The first article in this issue attempts to refute the concept of Altruism and calls it akin to Selfishness. The arguments are logically set in the way like that of Spinoza’s method of demonstration, with Axioms, Definitions, Propositions and Notes: so as to make them exact and precise. Interestingly, the writer introduces a new concept of Credit and through various other original propositions and examples rebuts the altruistic nature which is generally ascribed to humans.

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)

In the Critique of Pure Reason, Kant defends the mathematically deterministic world of physics by arguing that its essential features arise necessarily from innate forms of intuition and rules of understanding through combinatory acts of imagination. Knowing is active: it constructs the unity of nature by combining appearances in certain mandatory ways. What is mandated is that sensible awareness provide objects that conform to the structure of ostensive judgment: “This (S) is P.” -/- Sensibility alone provides no such objects, so (...) the imagination compensates by combining passing point-data into “pure” referents for the subject-position, predicate-position, and copula. The result is a cognitive encounter with a generic physical object whose characteristics—magnitude, substance, property, quality, and causality—are abstracted as the Kantian categories. Each characteristic is a product of “sensible synthesis” that has been “determined” by a “function of unity” in judgment. -/- Understanding the possibility of such determination by judgment is the chief difficulty for any rehabilitative reconstruction of Kant’s theory. I will show that Kant conceives of figurative synthesis as an act of line-drawing, and of the functions of unity as rules for attending to this act. The subject-position refers to substance, identified as the objective time-continuum; the predicate-position, to quality, identified as the continuum of property values (constituting the second-order type named by the predicate concept). The upshot is that both positions refer to continuous magnitudes, related so that one (time-value) is the condition of the other (property-value). -/- Kant’s theory of physically constructive grammar is thus equivalent to the analytic-geometric formalism at work in the practice of mathematical physics, which schematizes time and state as lines related by an algebraic formula. Kant theorizes the subject–predicate relation in ostensive judgment as an algebraic time–state function. When aimed towards sensibility, “S is P” functions as the algebraic relation “t → ƒ(t).”. (shrink)

This article analyzes the value of geometric models to understand matter with the examples of the Platonic model for the primary four elements (fire, air, water, and earth) and the models of carbon atomic structures in the new science of crystallography. How the geometry of these models is built in order to discover the properties of matter is explained: movement and stability for the primary elements, and hardness, softness and elasticity for the carbon atoms. These geometric models appear to (...) have a double quality: firstly, they exhibit visually the scientific properties of matter, and secondly they give us the possibility to visualize its whole nature. Geometrical models appear to be the expression of the mind in the understanding of physical matter. (shrink)

Newton frequently characterized his methodology as distinctive and capable of achieving greater evidential support than that of his contemporaries, due to its mathematical character. Newton's pronouncements reflect a striking position regarding the role of mathematics in natural philosophy. We can give an initial characterization of his position by considering two questions central to seventeenth century debates about the applicability of mathematics. First, how are we to understand the distinctive universality and necessity of mathematical reasoning? One common way to preserve the (...) demonstrative character of mathematics was to restrict its domain, as far as possible, to pure abstractions. The subject matter of mathematics is then taken to be abstracted from the changeable natural world, consisting of quantity and magnitude themselves rather than the objects bearing quantifiable properties. Yet restricting the domain in this way leaves it difficult to see how mathematics relates to natural phenomena. How could the book of nature be written in a language of pure abstractions? Second, what is the proper role of mathematical reasoning in natural philosophy? Many followed Aristotle in consigning mathematics to a subordinate role. Mathematics could not fulfill the main aim of natural philosophy, namely to provide demonstrations reflecting the essences of things and nature's causal order. A related concern was also pressing for the mechanical philosophers: a merely mathematical demonstration fails to provide an intelligible mechanical explanation. The very title of Newton's masterpiece, Philosophiae Naturalis Principia Mathematica, would have been extremely perplexing: how could natural philosophy be based on mathematical principles? My aim is to articulate Newton's position regarding the mathematization of nature in response to these two concerns. (shrink)

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

In a first investigation, a Lacan-motivated template of the Poe story is fitted to the data. A segmentation of the storyline is used in order to map out the diachrony. Based on this, it will be shown how synchronous aspects, potentially related to Lacanian registers, can be sought. This demonstrates the effectiveness of an approach based on a model template of the storyline narrative. In a second and more Comprehensive investigation, we develop an approach for revealing, that is, uncovering, (...) Lacanian register relationships. Objectives of this work include the wide and general application of our methodology. This methodology is strongly based on the “letting the data speak” Correspondence Analysis analytics Platform of Jean-Paul Benzécri, that is also the geometric data analysis, both qualitative and quantitative analytics, developed by Pierre Bourdieu. (shrink)

