Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type forallexistsforall, while the axiom system based on congruence and order can beformulated using only forallexists-axioms.

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

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

Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.

Using the axiom system provided by Carsten Augat in [1], it is shown that the only 6-variable statement among the axioms of the axiom system for plane hyperbolic geometry (in Tarski's language L B =), we had provided in [3], is superfluous. The resulting axiom system is the simplest possible one, in the sense that each axiom is a statement in prenex form about at most 5 points, and there is no axiom system consisting entirely of at most (...) 4-variable statements. (shrink)

The Thomas precession of relativity physics gives rise to important isometries in hyperbolic geometry that expose analogies with Euclidean geometry. These, in turn, suggest our bifurcation approach to hyperbolic geometry, according to which Euclidean geometry bifurcates into two mutually dual branches of hyperbolic geometry in its transition to non-Euclidean geometry. One of the two resulting branches turns out to be the standard hyperbolic geometry of Bolyai and Lobachevsky. The corresponding (...) bifurcation of Newtonian mechanics in the transition to Einsteinian mechanics indicates that there are two, mutually dual, kinds of uniform accelerations. Furthermore, while current hyperbolic geometry “does not use the notion of vector at all” (I. M. Yaglom, Geometric Transformations III, p. 135, trans. by Abe Shenitzer, Random House, New York, 1973), our bifurcation approach exposes the elusive hyperbolic vectors, that we call gyrovectors. (shrink)

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

Gyrogroup theory and its applications is introduced and explored, exposing the fascinating interplay between Thomas precession of special relativity theory and hyperbolic geometry. The abstract Thomas precession, called Thomas gyration, gives rise to grouplike objects called gyrogroups [A, A. Ungar, Am. J. Phys.59, 824 (1991)] the underlying axions of which are presented. The prefix gyro extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, and gyrosemidirect products, stems from their underlying abstract Thomas gyration. Thomas gyration is (...) tailor made for hyperbolic geometry. In a similar way that commutative groups underlie vector spaces, gyrocommutative gyrogroups underlie gyrovector spaces. Gyrovector spaces, in turn, provide a most natural setting for hyperbolic geometry in full analogy with vector spaces that provide the setting for Euclidean geometry. As such, their applicability to relativistic physics and its spacetime geometry is obvious. (shrink)

We show that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. The superiority of the use of the gyrogroup formalism over the use of the SL(2, C) formalism for dealing with the Lorentz group in some cases is indicated by (i) the validity of gyrogroups and gyrovector spaces in higher dimensions, by (ii) the analogies that they share with (...) groups and vector spaces, and by (iii) the demonstration that gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. As such, gyrogroups and gyrovector spaces provide powerful tools for the study of relativity physics. (shrink)

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

Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum computation theory. We bring the seemingly structureless set of all these density matrices under the umbrella of gyrovector spaces, where the Bloch vector is treated as a hyperbolic vector, called a gyrovector. As such, this article catalizes and supports interdisciplinary research spreading from mathematical physics to algebra and geometry. Gyrovector spaces are mathematical objects (...) that form the setting for the hyperbolic geometry of Bolyai and Lobachevski just as vector spaces form the setting for Euclidean geometry. It is thus interesting, in geometric quantum computation, to realize that the set of all qubit density matrices has rich structure with strong link to hyperbolic geometry. Concrete examples for the use of the gyro-structure to derive old and new, interesting identities for qubit density matrices are presented. (shrink)

The use of the axial vector representing a three-dimensional rotation makes the rotation representation much more compact by extending the trigonometric functions to vectorial arguments. Similarly, the pure Lorentz transformations are compactly treated by generalizing a scalar rapidity to a vector quantity in spatial three-dimensional cases and extending hyperbolic functions to vectorial arguments. A calculation of the Wigner rotation simplified by using the extended functions illustrates the fact that the rapidity vector space obeys hyperbolic geometry. New representations (...) bring a Lorentz-invariant fundamental equation of motion corresponding to the Galilei-invariant equation of Newtonian mechanics. (shrink)

As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we (...) explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18). (shrink)

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well (...) known that relativistic velocities are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π. (shrink)

