Le projet théorique et philosophique que Franck Fischbach poursuit depuis le début des années 2000 place en son cœur une volonté d’articuler la philosophie de Marx à la phénoménologie heideggérienne d’Être et Temps. Au travers de son triptyque publié entre 2005 et 2011, La production des hommes (2005), Sans objet (2009) et La privation de monde (2011) – mais également dans son dernier ouvrage, Après la production (2019) – on voit ainsi se déployer une forme de réactivation de ce programme (...) d’h... (shrink)

In recent years, with the development of society and the rapid development of the animation industry, people are paying more and more attention to and requirements for animation production. As an indispensable part of animation production, picture composition plays a major role in animation production. It can give full play to the application of color matching and light and shadow design and enhance the depth and space of the animation screen. Tone space conversion refers to the conversion or representation of (...) color data in one color space into corresponding data in another color space. Its purpose is to distinguish and process color components such as hue and saturation in an image. This article first introduces the domestic and foreign research status of digital image preprocessing and analyzes the basic principles of several color space conversions in detail. Then, several color space conversion algorithms are studied, and the performance of the algorithms is compared and analyzed. The paper focuses on the hardware implementation and optimization of the algorithm for converting RGB color space into HSI color space to meet the real-time requirements. This article focuses on the mutual conversion between the RGB tone space and the HSI tone space and describes in detail how each color component in the HSI tone space is converted from the three RGB color components from a geometric perspective, and then the conversion is derived, and several general conversion methods of RGB to HSI tone space are introduced; two conversion methods of geometric derivation method and standard modulus algorithm are implemented in the software, and the comparison verification is carried out, and the comparison is made from the perspective of hardware implementation. The pros and cons of the two methods are discussed. Finally, the paper summarizes the shortcomings in the design and proposes further research directions in the future. (shrink)

We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial (...) characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper. (shrink)

It has been claimed, recently, that the fact that all the non-gravitational fields are locally Poincaré invariant and that these invariances coincide, in a certain regime, with the symmetries of the spacetime metric is miraculous in general relativity. In this paper I show that, in the context of GR, it is possible to account for these so-called miracles of relativity. The way to do so involves integrating the realisation that the gravitational field equations impose constraints on the behaviour of (...) matter in a novel interpretation of the equivalence principle, which dictates the determination of local inertial frames through gravitational interaction. This proposed explanation of the miracles can also deal with the problematic cases for attempts at explaining them in the context of the standard geometrical perspective on relativity theory. (shrink)

Epicurus’ original version of atomism takes atoms to be physically indivisible but not completely unanalysable: each atom contains a finite number of minima. This paper explores the nature of the minima by focusing on a specific question: in which sense are the minima minimal? I do so by investigating the notions of parthood and divisibility into parts that are at play in paragraphs 56–59 of the Letter to Herodotus, where the theory of minima is introduced. By focusing on the analogy (...) (noticed by Francesco Verde) between Epicurean minima and Aristotelian limits, I argue that the minima should be understood as the minimal realiser of the atom’s physical functions. This allows me to keep together two very plausible but apparently incompatible claims: (i) the minima are supposed to block the paradoxes of theoretical divisibility, but (ii) their existence and their indivisibility can only be justified in physical (rather than geometrical) terms. (shrink)