The discussions of conatus – force, tendency, effort, and striving – in early modern metaphysics have roots in Aristotle’s understanding of life as an internal experience of living force. This paper examines the ways that Spinoza’s conatus is consonant with Aristotle on effort. By tracking effort from his psychology and ethics to aesthetics, I show there is a conatus at the heart of the activity of the ψυχή that involves an intensification of power in a way which anticipates many of (...) the central insights of early modern and 20th century European philosophy. The first section outlines how Aristotle’s developmental conception of the soul as geometrically ordered lays the foundation for his understanding of effort. The developmental series of powers of the soul are analogous to the series of shapes in mathematics. The second section links the striving of the soul to the gradual acquisition of virtues as a directed activity unifying multiplicity. The third examines the paradigm of self-awareness that Aristotelian effort involves. In the final section I show how ancient Greek theories of music were founded on the experience of striving. The “nature” of music is defined by Aristoxenus, and Theophrastus, in relation to the passion and intentionality of the soul. The geometrical order, as a synthesis of elements in geometry, music, or ethics, is a generative process in which past elements are retained and reintegrated in later stages of development. It requires effort to think geometrically, and the progress of knowledge itself is an integral aspect of all effort. Effort is the lived and self-aware cause which, moving step by step in an orderly and deliberate way, grows and advances upon itself. For both Spinoza and Aristotle, effort is the immanent intelligence which accomplishes what is in the purview of its understanding. Thus, will, in this conception of effort, is not something we already possess innately, but emerges gradually by an effort aimed at improvement. (shrink)

Baruch, or Benedictus, Spinoza (1632–77) is the author of works, especially the Ethics and the Theological-Political Treatise, that are a major source of the ideas of the European Enlightenment. The Ethics is a dense series of arguments on progressively narrower subjects – metaphysics, mind, the human affects, human bondage to passion, and human blessedness – presented in a geometrical order modeled on that of Euclid. In it, Spinoza begins by defending a metaphysics on which God is the only (...) substance and is bound by the laws of his own nature. Spinoza then builds a naturalistic ethics that is constrained by, and to some extent is a product of, his strong metaphysics. Human beings are individuals that causally interact with other individuals and are extremely vulnerable to external influence. They are not substances. Moreover, human beings are bound by the same laws that bind all other individuals in nature, so Spinoza presents accounts of goodness, virtue, and perfection that are consistent with these perfectly general laws. Spinoza’s principal influences include René Descartes, Thomas Hobbes (see hobbes, thomas), Moses Maimonides (see maimonides, moses), the Roman Stoics (see stoicism), and Aristotle (see aristotle). Although his innovative philosophical views undoubtedly contributed to the strong writ of cherem, or ostracism, that Spinoza received from the Portuguese Jewish community of Amsterdam in 1656, his work nevertheless also shows the influence of the study of Scripture and of Jewish law. (shrink)

In order to describe my findings/conclusions systematically, a new semantic system (i.e., a new language) has to be intentionally defined by the present article. Humans are limited in what they know by the technical limitation of their cortical language network. A reality is a situation model (SM). For example, the conventionally-called “physical reality” around my conventionally-called “physical body” is actually a “geometric” SM of my brain. The universe is an autonomous objective parallel computing automaton which evolves by itself automatically/unintentionally (...) – wave-particle duality and Heisenberg’s uncertainty principle can be explained under this “first-order” SM of my brain. Each elementary particle (as a building block of the universe) is an autonomous mathematical entity itself (i.e., a thing in itself). If we are happy to accept randomness, it is obviously possible that all other worlds in the many-worlds interpretation do not exist objectively. The conventionally-called “space” does not exist objectively. “Time” and “matter” are not physical. Consciousness is the subjective-form (aka quale) of the mathematical models (of the objective universe) which are intracorporeally/subjectively used by the control logic of a Turing machine’s program objectively-fatedly. A Turing machine’s consciousness or deliberate decisions/choices should not be able to actually/objectively change/control/drive the (autonomous or objectively-fated) world line of any elementary particle within this world. Besides the Schrodinger equation (or its real-world counterpart) which is a valid/correct/factual causality of the universe, every other causality (of the universe) is either invalid/incorrect/counterfactual or can be proved by deductive inference based on the Schrodinger equation only. If the “loop quantum gravity” theory is correct, time/space does not actually/objectively exist in the objective-evolution of the objective-reality, or in other words, we should not use the subjective/mental concept of “time”, “state” or “space” to describe/imagine the objective-evolution of the universe. (shrink)

On 6 January 1795, the twenty-year-old Schelling—still a student at the Tübinger Stift—wrote to his friend and former roommate, Hegel: “Now I am working on an Ethics à la Spinoza. It is designed to establish the highest principles of all philosophy, in which theoretical and practical reason are united”. A month later, he announced in another letter to Hegel: “I have become a Spinozist! Don’t be astonished. You will soon hear how”. At this period in his philosophical development, Schelling had (...) been deeply under the spell of Fichte’s new philosophy and the Wissenschaftslehre. The text Schelling was writing at the time was the early Vom Ich als Prinzip der Philosophie, though his characterization of this text would much better fit the somewhat later work which is the focus of the current paper: Schelling’s 1801 Darstellung meines System der Philosophie (hereafter: Presentation). The Presentation is a text written more geometrico, following the style of Spinoza’s Ethics. While Spinoza’s influence and inspiration is stated explicitly and unmistakably in Schelling’s preface, the content of this composition might seem quite foreign to Spinoza’s philosophy, so much so, in fact, that Michael Vater—the astute translator and editor of the recent English translation of the text—has contended that “despite the formal similarities between Spinoza’s geometrical method and Schelling’s numbered mathematical-geometrical constructions, Schelling’s direct debts to Spinoza are few”. The Presentation is an extremely dense and difficult text, and while I agree that at first glance Schelling’s engagement with the concept of reason (Vernunft) and the identity formula ‘A=A’ seems to have little if anything to do with Spinoza (especially since Spinoza’s key terminology of ‘God’, ‘causa sui’, ‘substance’, ‘attribute’, and ‘mode’ is barely mentioned in the Presentation), I suspect that at a deeper level Schelling is attempting to transform Spinoza’s system by replacing God, Spinoza’s ultimate reality, with reason. Though this might at first seem bizarre, I believe it can be profitably motivated and explained upon further reflection. It is this transformation of Spinoza’s God into (the early) Schelling’s reason that is the primary subject of this study. I develop this paper in the following order. In the first part I provide a very brief overview of Schelling’s lifelong engagement with Spinoza’s philosophy, which will prepare us for my study of the 1801 Presentation. In the second part, I consider the formal structure and rhetoric of the Presentation against the background of Spinoza’s Ethics, and show how Schelling regularly imitates Spinoza’s tiniest rhetorical gestures. In the third and final part I turn to the opening of the Presentation, and argue that Schelling attempts there to distance himself from Fichte by developing a conception of reason as the absolute, or the identity of the subject and object, just as the thinking substance and the extended substance are identified in Spinoza’s God. (shrink)