En 1676, alors qu'il sejourne encore a Paris, Leibniz entreprend de composer un volumineux traite qui restera sans doute l'un de ses ecrits mathematiques les plus fortement charpentes: La quadrature arithmetique du cercle, de l'ellipse et de l'hyperbole et la trigonometrie sans tables qui en est le corollaire. Ce traite se presente comme un abrege exhaustif de la geometrie infinitesimale, dont Leibniz avait pu esperer qu'elle lui ouvrirait les portes de l'Academie des Sciences. Cependant, contraint de quitter la capitale avant (...) sa publication, il abandonnera derriere lui un ouvrage qui n'allait voir le jour qu'en 1993. Le probleme de la quadrature est pretexte a soulever plusieurs enjeux capitaux tant pour les mathematiques que pour la philosophie. Il permet notamment a Leibniz d'examiner avec une grande precision la question de la methode, des fondements, ainsi que des notions cruciales de rigueur et d'infini, tout en mettant en evidence leurs applications pratiques. Si la presente edition de ce texte foisonnant ne reproduit pas l'integralite des variantes, elle en propose une version corrigee et annotee, qui s'accompagne d'une traduction francaise en regard. (shrink)

Geometric statements are expressions of natural language with descriptive meaning, since they refer to things such as triangles, and to their characteristics. It is shown, following statements by authors such as Carnap (Philosophical foundations of physics) that geometric meanings are not factual, but mathematical, so that it is not geometric theories that apply, but geometrically interpreted factual theories. Under the Euclidean paradigm, the object of study of geometry was space, but the theoretical pluralism that replaced it resulted in the (...) creation in the discipline of a new plural ontology according to which there are different spaces: Euclidean, hyperbolic and elliptic. Thus, geometrical knowledge no longer seems to be a knowledge of the universal (of universal characteristics of space), but of the particular: of individual spaces. If geometrical theories are mathematical, these questions constitute obstacles both to the thesis that mathematical knowledge is universal, and to the thesis that it is universally applicable. (shrink)

Atiyah and Sutcliffe made a number of conjectures about configurations of N distinct points in hyperbolic 3-space, arising from ideas of Berry and Robbins. In this paper we prove all these conjectures, purely geometrically, but we also provide a physical interpretation in terms of Electrons.

Hyperbolic vectors are called gyrovectors. We show that the Bloch vector of quantum mechanics is a gyrovector. The Bures fidelity between two states of a qubit is generated by two Bloch vectors. Treating these as gyrovectors rather than vectors results in our novel expression for the Bures fidelity, expressed in terms of its two generating Bloch gyrovectors. Taming the Thomas precession of Einstein's special theory of relativity led to the advent of the theory of gyrogroups and gyrovector spaces. Gyrovector (...) spaces, in turn, form the setting for various models of the hyperbolic geometry of Bolyai and Lobachevski just as vector spaces form the setting for the standard model of Euclidean geometry. It is the recent advent of the theory of gyrogroups and gyrovector spaces that allows the Bures fidelity to be studied in its natural context, hyperbolic geometry, resulting in our new representation of the Bures fidelity, that reveals simplicity, elegance, and hyperbolic geometric significance. (shrink)

Consider the set Q of quantum correlation vectors for two observers, each with two possible binary measurements. Quadric (hyperbolic) inequalities which are satis…ed by every q 2 Q are proved, and equality holds on a two dimensional manifold consisting of the local boxes, and all..

The paper distinguishes two types of Platonist approach, namely the Traditional one and the Robust one. In relation to this distinction I am going to argue that if the ontology of mathematics is intended to be defended plausibly in a Platonist way then this cannot be done according to the Traditional version. This will draw our attention to the plausibility of the Robust version. The plausibility of the two versions of Platonism will be examined in relation to one of the (...) central problems of the philosophy of mathematics, namely the truth-proof problem. The surveying of the truth-proof problem will bring to the surface the prima facie plausibility of the Platonist approach, as well as the apparent accessibility problem of it. Focusing on the accessibility problem will help us to identify two conditions that have to be met by any particular access theory of Platonism. These will be the reducibility condition, and the matching one. The Traditional version will appear an insufficient philosophical theory in relation to the two former conditions. The insufficiency will be demonstrated in the area of the incompatible mathematical theories, namely in the area of Euclidean and hyperbolic geometries. It will turn out that Robust Platonism can escape the squeeze of these conditions, so can it save the original prima facie plausibility of the Platonist approach. (shrink)

In much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.