Jakob Friedrich Fries (1773-1843): A Philosophy of the Exact Sciences -/- Shortened version of the article of the same name in: Tabula Rasa. Jenenser magazine for critical thinking. 6th of November 1994 edition -/- 1. Biography -/- Jakob Friedrich Fries was born on the 23rd of August, 1773 in Barby on the Elbe. Because Fries' father had little time, on account of his journeying, he gave up both his sons, of whom Jakob Friedrich was the elder, to the Herrnhut Teaching (...) Institution in Niesky in 1778. Fries attended the theological seminar in Niesky in autumn 1792, which lasted for three years. There he (secretly) began to study Kant. The reading of Kant's works led Fries, for the first time, to a deep philosophical satisfaction. His enthusiasm for Kant is to be understood against the background that a considerable measure of Kant's philosophy is based on a firm foundation of what happens in an analogous and similar manner in mathematics. -/- During this period he also read Heinrich Jacobi's novels, as well as works of the awakening classic German literature; in particular Friedrich Schiller's works. In 1795, Fries arrived at Leipzig University to study law. During his time in Leipzig he became acquainted with Fichte's philosophy. In autumn of the same year he moved to Jena to hear Fichte at first hand, but was soon disappointed. -/- During his first sojourn in Jenaer (1796), Fries got to know the chemist A. N. Scherer who was very influenced by the work of the chemist A. L. Lavoisier. Fries discovered, at Scherer's suggestion, the law of stoichiometric composition. Because he felt that his work still need some time before completion, he withdrew as a private tutor to Zofingen (in Switzerland). There Fries worked on his main critical work, and studied Newton's "Philosophiae naturalis principia mathematica". He remained a lifelong admirer of Newton, whom he praised as a perfectionist of astronomy. Fries saw the final aim of his mathematical natural philosophy in the union of Newton's Principia with Kant's philosophy. -/- With the aim of qualifying as a lecturer, he returned to Jena in 1800. Now Fries was known from his independent writings, such as "Reinhold, Fichte and Schelling" (1st edition in 1803), and "Systems of Philosophy as an Evident Science" (1804). The relationship between G. W. F. Hegel and Fries did not develop favourably. Hegel speaks of "the leader of the superficial army", and at other places he expresses: "he is an extremely narrow-minded bragger". On the other hand, Fries also has an unfavourable take on Hegel. He writes of the "Redundancy of the Hegelistic dialectic" (1828). In his History of Philosophy (1837/40) he writes of Hegel, amongst other things: "Your way of philosophising seems just to give expression to nonsense in the shortest possible way". In this work, Fries appears to argue with Hegel in an objective manner, and expresses a positive attitude to his work. -/- In 1805, Fries was appointed professor for philosophy in Heidelberg. In his time spent in Heidelberg, he married Caroline Erdmann. He also sealed his friendships with W. M. L. de Wette and F. H. Jacobi. Jacobi was amongst the contemporaries who most impressed Fries during this period. In Heidelberg, Fries wrote, amongst other things, his three-volume main work New Critique of Reason (1807). -/- In 1816 Fries returned to Jena. When in 1817 the Wartburg festival took place, Fries was among the guests, and made a small speech. 