We present a novel approach to defining the mathematical constant π through the infinite den- sification of a sweeping net, which approximates a circle as the net becomes infinitely dense. By developing and enhancing notation related to sweeping nets and saddle maps, we establish a rigor- ous framework for expressing π in terms of the densification process using reverse integration. This method, inspired by the concept that numbers ”come from infinity,” leverages a reverse integral approach to model the transition from (...) infinite densification to the finite circle. Our work not only offers a new perspective on the geometric interpretation of π but also provides insights into reverse integration techniques and their applications in mathematical analysis. (shrink)

Gjirokastra’s buildings occupy a special place in the housing typology of Albanian popular dwellings in the feudal period. The “popular tower" is linked with its defensive character, therefore in many cases, it takes the name of a castle or defensive tower. This paper takes into consideration a typical example of the historical fortified dwelling in a well-known city of Albania, Gjirokastra. The methodology used in order to improve the way of thinking, the way of implementing, and the way of (...) designing future dwellings is very related to the historical attitude of the dwelling. Based on the historical documentation, of the fortified dwelling a real analysis is performed according to the logic of modulations. The purpose of this paper is to create a methodology for future generations of architects in order to better engage in the design process, respecting the valuable architectural history. The modern dwellings of Gjirokastra historical-urban context must adopt the historical architectural elements, the historical manners of construction, the typical construction materials, the operational mode, and people’s lifestyle, in order to preserve the city's historical values via a modern touch. KEYWORDS: Architectural elements, historical preservation values, geometric ratios, modular. (shrink)

Hermann von Helmholtz’s geometrical papers have been typically deemed to provide an implicitly group-theoretical analysis of space, as articulated later by Felix Klein, Sophus Lie, and Henri Poincaré. However, there is less agreement as to what properties exactly in such a view would pertain to space, as opposed to abstract mathematical structures, on the one hand, and empirical contents, on the other. According to Moritz Schlick, the puzzle can be resolved only by clearly distinguishing the empirical qualities of spatial (...) perception from those describable in terms of axiomatic geometry. This paper offers a partial defense of the group-theoretical reading of Helmholtz along the lines of Ernst Cassirer in the fourth volume of The Problem of Knowledge of 1940. In order to avoid the problem raised by Schlick, Cassirer relied on a Kantian view of space not so much as an object of geometry, but as a precondition for the possibility of measurement. Although the concept of group does not provide a description of space, the modern way to articulate the concept of space in terms of transformation groups reveals something about the structure and the transformation of spatial concepts in mathematical and natural sciences. (shrink)

In this paper, we propose a mathematical model of subjective experience in terms of classes of hierarchical geometries of representations (“n-awareness”). We first outline a general framework by recalling concepts from higher category theory, homotopy theory, and the theory of (infinity,1)-topoi. We then state three conjectures that enrich this framework. We first propose that the (infinity,1)-category of a geometric structure known as perfectoid diamond is an (infinity,1)-topos. In order to construct a topology on the (infinity,1)-category of diamonds we then (...) propose that topological localization, in the sense of Grothendieck-Rezk-Lurie (infinity,1)-topoi, extends to the (infinity,1)-category of diamonds. We provide a small-scale model using triangulated categories. Finally, our meta-model takes the form of Efimov K-theory of the (infinity,1)-category of perfectoid diamonds, which illustrates structural equivalences between the category of diamonds and subjective experience (i.e.its privacy, self-containedness, and self-reflexivity). Based on this, we investigate implications of the model. We posit a grammar (“n-declension”) for a novel language to express n-awareness, accompanied by a new temporal scheme (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? We also examine the notion of “self” within our framework. A new model of personal identity is introduced which resembles a categorical version of the “bundle theory”: selves are not substances in which properties inhere but (weakly) persistent moduli spaces in the K-theory of perfectoid diamonds. (shrink)

How should your opinion change in light of an epistemic peer's? We show that the pooling rule known as "upco" is the unique answer satisfying some natural desiderata. If your revised opinion will impact your other views by Jeffrey conditionalization, then upco is the only standard pooling rule that ensures the order in which peers are consulted makes no difference. Popular alternatives like linear pooling, geometric pooling, and harmonic pooling cannot boast the same. In fact, no alternative can that (...) possesses four minimal properties these proposals all share. (shrink)

