provides a comprehensive conceptual framework for physics, and yet interferes with the exploitation of its dynamical and temporal aspects. Most significantly, it bars a fuller unification of mathematics and physics, despite Descartes' claims to quantify nature. The work of his contemporaries Galileo and Torricelli, and of (...) his successor Newton, illustrates conceptual possibilities Descartes left aside, due to his attachment to method. (shrink)

The first edition of Newton’s Principia opens with a “Praefatio ad Lectorem.” The first lines of this Preface have received scant attention from historians, even though they contain the very first words addressed to the reader of one of the greatest classics of science. Instead, it is the second half of the Preface that historians have often referred to in connection with their treatments of Newton’s scientific methodology. Roughly in the middle of the Preface, Newton defines the purpose of philosophy (...) as a twofold task: to investigate the forces of the phenomena of nature and, once having established the forces, to demonstrate the remaining phenomena. Newton then introduces a distinction between the first two books, which deal with general propositions, and the third, where the propositions are applied to particular instances of celestial phenomena. From these phenomena, Newton claims, the force of gravity, thanks to which bodies tend toward the Sun, is derived. By assuming this force in mathematical propositions, other motions are deduced: the motions of the planets, the comets, the moon, and the sea. Newton then declares his hope that phenomena relative to small particles will also be explained, as per the celestial ones, thanks to the understanding of attractive forces hitherto unknown to philosophers. The Preface ends with a well-deserved laudatio of Edmond Halley and an apologetic passage concerning certain imperfections in the presentation of advanced subjects, such as the motions of the moon. It is these lines to which historians have given most attention in their various studies. (shrink)

Abstract The first fractal constructions appeared in mathematics in the second half of the 19th century. Their history is divided into two periods. The first period lasted 100 years and is a good example of the method of proofs and refutations discovered by Lakatos. The modern history of these objects started 20 years ago, when Mandelbrot decided to create fractal geometry, a general theory concentrated on specific properties of fractals. His approach has been surprisingly effective. The aim of this (...) paper is to examine the reasons for Mandelbrot's success and for the present popularity of fractals. They are now known not only in mathematics but also in many fields of natural, social and applied sciences, and even in the arts. (shrink)

For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems (...) for representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems. (shrink)

The paper discusses the similarity between geometry, arithmetic, and logic, specifically with respect to the question of whether applied theories of each may be revised. It argues that they can - even when the revised logic is a paraconsistent one, or the revised arithmetic is an inconsistent one. Indeed, in the case of logic, it argues that logic is not only revisable, but, during its history, it has been revised. The paper also discusses Quine's well known argument against (...) the possibility of logical deviancy. (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 paper discusses the similarity between geometry, arithmetic, and logic, specifically with respect to the question of whether applied theories of each may be revised. It argues that they can - even when the revised logic is a paraconsistent one, or the revised arithmetic is an inconsistent one. Indeed, in the case of logic, it argues that logic is not only revisable, but, during its history, it has been revised. The paper also discusses Quine's well known argument against (...) the possibility of “logical deviancy”. (shrink)

We establish the Robinson joint consistency theorem for the infinite-valued propositional logic of Łukasiewicz. As a corollary we easily obtain the amalgamation property for MV-algebras—the algebras of Łukasiewicz logic: all pre-existing proofs of this latter result make essential use of the Pierce amalgamation theorem for abelian lattice-ordered groups together with the categorical equivalence Γ between these groups and MV-algebras. Our main tools are elementary and geometric.

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical (...) geometry, some of which are basically explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

An intermediate stage in Hrushovski’s construction of flat strongly minimal structures in a relational language L produces ω-stable structures of rank ω. We analyze the pregeometries given by forking on the regular type of rank ω in these structures. We show that varying L can affect the isomorphism type of the pregeometry, but not its finite subpregeometries. A sequel will compare these to the pregeometries of the strongly minimal structures.