The volume starts with the paper of Lynn Maurice Ferguson Arnold, former Premier of South Australia and former Minister of Education of Australia, concerning the Exposition Internationale des Arts et Techniques dans la Vie Moderne (International Exposition of Art and Technology in Modern Life) that was held from 25 May to 25 November 1937 in Paris, France. The organization of the world exhibition had placed the Nazi German and the Soviet pavilions directly across from each other. Many papers are devoted (...) to the interpretation of this opposition. Arnold’s paper considers the differences in the two totalitarian states’ architectural and design discourse styles. Although each of them communicated a totalitarian language of purposes, permissions, and boundaries, they essentially differed in the styles of discourse represented by the architecture and design of their respective pavilions. They were opposites of each other and the liberal ideals they contested. The internationalist viewpoint reflecting the multi-ethnic mix of the USSR is contrasted to the ein Volk homogeneity represented in the German pavilion. The Soviet pavilion opted for a utopian future to be arrived at by “benign” leadership, whereas the German pavilion anchored itself in the myth of Teutonic history with the nostalgic pride protected by the swastika-bearing eagle. Jean-Yves Béziau is best known as a logician and founder of Universal Logic. However, in this paper, he poses a new philosophical question: why philosophical discourse does not use images? The paper begins with a general analysis of different types of philosophical discourses. Then, he focuses on why images have been and are still rejected in philosophical discourse. He explains various ways to use images in a fruitful way to develop philosophical thinking and discourse and illustrates his view by providing several examples. He concludes with a promising programmatic declaration about founding a new journal entitled World Journal of Pictorial Philosophy to stimulate the usage of images for developing philosophical thinking. Although complaints about the obscurity of many philosophers’ discourse are widespread, Tatiana Denisova, ex-professor of the Surgut University and a research associate of the University of the Aegean undertakes a positive attitude to this problem. She explains that the reasons for the obscurity of philosophical texts, the subsequent complexity in communicating philosophers’ meanings and their eventual incomplete understanding is not a sign of their inferiority. On the contrary, it is a sign of the fruitfulness of philosophical discourse, which can generate new meanings. Thus, the darkness of philosophical discourse is like the life-giving chaos, and the obscurity that it inevitably contains can be the keeper of implicit meanings and even their generator. Katarzyna Gan-Krzywoszyńska and Piotr Leśniewski, authors of a monograph on Polish philosopher Kazimierz Ajdukiewicz (1890 – 1963), make a comparative analysis of his educational style with that of Paulo Reglus Neves Freire (1921 – 1997). The authors highlight that these scholars have distinctly different backgrounds. The former was an educator, philosopher and a leading advocate of critical pedagogy connected with the dialogical Latin-American tradition. The latter was a philosopher and logician, a notable representative of the Lwów–Warsaw school of logic with significant contributions to semantics, model theory and the philosophy of science. Despite their apparent difference, the authors identify some striking similarities regarding their attitudes towards education; notably, their approaches are essentially dialogical. Gilah Yelin Hirsch is a multidisciplinary artist who works as a painter, writer, curator, educator, and filmmaker. In her experiential paper, he explains how and why she uses four channels of communication in her creative expression: writing, painting, filmmaking, and teaching, as dialogic inquiry. She considers that these channels compose a continuum of discourse styles, each reaching a different facet of kaleidoscopic consciousness. She claims that her choice of medium is prompted by a need to communicate her insights profoundly. Thus, for instance, she created painted allegories to express narratives based on dreams and visions emerging from her subconscious. Art and healing is another Tibetan-rooted type of discourse practised by the author. Creating and observing a healing image produces a positive psychophysiological change in both the artist and the viewer. The discourse here is tripartite between creator, image, and viewer. Filmmaking is another type of discourse she practices between the filmmaker/artist as shaman or healer and the viewer as respondent or participant. These films are meant to be experienced frame by frame, physically and emotionally in both the body and mind. Jocelyn Ireson-Paine is a cartoonist and programmer. His paper offers an original attempt to use category theory in cartooning. Category theory is a branch of mathematics that provides concepts and methods to describe general abstract structures in terms of a labelled directed graph called category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms or transformations or mappings). The author explores different ways to define “style” in art using category theory and a relational view of art. Moreover, he examines how “translation” (transformation) between styles could be defined and reveals the difficulties of how it could be implemented in a computer using special software. Jens Lemanski is a philosopher who has revived research in Arthur Schopenhauer’s legacy by exploring his work on mathematical evidence, logic diagrams, and problems of semantics. In this paper, he advances a new approach to eristic as an art of protecting oneself from the one who deliberately violates norms of discourse ethics to win an argument. The author attributes the origins of this view to Schopenhauer, suggesting a new reading of his work. According to the author, eristic is a prohibitive technique that takes effect when the norms of discourse ethics are transgressed and violated. Thus, eristic is viewed as a discipline of Enlightenment philosophy and a correlate of discourse ethics. Roshdi Rashed is an authority on the history of Arabic mathematical sciences. Proceeding from Gilles-Gaston Granger’s definition of mathematical style, he studies the question of whether a mathematical work can be characterized by a single style or by a multitude of styles. This question is explored within the style of a single work, namely Menelaus’s Sphaerica, and through the study of the development of a single problem over time, namely the isoperimetric problem. He indicates Menelaus’s divergence from the Euclidean style of (plane and stereometric) geometry caused by the drop of the Parallel Postulate, which associated it with hyperbolic geometry. Hence, the author concludes that Menelaus’s Sphaerica incorporates the well-known Euclidean style with a variation of the non-Euclidean one. On the other hand, examining the isoperimetric problem reveals another interesting historical picture. A succession of different styles (cosmological, geometric, infinitesimalistic, style of the calculus of variations, of synthetic geometry) can be observed due to the transformation of the research object over time. Boris Shalyutin, a Russian social philosopher, and Ombudsman for Human Rights in the Kurgan Region, explores the origins of discourse in combination with the birth of society and law. He claims that legal discourse marks the historical emergence of discourse in general. Homo Juridicus generated Homo Sapiens, which have created new spheres of discourse, notably moral discourse and further philosophical, political, and scientific discourse. Petros Stefaneas, a logician, computer scientist and novelist, suggests an alternative to the traditional narratology approach for studying (interactive) social media discourse. He claims that style describes how the parts of a narrative are blended into a whole. The author relies upon Goguen’s and Harrel’s concept of style as a choice of blending principles and transfers to the study of the social media narratives elements from the methodology of studying Web-based collaborative search for mathematical proofs like those implemented within the Polymath project. The last paper belongs to Ghil‘ad Zuckermann, a linguist, language revivalist, and proponent of a model of the emergence of Israeli Hebrew, according to which Hebrew and Yiddish were the primary sources of Modern Hebrew. This paper explores the fascinating and multifaceted Yiddish language and its survival in Israeli. Yiddish is characterized by a unique style that embeds psycho-ostensive expressions throughout its discourse. The author highlights the cross-fertilization between Hebrew and Yiddish as it manifests itself in any aspect of the Israeli language. He claims that Yiddish survives beneath Israeli phonetics, phonology, discourse, syntax, semantics, lexis, and even morphology, although traditional and institutional linguists have been most reluctant to admit it. (shrink)