Philosophy, it is said, has always been concerned with space and nature. Since Aristotle, even since the pre-Socratics, and up to Descartes and Leibniz at least, no one was worthy of the title of philosopher if he had not written something about meteors. And not only about meteors: observations on the movements of the Earth and changes in nature, changes of state or speed, transitions and contacts between strange elements, spatial geometrical properties, the order of coexistences - their (...) limits - and the correspondences between scales; force and matter, constructive typologies and geomorphology were an indissoluble part of the philosophical field. It is not that philosophy reflected on these external themes, but that these themes formed the fabric of the philosophical body. What happened, then, between the end of the seventeenth century and the first half of the eighteenth century? How did the themes related to space and nature become diluted and disappear from philosophical discourses? It is not a question of the split between man and nature, ancient and incommensurable, but of the separation of nature itself, modern and discontinuous, by means of which it was placed alongside the barren, the inert, as a category. -/- La filosofía, se dice, se ocupó siempre de los espacios y de la naturaleza. Desde Aristóteles, incluso desde los presocráticos, y hasta Descartes y Leibniz por lo menos, nadie era digno del título de filósofo si no había escrito algo sobre los meteoros. Y no sólo sobre los meteoros: las observaciones sobre los movimientos de la Tierra y la mudanza en la naturaleza, los cambios de estado o de velocidad, las transiciones y los contactos entre elementos extraños, las propiedades geométricas espaciales, el orden de las coexistencias —sus límites— y las correspondencias entre las escalas; la fuerza y la materia, las tipologías constructivas y la geomorfología formaban parte indisoluble del campo filosófico. No es que la filosofía reflexionase sobre esos temas externos, sino que esos temas formaban el tejido del cuerpo filosófico. ¿Qué sucedió, entonces, entre el final del siglo XVII y la primera mitad del siglo XVIII? ¿Cómo fue que los temas relacionados con el espacio y la naturaleza se diluyeron y desaparecieron de los discursos filosóficos? No se trata de la escisión entre hombre y naturaleza, antigua e inconmensurable, sino de la separación de sí de la naturaleza misma, moderna y discontinua, por medio de la cual fue puesta al lado de lo yerto, lo inerte, como una categoría. (shrink)

It is now well-known that Newton–Cartan theory is the correct geometrical setting for modelling the quantum Hall effect. In addition, in recent years edge modes for the Newton–Cartan quantum Hall effect have been derived. However, the existence of these edge modes has, as of yet, been derived using only orthodox methodologies involving the breaking of gauge-invariance; it would be preferable to derive the existence of such edge modes in a gauge-invariant manner. In this article, we employ recent work by (...) Donnelly and Freidel in order to accomplish exactly this task. Our results agree with known physics, but afford greater conceptual insight into the existence of these edge modes: in particular, they connect them to subtle aspects of Newton–Cartan geometry and pave the way for further applications of Newton–Cartan theory in condensed matter physics. (shrink)

Abstract. The aim of this paper is to present a topological method for constructing discretizations (tessellations) of conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. Alexandroff spaces, as they are called today, have many interesting properties that distinguish them from other topological spaces. In particular, they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, a special type of Alexandroff spaces was (...) used by Ian Rumfitt to elucidate the logic of vague concepts in a new way. According to his approach, conceptual spaces such as the color spectrum give rise to classical systems of concepts that have the structure of atomic Boolean algebras. More precisely, concepts are represented as regular open regions of an underlying conceptual space endowed with a topological structure. Something is subsumed under a concept iff it is represented by an element of the conceptual space that is maximally close to the prototypical element p that defines that concept. This topological representation of concepts comes along with a representation of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical operations that characterize regular open interpretations of classical Boolean propositional logic. In the last 20 years, conceptual spaces have become a popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using prototypes and metrics of similarity spaces, one obtains geometrical discretizations of conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces. Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. This class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher2 order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The crucial role of order theory for Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical stimuli in favor of a more fine-grained gradual distinction between more-orless prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called “thickness problem” (Douven et al.) and problems of selecting a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines. Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Voronoi Tessellations, Digital Topology. (shrink)

This paper attempts to demonstrate that the conviction about the harmony and order of the world was a fundamental metaphysical principle of the Pythagoreans. This harmony and order were primarily sought in the structures of arithmetics, yet following the discovery of incommensurable magnitudes (irrational numbers, as we now call them), the Pythagoreans began to see geometrical structure as a fundamental part of the world. On the example of the Pythagoreans’ metaphysics and science, the paper shows the mutual (...) relations between metaphysics and science. It demonstrates— on the one hand—the necessity of the first as a guide for the latter, and—on the other—how our scientific research can change its basic metaphysical principles when these are found to be inappropriate. The paper also tries to show the need for a realistic approach in our scientific research by means of the same example of the Pythagoreans, that is, the need to discern something which is below the surface appearance. (shrink)

There are many reasons we might want to take the opinions of various individuals and pool them to give the opinions of the group they constitute. If all the individuals in the group have probabilistic opinions about the same propositions, there is a host of pooling functions we might deploy, such as linear or geometric pooling. However, there are also cases where different members of the group assign probabilities to different sets of propositions, which might overlap a lot, a little, (...) or not at all. There are far fewer proposals for how to proceed in these cases, and those there are have undesirable features. I begin by considering four proposals and arguing that they don't work. Then I'll describe my own proposal, which is intended to cover the situation in which we want to pool the individual opinions in order to ascribe an opinion to the group considered as an agent in its own right. (shrink)

Although Kant (1998) envisaged a prominent role for logic in the argumentative structure of his Critique of Pure Reason, logicians and philosophers have generally judged Kantgeneralformaltranscendental logics is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first-order logic. The main technical application of the formalism developed here is a formal proof that Kants logic is after all a distinguished subsystem of first-order logic, (...) namely what is known as geometric logic. (shrink)