There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are “the other way around”). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is understoood as (...) involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators—Post’s and Kleene’s—and we provide a simple group-theoretic argument demonstrating their generative power. (shrink)

Those familiar with the Critique of Pure Reason will not at all be surprised that Thomas C. Vinci has found it fitting to dedicate an entire book to the Transcendental Deduction of the Categories, a chapter of the CPR that is as important to Kant’s argument for Transcendental Idealism as it is difficult to decipher. The purpose of that section is to establish the objective validity of the categories—to show, that is, that the pure concepts of the understanding apply to (...) all objects of human experience. While the general goal of the TD may be easy enough to state, the argument strategy that Kant uses to establish the objective validity of the categories is hardly easy to understand.Vinci focuses on the.. (shrink)

Most commentators agree that (part of what) Kant means by characterizing the propositions of geometry as synthetic is that they are not true merely in virtue of logic or meaning, and that this characterization has something to do with his views about the construction of geometrical concepts in intuition. Many commentators regard construction in intuition as an essential part of geometrical proofs on Kant’s view. On this reading, the propositions of geometry are synthetic because the geometrical theorems cannot (...) be proved in purely conceptual or logical terms. Other commentators see the main role of pure intuition and the figures constructed in pure intuition in that they provide a model for Euclidean geometry. On views of this kind, the propositions of geometry are synthetic because the geometrical axioms are substantive truths about one of our forms of intuition. On the interpretation proposed in this essay, what Kant means by claiming that the propositions of geometry are synthetic is not only that the Euclidean axioms and theorems cannot be reduced to tautologies or logical truths, but also that they apply to really possible objects. Construction in intuition plays no essential role in (what we now call) ‘pure’ geometry on Kant’s view. But the fact that the concepts of geometry can be constructed in intuition is of crucial importance in the context of Kant’s transcendental philosophy of geometry, because, among other things, it allows him to explain how Euclidean geometry is possible as an a priori synthetic science in the sense just indicated. (shrink)

Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive (...) axiomatization allows solutions to geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (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)

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)

In lieu of an abstract, here is a brief excerpt of the content:66. ' HUME ON SPACE AND GEOMETRY * : A REJOINDER TO FLEW ' S 'ONE RESERVATION '.? Flew' s reservation about my assertion that the Enquiry contains no significant revision of the Treatise conception of geometry as a body of necessary and synthetic knowledge, appears to involve two charges. Firstly, he alleges that I dismiss but offer no substantial argument against his own view that the (...) Enquiry restores pure geometry "to its place alongside the other two elements of the trinity ". Secondly, he thinks that I have committed a serious sin of omission in failing to take into consideration a certain passage in the Enquiry (page 31, section 27) which, in his opinion, constitutes "a quite decisive reason" for supposing that Hume abandoned the Treatise position, in that it contains an account of applied mathematics, a subject on which the Treatise is silent. To the charge of omission I plead guilty, and I shall presently seek to remedy this defect and to state why I do not share Flew' s view of the significance of the said passage. To the former charge I am not so ready to plead guilty, and I shall counter-charge that Flew has failed not only to do justice to my proposed reading of Enquiry page 25, section 20, but also to provide clear and convincing support for his own. One difficulty encountered in answering Flew lies in understanding whether or not his claim about the status of geometry as a pure science in the Enquiry is put forward independently of his interpretation of E31. As originally presented in Hume's Philosophy of Belief this appears to be the case, and if so, on what evidence does it rest? 2 In his book, Flew' s statement about Hume replacing pure geometry alongside arithmetic and algebra follows directly upon a paragraph in which he comments that Hume's references to geometrical propositions at E25 signify "an important advance from the position of the Treatise". For whereas the status of perfect precision and certainty was 67. there withheld from geometry (T71), Hume now speaks of the certainty and evidence of the propositions of Euclidean geometry, and declares such propositions to be discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. (E25) Now, I argued at some length in my paper (pp. 21-24) that Hume is prepared to grant scientific status to geometry in the Enquiry because he has come to regard its certainty as that proper to a mathematics of space. He has abandoned the Treatise contrast between an ideal or theoretical standard of equality, viewed therein as necessary to geometry considered as a science proper, and the merely sensible standard of equality upon which geometry as practised depends. It should not be forgotten that Hume recognises two kinds of certainty where "Relations of Ideas" are concerned, intuitive and demonstrative. Since he fails to provide any detail of the place of each of these in mathematics, the door is left open for speculation along the lines followed in pp. 28-29 of my paper. For there is nothing in?25 which implies that the certainty of all geometrical propositions is of the same kind, or that the process of geometrical demonstration is of the same logical nature as demonstrative reasoning in arithmetic and algebra. In his comment on my position Flew makes no reference to my proposed reason why Hume classes geometry as a science in the Enquiry.· Yet, even if this reclassification does mark an advance of a kind, the fact remains that E25 does not furnish incontrovertible evidence for that advance which Flew thinks he finds there. Nor, therefore, is it conclusive against my assertion that Hume in both works regards the propositions of geometry as necessary and synthetic truths. Given that Flew does not wish to deny the inclusion of a synthetic element even in the Enquiry account of mathematics (see H. P. B. pp. 64-66), I find it difficult to see why he supposes that the logical status of the propositions of geometry is different in the Enquiry from that in the Treatise, since in the latter too they are held 68. to be truths which depend... (shrink)