1819 was the so-called "Great Year" for Fries: His wife Caroline died, and Karl Sand, a member of a student fraternity, and one of Fries' former students stabbed the author August von Kotzebue to death. Fries was punished with a philosophy teaching ban but still received a professorship for physics and mathematics. Only after a period of years, and under restrictions, he was again allowed to read philosophy. From now on, Fries was excluded from political influence. The rest of his life he devoted himself once again to philosophical and natural studies. During this period, he wrote "Mathematical Natural Philosophy" (1822) and the "History of Philosophy" (1837/40). -/- Fries suffered from a stroke on New Year's Day 1843, and a second stroke, on the 10th of August 1843 ended his life. -/- 2. Fries' Work Fries left an extensive body of work. A look at the subject areas he worked on makes us aware of the universality of his thinking. Amongst these subjects are: Psychic anthropology, psychology, pure philosophy, logic, metaphysics, ethics, politics, religious philosophy, aesthetics, natural philosophy, mathematics, physics and medical subjects, to which, e.g., the text "Regarding the optical centre in the eye together with general remarks about the theory of seeing" (1839) bear witness. With popular philosophical writings like the novel "Julius and Evagoras" (1822), or the arabesque "Longing, and a Trip to the Middle of Nowhere" (1820), he tried to make his philosophy accessible to a broader public. Anthropological considerations are shown in the methodical basis of his philosophy, and to this end, he provides the following didactic instruction for the study of his work: "If somebody wishes to study philosophy on the basis of this guide, I would recommend that after studying natural philosophy, a strict study of logic should follow in order to peruse metaphysics and its applied teachings more rapidly, followed by a strict study of criticism, followed once again by a return to an even closer study of metaphysics and its applied teachings." -/- 3. Continuation of Fries' work through the Friesian School -/- Fries' ideas found general acceptance amongst scientists and mathematicians. A large part of the followers of the "Fries School of Thought" had a scientific or mathematical background. Amongst them were biologist Matthias Jakob Schleiden, mathematics and science specialist philosopher Ernst Friedrich Apelt, the zoologist Oscar Schmidt, and the mathematician Oscar Xavier Schlömilch. Between the years 1847 and 1849, the treatises of the "Fries School of Thought", with which the publishers aimed to pursue philosophy according to the model of the natural sciences appeared. In the Kant-Fries philosophy, they saw the realisation of this ideal. The history of the "New Fries School of Thought" began in 1903. It was in this year that the philosopher Leonard Nelson gathered together a small discussion circle in Goettingen. Amongst the founding members of this circle were: A. Rüstow, C. Brinkmann and H. Goesch. In 1904 L. Nelson, A. Rüstow, H. Goesch and the student W. Mecklenburg travelled to Thuringia to find the missing Fries writings. In the same year, G. Hessenberg, K. Kaiser and Nelson published the first pamphlet from their first volume of the "Treatises of the Fries School of Thought, New Edition". -/- The school set out with the aim of searching for the missing Fries' texts, and re-publishing them with a view to re-opening discussion of Fries' brand of philosophy. The members of the circle met regularly for discussions. Additionally, larger conferences took place, mostly during the holidays. Featuring as speakers were: Otto Apelt, Otto Berg, Paul Bernays, G. Fraenkel, K. Grelling, G. Hessenberg, A. Kronfeld, O. Meyerhof, L. Nelson and R. Otto. On the 1st of March 1913, the Jakob-Friedrich-Fries society was founded. Whilst the Fries' school of thought dealt in continuum with the advancement of the Kant-Fries philosophy, the members of the Jakob-Friedrich-Fries society's main task was the dissemination of the Fries' school publications. In May/June, 1914, the organisations took part in their last common conference before the gulf created by the outbreak of the First World War. Several members died during the war. Others returned disabled. The next conference took place in 1919. A second conference followed in 1921. Nevertheless, such intensive work as had been undertaken between 1903 and 1914 was no longer possible. -/- Leonard Nelson died in October 1927. In the 1930's, the 6th and final volume of "Treatises of the Fries School of Thought, New Edition" was published. Franz Oppenheimer, Otto Meyerhof, Minna