This paper formalizes part of the cognitive architecture that Kant develops in the Critique of Pure Reason. The central Kantian notion that we formalize is the rule. As we interpret Kant, a rule is not a declarative conditional stating what would be true if such and such conditions hold. Rather, a Kantian rule is a general procedure, represented by a conditional imperative or permissive, indicating which acts must or may be performed, given certain acts that are already being performed. These (...) acts are not propositions; they do not have truth-values. Our formalization is related to the input/ output logics, a family of logics designed to capture relations between elements that need not have truth-values. In this paper, we introduce KL3 as a formalization of Kant’s conception of rules as conditional imperatives and permissives. We explain how it differs from standard input/output logics, geometric logic, and first-order logic, as well as how it translates natural language sentences not well captured by first-order logic. Finally, we show how the various distinctions in Kant’s much-maligned Table of Judgements emerge as the most natural way of dividing up the various types and sub-types of rule in KL3. Our analysis sheds new light on the way in which normative notions play a fundamental role in the conception of logic at the heart of Kant’s theoretical philosophy. (shrink)

While there are satisfactory answers to the question “How should we teach children mathematics?”, there are no satisfactory answers to the question “What mathematics should we teach children?”. This paper provides an answer to the last question for preschool children (early childhood), although the answer is also applicable to older children. This answer, together with an appropriate methodology on how to teach mathematics, gives a clear conception of the place of mathematics in the children’s world and our role in helping (...) children develop their mathematical abilities. Briefly, children’s mathematics consists of the world of children’s internal activities that they eventually purposefully organize in order to understand and control the outside world and organize their overall activities in it. We need to support a child in mathematical activities that she does spontaneously and in which she shows interest, and we need to teach her mathematics that she is interested in developing through these activities. In doing so, we must be fully aware that the child’s mathematics is part of the child’s world of internal activities and is not outside of it. We help the child develop mathematical abilities by developing them in the context of her world and not outside of it. From the point of view of this conception, the standards established today are limiting and too focused on numbers and geometric figures: these topics are too prominent and elaborated, and other mathematical contents are subordinated to them. Adhering to the standards, we drastically limit the mathematics of the child’s world, hamper the correct mathematical development of a child, and we can turn her away from mathematics. (shrink)

This paper focuses on Leibniz’s conception of modality and its application to the issue of natural laws. The core of Leibniz’s investigation of the modality of natural laws lays in the distinction between necessary, geometrical laws on the one hand, and contingent, physical laws of nature on the other. For Leibniz, the contingency of physical laws entailed the assumption of the existence of an additional form of causality beyond mechanical or efficient ones. While geometrical truths, being necessary, do (...) not require the use of the principle of sufficient reason, physical laws are not strictly determined by geometry and therefore are logically distinct from geometrical laws. As a consequence, the set of laws that regulate the physical laws could have been created otherwise by God. However, in addition to this, the contingency of natural laws does not consist only in the fact that God has chosen them over other possible ones. On the contrary, Leibniz understood the status of natural laws as arising from the action internal to physical substances. Hence the actuality of physical laws results from a causal power that is inherent to substances rather than being the mere consequence of the way God arranged the relations between physical objects. Focusing on three instances of Leibniz’s treatment of contingency in physics, this paper argues that, in order to account for the contingency of physical laws, Leibniz maintained that final causes, in addition to efficient and mechanical ones, must operate in physical processes and operations. (shrink)

Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing (...) at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

This is a contribution to the idea that some proofs in first-order logic are synthetic. Syntheticity is understood here in its classical geometrical sense. Starting from Jaakko Hintikka’s original idea and Allen Hazen’s insights, this paper develops a method to define the ‘graphical form’ of formulae in monadic and dyadic fraction of first-order logic. Then a synthetic inferential step in Natural Deduction is defined. A proof is defined as synthetic if it includes at least one synthetic inferential (...) step. Finally, it will be shown that the proposed definition is not sensitive to different ways of driving the same conclusion from the same assumptions. (shrink)

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the (...) NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. (shrink)

This article reexamines Vico’s early critique of Cartesian reasoning and of how the Cartesian method, which comes from epistemology, creates problems for the sciences once embedded into their methodologies and given a foundational role. The focus will be on De nostri temporis studiorum ratione (1709), where Vico argues against generalizing the Cartesian method and overemphasizing clarity and distinctness in the search for truth. To this end, Vico’s relation to Cartesianism is first carefully contextualized. Then, Vico is presented as a hylomorphist (...) when it comes to how scientific disciplines and their instruments interrelate, something key for understanding why he questions the “instrumentalization” of the Cartesian method in the first place. Afterwards, Vico’s account of the impossibility of maximal scientific progress when the Cartesian method is prioritized across the sciences is presented, as probabilistic reasoning and the importance of the imagination and memory take centerstage. Lastly, the article looks at Vico’s opposition to the geometrical method’s role in physics and his defense of visual thinking in mathematics. The hope throughout is to advance a picture of the early Vico as being not only a neo- Baconian pragmatist, but a hypermodern ”meta-Cartesian” thinker concerned with humanity’s optimizing its creative potential in order to achieve scientific advances. (shrink)