The article is devoted to the justification of the law of inertia. It is often called Newton's first law. It was established that this is not a law, but a postulate. Modern definitions of this law are given. It turned out that well -known definitions of this law are similar to each other. It is shown that this law before Newton was formulated by Descartes, Balillians, Ballo and Galileo. The ontology and philosophical significance of the category "cause" are considered. It (...) is shown that any thing or phenomenon arises for some reason, and nothing happens for no reason and nothing exists. And Democritus understood that there was nothing causeless and could not be. Democritus’s mistake is that he did not recognize a chance. Causality is a philosophical category showing the ontological connection of such two phenomena, when the second phenomenon arises due to the first. Moreover, the first phenomenon is called the cause, and the second is the consequence. In this connection of phenomena, the first phenomenon (cause) necessarily predicts the second phenomenon (investigation). And the mutual relationship between the first and second phenomena is called a causal relationship. is present in all known phenomena of the world, and it is usually understood as the occurrence of the same effects with the same causes. It is shown that in history many ideas and scientific achievements attributed to a certain author have often been discovered before them by other authors. For example, Lobachevsky and Boiai are considered to be the founders of hyperbolic geometry. In fact, this discovery was actually made before them by the now almost forgotten F.A. Taurinus (Taurin). The same can be said about the law of refraction of light. This law was discovered by the Dutch scientist Willebrord Snellius. In fact, 600 years before him, this law had been discovered by the great scientist and philosopher Bahmanyar al-Azerbaijani. And the same mistake is Newton's first law. Therefore, it is best to henceforth always refer to this law as the law of inertia. (shrink)