Specht and Grete Hermann were involved in their publication. -/- 4. About Mathematical Natural Philosophy -/- In 1822, Fries' "Mathematical Natural Philosophy" appeared. Fries rejects the speculative natural philosophy of his time - above all Schelling's natural philosophy. A natural study, founded on speculative philosophy, ceases with its collection, arrangement and order of well-known facts. Only a mathematical natural philosophy can deliver the necessary explanatory reasoning. The basic dictum of his mathematical natural philosophy is: "All natural theories must be definable using purely mathematically determinable reasons of explanation." Fries is of the opinion that science can attain completeness only by the subordination of the empirical facts to the metaphysical categories and mathematical laws. -/- The crux of Fries' natural philosophy is the thought that mathematics must be made fertile for use by the natural sciences. However, pure mathematics displays solely empty abstraction. To be able to apply them to the sensory world, an intermediatory connection is required. Mathematics must be connected to metaphysics. The pure mechanics, consisting of three parts are these: a) A study of geometrical movement, which considers solely the direction of the movement, b) A study of kinematics, which considers velocity in Addition, c) A study of dynamic movement, which also incorporates mass and power, as well as direction and velocity. -/- Of great interest is Fries' natural philosophy in view of its methodology, particularly with regard to the doctrine "leading maxims". Fries calls these "leading maxims" "heuristic", "because they are principal rules for scientific invention". -/- Fries' philosophy found great recognition with Carl Friedrich Gauss, amongst others. Fries asked for Gauss's opinion on his work "An Attempt at a Criticism based on the Principles of the Probability Calculus" (1842). Gauss also provided his opinions on "Mathematical Natural Philosophy" (1822) and on Fries' "History of Philosophy". Gauss acknowledged Fries' philosophy and wrote in a letter to Fries: "I have always had a great predilection for philosophical speculation, and now I am all the more happy to have a reliable teacher in you in the study of the destinies of science, from the most ancient up to the latest times, as I have not always found the desired satisfaction in my own reading of the writings of some of the philosophers. In particular, the writings of several famous (maybe better, so-called famous) philosophers who have appeared since Kant have reminded me of the sieve of a goat-milker, or to use a modern image instead of an old-fashioned one, of Münchhausen's plait, with which he pulled himself from out of the water. These amateurs would not dare make such a confession before their Masters; it would not happen were they were to consider the case upon its merits. I have often regretted not living in your locality, so as to be able to glean much pleasurable entertainment from philosophical verbal discourse." -/- The starting point of the new adoption of Fries was Nelson's article "The critical method and the relation of psychology to philosophy" (1904). Nelson dedicates special attention to Fries' re-interpretation of Kant's deduction concept. Fries awards Kant's criticism the rationale of anthropological idiom, in that he is guided by the idea that one can examine in a psychological way which knowledge we have "a priori", and how this is created, so that we can therefore recognise our own knowledge "a priori" in an empirical way. Fries understands deduction to mean an "awareness residing darkly in us is, and only open to basic metaphysical principles through conscious reflection.". -/- Nelson has pointed to an analogy between Fries' deduction and modern metamathematics. In the same manner, as with the anthropological deduction of the content of the critical investigation into the metaphysical object show, the content of mathematics become, in David Hilbert's view, the object of metamathematics. -/-. (shrink)