In Hegel ou Spinoza,1 Pierre Macherey challenges the influence of Hegel’s reading of Spinoza by stressing the degree to which Spinoza eludes the grasp of the Hegelian dialectical progression of the history of philosophy. He argues that Hegel provides a defensive misreading of Spinoza, and that he had to “misread him” in order to maintain his subjective idealism. The suggestion being that Spinoza’s philosophy represents, not a moment that can simply be sublated and subsumed within the dialectical progression of (...) the history of philosophy, but rather an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. Gilles Deleuze also considers Spinoza’s philosophy to resist the totalising effects of the dialectic. Indeed, Deleuze demonstrates, by means of Spinoza, that a more complex philosophy antedates Hegel’s, which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuze’s project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference. It is Spinoza’s role in this project that will be demonstrated in this paper by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) in Expressionism in Philosophy, Spinoza,2 from that which Hegel presents in the Science of Logic.3. (shrink)

Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in order to show that (...) both Euclidean and non-Euclidean geometry are bodies of synthetic a priori truths. Michael Friedman has argued that this defence presumes a polyadic conception of logic that was foreign to Kant. According to Friedman, Kant held that geometrical reasoning itself relies essentially on intuition, and that this precludes the very possibility of non-Euclidean geometry. While Friedman's characterization of Kant's views on geometrical reasoning is correct, I argue that Friedman's conclusion that non-Euclidean geometries are logically impossible for Kant is not. I argue that Kant is best understood as a proto-constructivist and that modern constructive axiomatizations (unlike Hilbert-style axiomatizations) of both Euclidean and non-Euclidean geometry capture Kant's views on the essentially constructive nature of geometrical reasoning well. (shrink)

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

The paper examines Posterior Analytics II 11, 94a20-36 and makes three points. (1) The confusing formula ‘given what things, is it necessary for this to be’ [τίνων ὄντων ἀνάγκη τοῦτ᾿ εἶναι] at a21-22 introduces material cause, not syllogistic necessity. (2) When biological material necessitation is the only causal factor, Aristotle is reluctant to formalize it in syllogistic terms, and this helps to explain why, in II 11, he turns to geometry in order to illustrate a kind of material cause (...) that can be expressed as the middle term of an explanatory syllogism. (3) If geometrical proof is viewed as a complex construction built on simpler constructions, it can in effect be described as a case of purely material constitution. (shrink)

This article addresses Rancière’s critique of Aristotle’s political theory as parapolitics in order to show that Aristotle is a resource for developing an inclusionary notion of political community. Rancière argues that Aristotle attempts to cut off politics and merely police (maintain) the community by eliminating the political claim of the poor by including it. I respond to three critiques that Rancière makes of Aristotle: that he ends the political dispute by including the demos in the government; that he includes (...) the free masses only incidentally because, by chance, the rule is political, not based on mastery; and that he attempts to close off politics by determining in advance what speech counts as speech. I argue that Aristotle attempts to institute this contest by accepting the incommensurable claims of arithmetic and geometric equality. Aristotle puts this conflict at the root of political life in such a way that serves Rancière’s larger project of perpetuating politics. (shrink)

It will be a wrong judgment to consider grave stones as an ordinary tradition. When it is viewed in terms of history, art and culture, it can be seen that especially Turkish grave stones are record drawings that include many types of arts and artists’ labor, shed our culture and history and that is precious and unique. Grave stones are the documents that transfer not only the national culture but also transfer people’s beliefs, problems, fears, sadness and different feelings, who (...) breathe the same air and live in the same places. Because of this reason, grave stones are the cultural features that are shaped according to the differences in culture and decorated with life styles. Of course, Of course, having been applied plenty of traditional Turkish plastic arts on the tombstones, have all contributed to the diversity of tombstones in terms of composition and motifs. Rich craftsmanship of gravestones and sarcophagus in the second Bayezid graveyard is almost like an open air museum that shows Turkish motifs. While scarping was used for making gravestones are scarping, and painting were used as the main decoration tecnique is seen on foot stone with the painting technique, inscriptions of stones were coloured. Most of the inscriptions and decorations are relieved. Decoration subjects seen on gravestones are phytomorphic, geometric, objective and calligraphic. On the other side, these tombstones include wide knowledge about the socio-economic, cultural and regional headgears. When evaluating the information on the inscriptions, it can be obtained knowledge about the health conditions or epidemics in that period. Adolf Ebeling, article titled “Turkish cemeteries” in the 1882 the year Gartenlaube journal, when talking about the importance given to the graves by Turks, to say that the Turkish burial tomb can be an example to the world is no need to say more about it. The thomstones as the verbal means service to rebuilt the material world where the man has saved so far and the culture developing in accordance with this. The written tombstones which were formed by the individuals’ cherishing and remembering desires that the cultural, religious and traditional ones shaped, are the carrier elements of the values wich will provide to understand the sterotype requirements and ideologies within the cultural structure. Looking at the overall structure of the Turkish tombstones form shows major differences in male and female gravestones. In the Turkish tombstone tradition, women's elegance, finesse and beauty is reflected as a form of artwork. In fact, women in the Ottoman Empire, although excluded from the social life in very marginalized, especially in women flashy decorations tombstones, can be considered as an indication value which is given to women but unspoken. Since the tombstones have dates, they hold the distinction of being a document for ethnographic and art works. Moreover, inscribed tombstones are living evidences in order to prove that Turkish people lived in this land and also they have ethnographic value. Therefore, tombstones are important while the histories of cities, villages or boroughs are preparing. In short, tombstones are the common products of the environment that are built, religious beliefs, traditions, art, natural, social and economic conditions of the century. In this case they hold key for our culture history in addition to our art history. In this study, besides the effects of Turkish grave stones on our beliefs and daily lives, fruit patterns on the stones are also examined. The reflections traditional architectural and handicrafts motifs and the religious believes on the Turkish ornament art are explained by the examples chosen among gravestones in İstanbul II. Beyazıt Cemetery. In the colloquial language what fruit motifs mean in folk songs ang manias and how it is reflected on the ornaments of grave stones are investigated. (shrink)