Originally, the expression “Teichmüller theory” referred to the theory that Oswald Teichmüller developed on deformations and on moduli spaces of marked Riemann surfaces. This theory is not an isolated field in mathematics. At different stages of its development, it received strong impetuses from analysis, geometry, and algebraic topology, and it had a major impact on other fields, including low-dimensional topology, algebraic topology, hyperbolic geometry, geometric group theory, representations of discrete groups in Lie groups, symplectic geometry, topological (...) quantum field theory, theoretical physics, and there are certainly others. Of course, the impacts on these various fields are not equally important, but in some cases (namely, low-dimensional topology, algebraic geometry, and physics) the impact was crucial. At the same time, Teichmüller theory established important connections between the fields mentioned. This, in part, is a consequence of the diversity and the richness of the structure that Teichmüller space itself carries. From a more subjective point of view, the result of pondering on these connections and applications demonstrates the unity of mathematics. The aim of this paper is to survey the origin of Teichmüller theory and the development of its early major ideas. (shrink)

Inspired by Peirce’s repeated claim in the final decade of his life that Spinoza was a pragmati(ci)st, this article examines whether or not Peirce also believed that Spinoza’s metaphysics leaves room for Firstness. He engaged this issue explicitly in his third “Lecture on Pragmatism” (1903), listing Spinoza’s among the metaphysics that include Firstness, Secondness and Thirdness. Moreover, over a decade earlier, in the context of his exploration of hyperbolic geometry and the evolutionary cosmology that he regarded as corresponding (...) to it, Peirce repeatedly (if obliquely) identified Spinoza with the cosmological model that embraces all three of the categories. The article concludes by sketching the ambitious thesis that Spinoza was not only, as is usually held, a necessitarian, but also a Peircean possibilist. (shrink)

The obscured Thomas precessionof the special theory of relativity (STR) has been soared into prominence by exposing the mathematical structure, called a gyrogroup,to which it gives rise [A. A. Ungar, Amer. J. Phys.59,824 (1991)], and the role that it plays in the study of Lorentz groups [A. A. Ungar, Amer. J. Phys.60,815 (1992); A. A. Ungar, J. Math. Phys.35,1408 (1994); A. A. Ungar, J. Math. Phys.35,1881 (1994)]. Thomas gyrationresults from the abstraction of Thomas precession.As such, its study sheds light on (...) relativistic velocity spaces and their symmetries which are concealed in Thomas precession. In order to uncover new properties of relativistic gyrogroups, we employ in this article the group theoretic concepts of divisible groupsand two-torsion free groupsto construct midpointsin gyrogroups. Systems of successive midpoints then describe straight gyrolinesand suggest a new, natural distance function that involves a Thomas gyration. These, in turn, reveal a new, interesting geometry that underlies relativistic velocity spaces. In this resulting gyrogeometrythe straight gyrolines form geodesics under a distance function which turns out to be the Poincaré hyperbolic distance function relaxed by a Thomas gyration. These geodesics do obey the parallel axiom of Euclidean geometry. Ironically, (i) attempts to understand the parallel postulate of Euclidean geometry gave rise to hyperbolic geometry in which the parallel postulate disappears;(ii) hyperbolic geometry gave rise to Einstein's STR; (iii) Einstein's STR established the bizarre and counterintuitive relativistic effect called Thomas precession, the abstraction of which is called Thomas gyration; and (iv) Thomas gyration now repairs in this article the Poincaré distance function of hyperbolic geometry to the point where the parallel postulate reappears. (shrink)

For the classical Geometry, in a geometrical space, all items (concepts, axioms, theorems, etc.) are totally (100%) true. But, in the real world, many items are not totally true. The NeutroGeometry is a geometrical space that has some items that are only partially true (and partially indeterminate, and partially false), and no item that is totally false. The AntiGeometry is a geometrical space that has some item that are totally (100%) false. While the Non-Euclidean Geometries [hyperbolic and elliptic (...) geometries] resulted from the total negation of only one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom [and in general: theorem, concept, idea etc.] and even of more axioms [theorem, concept, idea, etc.] and in general from any geometric axiomatic system (Euclid’s five postulates, Hilbert’s 20 axioms, etc.), and the NeutroAxiom results from the partial negation of any axiom (or concept, theorem, idea, etc.). Clearly, the AntiGeometry is a generalization of Non-Euclidean Geometries. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" (...) are devoted. At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