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, (...) in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. (shrink)

Inquiries of Wellstein, Grünbaum and others have proved that there are indefinitely many different spatial models of Euklidian geometry. The points, lines and planes of these models are related to each other as the points, straight lines and planes of Euklidian geometry, but they are obviously totally different from them. That means that the axiomatic Euklidian geometry does not clearly determine the spatial forms of their planes and straight lines. The constructive geometry basing on approaches of Hugo Dingler tries to (...) solve this problem of disambiguity so that a clear realization of the geometrical basic objects plane, straight line and rectangle and others is possible because of certain rules. On the one hand these rules are to pledge the techniques of realisation and on the other hand they are the basic for the deduction of statements on the technically realized geometrical basic objects. The rules are especially to pledge the selection of apt material that keep its form. The following will show that the disambiguity cannot be pledged by the mere laws established by the constructive geometry and especially that the problem of electing the material cannot be solved by these laws. Furthermore it proves to be necessary to analyze and reconstruct more circumstances describing the selection of apt material which exceed the present standard of constructive geometry. (shrink)

A possible relevant meaning of Hilbert’s program is the following one: “give a constructive semantic for classical mathematics”. More precisely, give a systematic interpretation of classical abstract proofs about abstract objects, as constructive proofs about constructive versions of these objects.If this program is fulfilled we are able “at the end of the tale” to extract constructive proofs of concrete results from classical abstract proofs of these results.Dynamical algebraic structures or geometric theories seem to be a good tool for doing (...) this job. In this setting, classical abstract objects are interpreted through incomplete concrete specifications of these objects.The structure of axioms in geometric theories gives rise in a natural way to distributive lattices and pointfree topological spaces. Abstract objects correspond to classical points of these pointfree spaces.We shall insist on the Zariski spectrum of distributive lattices and commutative rings and give a constructive interpretation of the Krull dimension.We underline the fact that many abstract objects in classical mathematics can be viewed as points of spectral spaces corresponding to distributive lattices whose definition is concrete and natural.Two important facts are to be stressed.First, abstract proofs about points of these pointfree spaces can very often be reread as constructive proofs about constructible subsets of these spaces.Second, function spaces on these pointfree spaces are often explicitly used in elegant abstract theories. The structure of these function spaces is fully constructive: indeed by the compactness theorem, “all is finite”. The constructive rereading of the abstract proofs is in this setting is nothing but the simple constatation that abstract proofs use correctly axioms.RésuméUne manière pertinente de revisiter le Programme de Hilbert serait la suivante: « donner une sémantique constructive pour les mathématiques classiques ». Plus précisément donner une interprétation systématique des preuves classiques abstraites au sujet des objets abstraits, en terme de preuves constructives au sujet de contreparties constructives de ces objets abstraits.Si ce programme est rempli, nous sommes capables « à la fin de l’histoire » d’extraire des preuves constructives de résultats concrets à partir des preuves classiques abstraites de ces résultats.Les structures algébriques dynamiques, ou ce qui revient à peu près au même les théories géométriques, semblent être un bon outil pour réaliser ce travail. Dans cette optique, les objets abstraits des mathématiques classiques sont remplacés par des spécifications incomplètes mais concrètes de ces mêmes objets.La structure des théories géométriques donne naissance de manière naturelle à des treillis distributifs et à des espaces topologiques sans points. Les objets abstraits utilisés par les mathématiques classiques correspondent aux points classiques de ces espaces sans points.Dans cet article, nous illustrerons ce phénomène principalement avec le spectre de Zariski des treillis distributifs et celui des anneaux commutatifs, en indiquant notamment un équivalent constructif de la notion de dimension de Krull.Nous insistons sur le caractère extrêmement général de l’interpétation des objets abstraits idéaux des mathématiques classiques comme des points d’espaces spectraux associés à des treillis distributifs qui sont définis de façon naturelle et concrète.Souligons deux faits d’expérience importants.Tout d’abord, les preuves abstraites au sujet des points de ces espaces sans points peuvent en général être relues comme des preuves concernant les parties constructibles de ces espaces.Enfin, les espaces de fonctions continues sur ces espaces sans points sont souvent utilisés dans d’élégantes théories abstraites. Ces espaces de fonctions sont bien définis constructivement. Cela tient au « théorème de compacité » qui nous dit que dans le cadre en question « tout est fini ». La relecture constructive des preuves abstraites n’est alors rien d’autre que la constatation que les axiomes géométriques sont utilisés de manière correcte. (shrink)

The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C 0,2 . This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, $\mathbb{F}^{a}$ of the Euclidean three-space. This (...) enables us to construct a Poisson Clifford algebra, ℍ F , of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realisation of ℍ F in terms of a Clifford algebra consisting of Hermitian operators. (shrink)

This paper discusses four diagrams in the Archimedes Palimpsest, a manuscript that provides among other texts the only extant witness to Archimedes’ Method. My study of the two diagrams for Method 12 aims to open up discussions about the following two questions. First, I want to question the assumed relationship between diagram and geometric configuration. Rather than a representation-represented relation, I argue that the two diagrams for Method 12 have a stronger independence from the geometric configuration they are (...) related to. Their connection with the theorem rather lies in their role in shaping the way in which the proof is written. Secondly, I want to examine the connection between geometric intuition and figure. Geometric intuition is the recognition of certain properties or relations in a geometric configuration not through deduction and is held to be parallel to, if not dependent on, the empirical observation of a physical figure. Method 12 provides a curious case where an intuitive fact is not only unrepresented, but even hindered by the diagrams, and the proof assumes that the reader has come to realise that fact, which is most easily achieved when the diagrams are ignored. (shrink)

In this paper, it is shown that if [Formula: see text] is a complete type of Lascar rank at least 2, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations [Formula: see text], [Formula: see text] such that p has a nonalgebraic forking extension over [Formula: see text]. Moreover, if A is contained in the field of constants then p already has a nonalgebraic forking extension over [Formula: see text]. The results are (...) also formulated in a more general setting. (shrink)