One of the mathematical topics examined in the Old Babylonian period consisted of calculating the size of a reed which was used to measure either a longitude or the perimeter of a rectangle or trapezium. These subjects were solved, probably, applying the geometric construction called completing the square. In this paper, we analyse the problem texts on the tablets AO 6770 (5), Str 368, VAT 7532, and VAT 7535.

This paper aims to show how the mathematical content of Hilbert's Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring (...) out how Hilbert's conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert's foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert's position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert's for the completeness of set theory. (shrink)

ABSTRACT A spatial logic is a modal logic of which the models are the mathematical models of space. Successively considering the mathematical models of space that are the incidence geometry and the projective geometry, we will successively establish the language, the semantical basis, the axiomatical presentation, the proof of the decidability and the proof of the completeness of INC, the modal multilogic of incidence geometry, and PRO, the modal multilogic of projective geometry.

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of (...) correct independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments. (shrink)

Berkeley in his Introduction to the Principles of Human knowledge uses geometrical examples to illustrate a way of generating “universal ideas,” which allegedly account for the existence of general terms. In doing proofs we might, for example, selectively attend to the triangular shape of a diagram. Presumably what we prove using just that property applies to all triangles.I contend, rather, that given Berkeley’s view of extension, no Euclidean triangles exist to attend to. Rather proof, as Berkeley would normally assume, requires (...) idealizing diagrams; treating them as if they obeyed Euclidean constraints. This convention solves the problem of representative generalization. View HTML Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

Following a theme which increasingly interests people concerned with problems of sentience, this paper describes how consciousness might encode information. The idea that awareness is inseparable from Bose-Einstein condensation in the brain is identified as the most promising of the ‘quantum consciousness’ notions. It can be inferred from this idea that brain Bose condensates will have a geometrical structure, analagous to that of tapestries, which can encode information and to which knot theory can be applied.These notions are able to (...) explain otherwise puzzling features of conscious information handling and a neuroanatomically testable prediction is derived from them. Finally it's pointed out that they imply that any ‘strong’ panpsychist position is unnecessary. (shrink)

This article is about a period of technology transfer – the late 1910s and 1920s – when wartime aerial reconnaissance techniques and operations were being adapted to a range of civilian uses, including urban planning, land use analysis, traffic control, tax equalization, and even archaeology. At the center of the discussion is the ‘photomosaic’: a patchwork of overlapping aerial photographs that have been rectified and fit together so as to form a continuous survey of a territory. Initially developed during the (...) First World War to provide coverage of fronts, photomosaic mapping was widely practiced and celebrated during the postwar years as an aid to urban development. The article traces both the refinements in photomosaic technology after the Armistice and the rhetorical means by which the form’s avant-garde wartime reputation was domesticated into an ‘applied realism’ that often effaced its site-specific perspective, its elaborately rectified optics, and the oppositionality of both its military and civilian uses. The article has a broader theoretical aim as well. Classic statements of both structuralist and post-structuralist spatial theory have produced an ossified geometry wherein the vertical is the axis of paradigm, top-down strategy, and manipulative distance and the horizontal the axis of syntagm, grassroots tactics, and resistant proximities and differences. In its close study of the technology and rhetoric surrounding interwar photogrammetry, the article provides an example of how one might reverse the long-standing misrecognition of high-altitude optics as effacing time, difference, and materiality – and what it might mean to view such optics as, instead, a resource in turning from abstract toward differential conceptions of both aerial photography and our theoretical habits. This turn I call ‘applied modernism’, a term that accesses both the wartime photomosaic’s affiliation with avant-garde painting and its insistence that portraits of the total are always projections from partial, specific vantages. (shrink)