Techne and Truth. The problem of techne in the dispute between Gorgias and Plato -/- The source of the problem matter of the book is the Plato’s dialogue „Gorgias”. One of the main subjects of the discussion carried out in this multi-aspect work is the issue of the art of rhetoric. In the dialogue the contemporary form of the art of rhetoric, represented by Gorgias, Polos and Callicles, is confronted with Plato’s proposal of rhetoric and concept of art (techne). The (...) ingenuous and dramatic structure of the dialogue composed of three acts, in each of which Socrates’ consecutive interlocutors emerge, is difficult to interpret. It seems, however, that even though the first conversation between Socrates and Gorgias constitutes a small part of the dialogue, it forms the foundations for the discussion. This part, dealing with such important issues as the subject of art, knowledge, power and relation between art and good, is the starting point for the conversations which follow. What do the differences between the Gorgias’ and Plato’s concepts of art consist in? Any attempt to answer the question requires presentation of the background for the discussion taking place in Plato’s work as well as the reconstruction of Gorgias’ vision of the art of rhetoric and art in general, defining the relation between art and knowledge. The latter may be undertaken on the basis of two paraphrases of the treatise “On Non-existence”, two epideictic speeches entitled “Helen” and “Palamedes” and some smaller fragments which have been preserved from Gorgias’ creative output. The treatise “On Non–existence”, very paradoxical in its philosophical meaning, testifies to new, sophistic thinking represented by Gorgias of Leontinoi. The treatise is the polemic and attack on Eleatism based on certain principles and methods elaborated by the Eleatics themselves. It also shows that Gorgias – using traditional Eleatic schemes – is opposed to this tradition. Even though the treatise does not discuss the problem of art directly, it anticipates the considerations from epideictic speeches. “Helen” and “Palamedes” constitute further stages in the development of Gorgias’ thought and concept of art combining typically epideictic elements with some philosophical convictions. Gorgias refers to the Eleatic theme considering the issues of truth, knowledge and belief, yet defining them in the empirical spirit. Solutions in the cognitive sphere determine the horizon of Gorgias’ rhetoric. The word is separated from the existence. Human cognition is limited and therefore the word and visual images affect the belief. This promotes the great importance of rhetoric, painting and sculpture, i.e. the areas which influence in a specific way – creating the “delusion” (ἀπάτη). Through idealization art provides pleasure, which is its objective. Therefore the task of art is to create delusion. The rhetor of Leontinoi discusses the issue of “how?” to use the word ‘rhetoric’. Gorgias is aware that rhetoric may be used to talk about what is unknown and that technical correctness of a rhetoric statement is not identical with the truthfulness of the proclaimed message. Mentioning it, Gorgias shows his respect to the truth as a value which should be respected in speech on the one hand, and on the other emphasizes that the truth in itself has no persuasive power. For this reason it is not enough to proclaim the truth, but, to convince the listeners, the orator should possess the skill of speaking. What is then the relation between truth and art emerging from epideictic speeches? Gorgias undoubtedly emphasizes the formal aspect of speech, as rhetoric is art, thus an ability based on certain principles. According to Gorgias, the truth is the value which should be respected in speeches. Rhetoric is able to create a fictive world, as a great range of reality remains outside direct human experience. Rhetoric does not oppose the truth but discussing not only what is directly given concerns the area outside the sphere of direct experience. This concept of rhetoric is the basis of the discussion about art carried out in the dialogue “Gorgias”. Even though the first of the discussions in the dialogue – the conversation between Socrates and Gorgias – is directly concerned with rhetoric itself, numerous remarks testify to its much wider character. This is best exemplified by the way of carrying out the conversation by Socrates, who, asking questions about rhetoric practised by Gorgias, throughout the whole conversation compares it with other skills. Additionally, the manner and the structure of the conversation prove that Plato considers the issue of rhetoric from a wider perspective determined by the issue of art. The specific feature of the conversation is ordering it around main issues – such notions as πρᾶγμα, ἐπιστήμη, δύναμις, each of which has its own meaning in Plato. In the first part of Gorgias Plato presents the principles previously occurring in other dialogues in a certain order, thus constructing the structure of the conversation with Gorgias. The notions are not novum in Plato but the fact that he clearly wants to systematise them and present a certain concept of art is a new and significant aspect. This tendency is of paramount importance for the discussion because Gorgias’ answers are evaluated by Plato from the perspective of this clearly set concept consisting of not only established notions but also certain principles. The discussion between Socrates and Gorgias introduces a theme which is crucial for the dialogue: the relation between art and good, i.e. the technical and ethical spheres. The issue is introduced by Gorgias, who mentions that rhetoric may be used for evil purposes. On this basis Socrates proves that if rhetoricians can use rhetoric for evil purposes, they have no knowledge concerning the subject of their art – justice, because one who knows what is just, always acts justly. Thus the relation between the technical and ethical spheres transpires to be the main point of contention. The question is whether a technician following the principles of the art he practices may act against good. This paradoxical statement based on the well-known Socrates’ principle "nemo sua sponte peccat" acquires philosophical foundations in the dialogue. In its remaining two parts Plato presents a justification, placing the principle within the framework of a certain concept of art, as a result of which not only rhetoric but also many other skills are criticised. The notions on which the conversation with Gorgias focuses are more precisely defined in the subsequent parts of the dialogue. Additionally, Plato presents twice the conditions which art should fulfill. They oblige the “technicians” to know the nature and purpose of the art and to know the causes of undertaken activities and to be able to justify them. These conditions determine the relation between the two spheres – technical and ethical. According to Plato, art may not act against good because it should be the objective of its activities. Certain concepts of good and knowledge outlined in the dialogue are the guarantees of “technicality”. Good, which should be respected by every art, is understood objectively and is contrasted with subjective pleasure. Knowledge, which should be the basis of art, opposes experience, which uses observation and recollection. Additionally, the definition of knowledge, significance of order in the world assuming the form of “geometrical equality”, significance of mathematics becoming the ideal of exactness indicate that Plato’s concept of knowledge is formed in Gorgias. These conclusions render it impossible that the results of any activity deserving the name of art should oppose good. And this is Plato’s main reproach of Gorgias’ concept, who wrote about false speeches possessing the power of conviction because they were written according to the principles proof of art or who defined tragedy as “a deceit, where he who cheats is more just than he who does not and the cheated is wiser than he who was not cheated”. Such determination of the relation between the technical and ethical sphere resulted from Gorgias’ concept of the world and fundamental philosophical principles. The whole concept of art described by Plato in the dialogue opposes the view presented by Gorgias. According to Plato, every technical activity has a defined object (ἔργον), is based on knowledge (ἐπιστήμη) and is always aimed at the good of human soul or body. The danger revealed and presented in the dialogue by Plato exists in the very concept of art presented by Gorgias, i.e. in its empirical foundations and the resulting understanding of the relation between art and good. Gorgias’ concept of art based on empirical cognition, which concerns the world of phenomena and rejects the possibility of perfect knowledge, leads, according to Plato, to a dangerous falsification of the relation between art and good. Plato proves that this weakness of Gorgias’ concept is revealed when confronted with these representatives of “the magnificent art” who are not protected against radicalism by the adherence to the moral tradition. The comparison of two, completely contrasting concepts of art based on different philosophical understanding of knowledge and good is decisive for the criticism of rhetoric in Gorgias. The concept presented by Plato criticises contemporary rhetoric and sophistry but determines the direction of research of the real art of rhetoric. (shrink)