We give a geometric realization of space-time spinors and associated representations, using the Jordan triple structure associated with the Cartan factors of type 4, the so-called spin factors. We construct certain representations of the Lorentz group, which at the same time realize bosonic spin-1 and fermionic spin- $${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{$2$}}$$ wave equations of relativistic field theory, showing some unexpected relations between various low-dimensional Lorentz representations. We include a geometrically and physically motivated introduction to Jordan triples and spin (...) factors. (shrink)

Gordon Belot investigates the distinctive notion of geometric possibility that relationalists rely upon. He examines the prospects for adapting to the geometric case the standard philosophical accounts of the related notion of physical possibility, with particular emphasis on Humean, primitivist, and necessitarian accounts of physical and geometric possibility. This contribution to the debate concerning the nature of space will be of interest not only to philosophers and metaphysicians concerned with space and time, but also to those interested (...) in laws of nature, modal notions, or more general issues in ontology. (shrink)

The paper argues that Roger Bacon adhered to a unique form of geometrical atomism, according to which elemental matter can be analysed into cubic (when at rest) or pyramidal (when in motion) portions. Bacon addressed geometrical atomism from the perspective of the Aristotelian review, using his interpretation of Aristotelian principles to render the theory plausible. He was mostly concerned with solving the contradiction between the angular shapes of the portions and the shape of the elemental spheres. His motivation for doing (...) so, I argue, was his conviction in the applicability of geometry onto natural philosophy, and his goal was to render matter capable of geometrical analysis. In this way, he was convinced, geometrical properties can be considered efficient causes of matter’s spatial formation and motion. (shrink)

When faced with uncertainty, consequentialists often advocate choosing the option with the largest expected utility, as calculated using the arithmetic average. I provide some arguments to suggest that instead, one should consider choosing the option with the largest geometric average of utility. I explore the difference between these two approaches in a variety of ethical dilemmas and argue that geometric averaging has some appealing properties as a normative decision-making tool.

Evans’ 1968 ANALOGY system was the first computer model of analogy. This paper demonstrates that the structure mapping model of analogy, when combined with high‐level visual processing and qualitative representations, can solve the same kinds of geometric analogy problems as were solved by ANALOGY. Importantly, the bulk of the computations are not particular to the model of this task but are general purpose: We use our existing sketch understanding system, CogSketch, to compute visual structure that is used by our (...) existing analogical matcher, Structure Mapping Engine (SME). We show how SME can be used to facilitate high‐level visual matching, proposing a role for structural alignment in mental rotation. We show how second‐order analogies over differences computed via analogies between pictures provide a more elegant model of the geometric analogy task. We compare our model against human data on a set of problems, showing that the model aligns well with both the answers chosen by people and the reaction times required to choose the answers. (shrink)

At some point in the Incessu Animalium, Aristotle appeals to some geometrical claims in order to explain why animal progression necessarily involves the bending (of the limbs), and this appeal to geometrical claims might be taking as violating the recommendation to avoid “kind-crossing” (as found in the Posterior Analytic). But a very unclear notion of kind-crossing has been assumed in most debates. I will argue that kind-crossing in the Posterior Analytics does not mean any employment of premises from a discipline (...) other than that to which the explanandum belongs. Kind-crossing was meant to cover a specific sort of employment of premises from a different discipline, namely, the case in which premises from a discipline X are taken as the most important explanatory factor that delivers the fullest appropriate explanation of an explanandum within discipline Y. If this is so, the employment of geometrical premises in the Incessu Animalium is not an instance of the prohibited kind-crossing, but something that is in line with the theory of the Posterior Analytics. (shrink)

It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (...) (Implication via spacetime. In: Mathematics, logic, and their philosophies: essays in honour of Mohammad Ardeshir, pp 161–216, 2021). Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category $${\mathcal {S}}$$ S, provided that $${\mathcal {S}}$$ S has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication. (shrink)

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the (...) issue. Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, which, en passant, provides a precise characterization of the idea of a "geometric object", I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime. I conclude by showing that my results also imply that, under a few natural conditions, the Einstein field equation is the unique equation relating gravitational phenomena to spatiotemporal structure, and discuss how this relates to the non-localizability of gravitational stress-energy. (shrink)

Large portions of mathematics such as algebra and geometry can be formalized using first-order axiomatizations. In many cases it is even possible to use a very well-behaved class of first-order axioms, namely, what are called coherent or geometric implications. Such class of axioms can be translated to inference rules that can be added to a sequent calculus while preserving its structural properties. In this work, this fundamental result is extended to their infinitary generalizations as extensions of sequent calculi for (...) both classical and intuitionistic infinitary logic. As an application, a simple proof of the infinitary Barr’s theorem without the axioms of choice is shown. (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.