How can we invent new certain knowledge in a methodical manner? This question stands at the heart of Salomon Maimon's theory of invention. Chikurel argues that Maimon's contribution to the ars inveniendi tradition lies in the methods of invention which he prescribes for mathematics. Influenced by Proclus' commentary on Elements, these methods are applied on examples taken from Euclid's Elements and Data. Centering around methodical invention and scientific genius, Maimon's philosophy is unique in an era glorifying the artistic genius, (...) known as Geniezeit. Invention, primarily defined as constructing syllogisms, has implications on the notion of being given in intuition as well as in symbolic cognition. Chikurel introduces Maimon's notion of analysis in the broader sense, grounded not only on the principle of contradiction but on intuition as well. In philosophy, ampliative analysis is based on Maimon's logical term of analysis of the object, a term that has yet to be discussed in Maimonian scholarship. Following its introduction, a new version of the question quid juris? arises. In mathematics, Chikurel demonstrates how this conception of analysis originates from practices of Greek geometrical analysis. (shrink)

We introduce and study a variety of modal logics of parallelism, orthogonality, and affine geometries, for which we establish several completeness, decidability and complexity results and state a number of related open, and apparently difficult problems. We also demonstrate that lack of the finite model property of modal logics for sufficiently rich affine or projective geometries (incl. the real affine and projective planes) is a rather common phenomenon.

Mathematics is an important foundation for the development of science education. In the past, when instructors taught mathematical concepts of geometry shapes, they usually used traditional textbooks and aids to conduct teaching activities, which resulted in students not being able to understand the principles completely. Nowadays, it has become a trend to integrate emerging technologies into mathematics courses and to use digital instructional aids. Emerging technologies can effectively enhance students’ sensory experience while strengthening their impressions and understandings of subject (...) concepts. In this paper, we apply virtual reality immersive technologies to develop a “virtual reality immersive learning mathematics geometry system,” which is used to teach mathematical geometry concepts. Teachers use the system to develop three basic mathematical geometry learning materials: “Triangular pyramid volume = 1/3 prism volume,” “Cone volume calculation,” and “Triangle center of gravity derivation.” In the experimental activity, the teacher uses virtual reality teaching aids to guide students to learn mathematical geometry concepts in a fun way so that they can achieve the effectiveness of immersive learning. This study explores the impact of using the virtual reality immersive learning mathematics geometry system on students’ technology acceptance, learning motivations, and learning performance. The experimental result showed that using the virtual reality immersive learning mathematics geometry system can improve the learning motivation and learning performance of students. The findings indicated that the experimental group had better learning outcomes after completing the learning tasks of three geometric units. The experimental group used the virtual reality immersive learning mathematics geometry system which can lead to better learning outcomes. According to the ARCS questionnaire, students in the experimental group were confident to understand new subjects. At the same time, the mode of completing the game can effectively give students a sense of accomplishment. The use of emerging technologies in the classroom can be an attractive learning mode for students. (shrink)

The aim of this paper is to argue that there existed relevant interactions between mechanics and geometry during the first half of the nineteenth century, following a path that goes from Gauss to Riemann through Jacobi. By presenting a rich historical context we hope to throw light on the philosophical change of epistemological categories applied by these authors to the fundamental principles of both disciplines. We intend to show that presentations of the changing status of the principles of (...) mechanics as a mere epiphenomenon of the emergence of non-Euclidean geometries are inaccurate, that the relations between the two disciplines were richer than what is usually considered in the literature. These claims will be based on historical and philosophical arguments, starting from the fact that disciplinary boundaries at the time were not rigid as we are used today. It is widely known that the main figures we target worked in different areas, which is a first piece of evidence for the plausibility of our main thesis. (shrink)