The generative and governing "idea" of radical modernity is spawned by the technique of mathematical construction deployed and interpreted by the major early-modern thinkers and their legatees. ;Chapter I is a survey of this legacy as it appears in Vico, Kant, Fichte, Marx and Nietzsche and in the post-Nietzschean inheritance of contemporary philosophy, hyperbolic in the case of Derrida et al., elliptical, in the case of Carnap and Goodman. ;In Chapter II I try to show how the pre-modern mathematical (...) tradition, represented by Euclid, aimed at keeping the enticements of technical facility in check by means of didactic phronesis and how the post-Kantian interpretation of "existence" in Euclid as constructibility betrays his usage and self-understanding. I suggest that his focus in the postulates and elsewhere is on the undistorted iterability of graphic evocations of the items already intelligible thanks to the definitions or to the pre-understanding shared by the teacher and student. ;In Chapter III, devoted to Descartes the principal claims of modern constructivism are brought to sight. After examining Descartes' fabulous autobiography and its emphasis on self-origination, I turn to the style, contents and under-pinnings of the Geometry in an effort to extract from that text what he once referred to as "the metaphysics of geometry." The latter yields the conditions of successful problem-solving, i.e., dimensional homogeneity and kinematic continuity. These conditions, in turn, find their justification in Descartes' theses in the Rules concerning order, measure and the uniformity of "mental" activity. In the final section I apply the lessons learned from the Geometry and the Rules to one critical issue in the later Meditations, the transition from essence to existence. Descartes' "solution" generates a sequence of perplexities with Hobbes, Leibniz, Kant and other radical moderns continue to wrestle. (shrink)

Discusses various works of Descartes's and their reception, including objections to them and his response to those objections. Météors deals with meteorology, which includes a corpuscular model of light, an account of refraction, and vision, and its links with optical instruments; the Dioptrique is a practical treatise on the construction of these optical instruments; and Géométrie compares arithmetic with geometry and extends Descartes's treatment of the Pappus problem and the classification of curves. The organization of material in the Discours (...) appears arbitrary unless interpreted autobiographically. The Meditationes contains a fuller treatment of scepticism, in comparison with Pyrrhonism, and hyperbolic doubt, a defence of mechanism, discussions of the cogito and the transcendence of God, the classification of ideas, a new version of the doctrine of clarity and distinctness, and the nature of the thinking self, the identification of the self with the mind and mind/body dualism. (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)

Derrida's introduction to his French translation of Husserl's essay "The Origin of Geometry," arguing that although Husserl privileges speech over writing in an account of meaning and the development of scientific knowledge, this privilege is in fact unstable.

Earlier in this century, many philosophers of science (for example, Rudolf Carnap) drew a fairly sharp distinction between theory and observation, between theoretical terms like 'mass' and 'electron', and observation terms like 'measures three meters in length' and 'is _2° Celsius'. By simply looking at our instruments we can ascertain what numbers our measurements yield. Creatures like mass are different: we determine mass by calculation; we never directly observe a mass. Nor an electron: this term is introduced in order to (...) explain what we observe. This (once standard) distinction between theory and observation was eventually found to be wanting. First, if the distinction holds, it is difficult to see what can characterize the relationship between theory :md observation. How can theoretical terms explain that which is itself in no way theorized? The second point leads out of the first: are not the instruments that provide us with observational material themselves creatures of theory? Is it really possible to have an observation language that is entirely barren of theory? The theory-Iadenness of observation languages is now an accept ed feature of the logic of science. Many regard such dependence of observation on theory as a virtue. If our instruments of observation do not derive their meaning from theories, whence comes that meaning? Surely - in science - we have nothing else but theories to tell us what to try to observe. (shrink)

Brown and Pooley’s ‘dynamical approach’ to physical theories asserts, in opposition to the orthodox position on physical geometry, that facts about physical geometry are grounded in, or explained by, facts about dynamical fields, not the other way round. John Norton has claimed that the proponent of the dynamical approach is illicitly committed to spatiotemporal presumptions in ‘constructing’ space-time from facts about dynamical symmetries. In this article, I present an abstract, algebraic formulation of field theories and demonstrate that the (...) proponent of the dynamical approach is not committed, in special relativity, to the illicit presumptions to which Norton refers. (shrink)

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

This paper gives a contextualized reading of Kant's theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid's definitions. I argue that Kant accepted Euclid's definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in (...) class='Hi'>geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions. (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)