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure of (...) processing complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations. (shrink)

Modern formal accounts of the constructive nature of elementary geometry do not aim to capture the intuitive or concrete character of geometrical construction. In line with the general abstract approach of modern axiomatics, nothing is presumed of the objects that a geometric construction produces. This study explores the possibility of a formal account of geometric construction where the basic geometric objects are understood from the outset to possess certain spatial properties. The discussion is centered around Eu , (...) a recently developed formal system of proof (presented in Mumma (Synthese 175:255–287, 2010 )) within which Euclid’s diagrammatic proofs can be represented. (shrink)

This paper is a direct successor to 12. Its aim is to introduce a new realisability interpretation for weak systems of explicit mathematics and use it in order to analyze extensions of the theory PET in 12 by the so-called join axiom of explicit mathematics.

There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential compass-and-straightedge drawings qua quantitative and (...) continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous. (shrink)

Ever since Plato, a tragic conception of the human self has been the point de depart of moral and political philosophy: the I and the we belong to one another yet oppose each other. Ancients such as Aristotle contended that the we is ontologically prior and moderns such as Hobbes that the I is ontologically prior. I make the case that Jesus Christ realised an ontology which collapses this dichotomy: the human self is neither I nor we, but fundamentally I-we. (...) I demonstrate that this is an ontology of gift-dynamics, made explicit in the mythical complex of the cult centring on Jesus Christ, and engraved unto this cult's heart through ritual. (shrink)

Ever since Plato, a tragic conception of the human self has been the point de depart of moral and political philosophy: the I and the we belong to one another yet oppose each other. Ancients such as Aristotle contended that the we is ontologically prior and moderns such as Hobbes that the I is ontologically prior. I make the case that Jesus Christ realised an ontology which collapses this dichotomy: the human self is neither I nor we, but fundamentally I-we. (...) I demonstrate that this is an ontology of gift-dynamics, made explicit in the mythical complex of the cult centring on Jesus Christ, and engraved unto this cult's heart through ritual. (shrink)

The aim of this paper is to provide a logic-based conceptual analysis of the twin paradox (TwP) theorem within a first-order logic framework. A geometrical characterization of TwP and its variants is given. It is shown that TwP is not logically equivalent to the assumption of the slowing down of moving clocks, and the lack of TwP is not logically equivalent to the Newtonian assumption of absolute time. The logical connection between TwP and a symmetry axiom of special relativity is (...) also studied. (shrink)

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investigation, we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, Geach laws (...) and Switching laws. (shrink)

We analyze the geometric foundations of classical Yang-Mills theory by studying the relationships between internal relativity, locality, global/local invariance, and background independence. We argue that internal relativity and background independence are the two independent defining principles of Yang-Mills theory. We show that local gauge invariance -heuristically implemented by means of the gauge argument- is a direct consequence of internal relativity. Finally, we analyze the conceptual meaning of BRST symmetry in terms of the invariance of the gauge fixed theory under (...) general local gauge transformations. (shrink)

I draw together some recent literature on the debate between dynamical versus geometrical approaches to spacetime theories, in order to argue that there exist defensible versions of the geometr...

We develop the theory of the nonadiabatic geometric phase, in both the Abelian and non-Abelian cases, in quaternionic Hilbert space.

The Geometrical Method The Geometrical Method is the style of proof that was used in Euclid’s proofs in geometry, and that was used in philosophy in Spinoza’s proofs in his Ethics. The term appeared first in 16th century Europe when mathematics was on an upswing due to the new science of mechanics. … Continue reading Geometrical Method →.

The object of this paper is to show the predominance of the influence of geometrical ideas in Aristotle's account of first principles in the Posterior Analytics— to show that his analysis of first principles is in its essentials an analysis of the first principles of geometry as he conceived them. My proof of this falls into two parts. I. A consideration of the parallel between Aristotle's and Euclid's account of first principles. II. A comparison between the general movement of thought (...) in the Aristotelian παγωγ and πδειξις and in the δινοια and νησις of the Platonic dialectic. (shrink)

In a recent paper, Takacs and Bourrat (Biol Philos 37:12, 2022) examine the use of geometric mean reproductive output as a measure of biological fitness. We welcome Takacs and Bourrat’s scrutiny of a fitness definition that some philosophers have adopted uncritically. We also welcome Takacs and Bourrat’s attempt to marry the philosophical literature on fitness with the biological literature on mathematical measures of fitness. However, some of the main claims made by Takacs and Bourrat are not correct, while others (...) are correct but not for the reasons they give. (shrink)

Les auteurs présentent les pratiques des collectifs d’architectes, designers, artistes et citoyens qui occupent le devant de la scène depuis la fin des années 1990 dans les espaces urbains, en formulant l’hypothèse qu’ils s’inspirent (de façon plus ou moins consciente et explicite) des écrits et des expérimentations menées par les représentants de la mouvance architecturale dite « radicale » qui les a précédés dans les années 1970. Dans cette perspective, ils analysent en particulier l’héritage de la notion d’« utopie réalisable (...) » développée par l’architecte Yona Friedman. (shrink)

Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.

This article analyzes Galileo's mathematization of motion, focusing in particular on his use of geometrical diagrams. It argues that Galileo regarded his diagrams of acceleration not just as a complement to his mathematical demonstrations, but as a powerful heuristic tool. Galileo probably abandoned the wrong assumption of the proportionality between the degree of velocity and the space traversed in accelerated motion when he realized that it was impossible, on the basis of that hypothesis, to build a diagram of the law (...) of fall. The article also shows how Galileo's discussion of the paradoxes of infinity in the First Day of the Two New Sciences is meant to provide a visual solution to problems linked to the theory of acceleration presented in Day Three of the work. Finally, it explores the reasons why Cavalieri and Gassendi, although endorsing Galileo's law of free fall, replaced Galileo's diagrams of acceleration with alternative ones. (shrink)

L'A. établit la biographie de Jean le Géomètre en se référant à ses poèmes et à ses épigrammes. Ces poèmes à Basile le Nothos apportent des éclaircissements sur sa carrière comme poète lauréat et sur ses attachements politiques. Il a notamment été renvoyé du service militaire juste après 985 à cause de son attachement à la faction de Basile le Nothos. Ces poèmes à Psenas prouvent indirectement qu'il a vécu au monastère de Kyros vers la fin de sa vie.

Joannes Geometres gehörte als einer der führenden Rhetoriker und Dichter seiner Zeit und als Offizier in der byzantinischen Armee – ϰαί σοφίη θάλλων ϰαί τόλμη ϰϱαδίης – zur politischen und literarischen Elite Konstantinopels. Nicht lange vor dem Jahr 986 wurde er aus dem Militärdienst entlassen; wie er selbst zu verstehen gibt, lag seine Tätigkeit als Soldat und Dichter, die bei seinen Zeitgenossen Neid ausgelöst hätte, diesem Ereignis zugrunde; die wirkliche Ursache war aller Wahrscheinlichkeit nach jedoch seine Sympathie für Basileios Nothos, (...) der seit 976 de facto als Kaiser des Reiches aufgetreten war, 985 aber gestürzt wurde. Wie dem auch sei, Joannes verließ seine luxuriöse Wohnung im Mesomphalos-Bezirk der Metropole und zog sich als Mönch ins Kloster Τα Κύϱου zurück; möglicherweise bekam er infolgedessen seinen zweiten Namen: Kyriotes. (shrink)