An ideal constructor produces geometry from scratch, modelled through the bottom-up assembly of a graph-like lattice within a space that is defined, bootstrap-wise, by that lattice. Construction becomes the problem of assembling a homogeneous lattice in three-dimensional space; that becomes the problem of resolving geometrical frustration in quasicrystalline structure; achieved by reconceiving the lattice as a dynamical system. The resulting construction is presented as the introductory model sufficient to motivate the formal argument that it is a fundamental structure; based (...) on which, it is proposed that where mathematics’ numbers conventionally correspond to dimensionless points on the stateless number line, numbers more fundamentally correspond to an ordering of discrete objects constructed within the stateful number lattice. A second observation is that this fundamental lattice structure is helically configured with fractal character, which, as it relates to the geometry underlying spacetime, has relevance to questions in physics, particularly those involving wave-particle duality. (shrink)

Many global catastrophic risks are threatening human civilization, and a number of ideas have been suggested for preventing or surviving them. However, if these interventions fail, society could preserve information about the human race and human DNA samples in the hopes that the next civilization on Earth will be able to reconstruct Homo sapiens and our culture. This requires information preservation of an order of magnitude of 100 million years, a little-explored topic thus far. It is important that a (...) potential future civilization will discover this information as early as possible, thus a beacon should accompany the message in order to increase visibility. The message should ideally contain information about how humanity was destroyed, perhaps including a continuous recording until the end. This could help the potential future civilization to survive. The best place for long-term data storage is under the surface of the Moon, with the beacon constructed as a complex geometric figure drawn by small craters or trenches around a central point. There are several cost-effective options for sending the message as opportunistic payloads on different planned landers. (shrink)

Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves (...) two questions unanswered: (i) Why is x^n +y^n = z^n solvable only for n < 3 if x, y, z, n are natural numbers? (ii) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? Prevailing post-Wiles wisdom---leaving (i) essentially unaddressed---dismisses Fermat's claim as a conjecture without a plausible proof of FLT. -/- However, we posit that providing evidence-based answers to both queries is necessary not only for treating FLT as significant, but also for understanding why FLT can be treated as a true arithmetical proposition. We thus argue that proving a theorem formally from explicit, and implicit, premises/axioms using rules of deduction---as currently accepted---is a meaningless game, of little scientific value, in the absence of evidence that has already established---unambiguously---why the premises/axioms and rules of deduction can be treated, and categorically communicated, as pre-formal truths in Marcus Pantsar's sense. Consequently, only evidence-based, pre-formal, truth can entail formal provability; and the formal proof of any significant mathematical theorem cannot entail its pre-formal truth as evidence-based. It can only identify the explicit/implicit premises that have been used to evidence the, already established, pre-formal truth of a mathematical proposition. Hence visualising and understanding the evidence-based, pre-formal, truth of a mathematical proposition is the only raison d'etre for subsequently seeking a formal proof of the proposition within a formal mathematical language (whether first-order or second order set theory, arithmetic, geometry, etc.) By this yardstick Andrew Wiles' proof of FLT fails to meet the required, evidence-based, criteria for entailing a true arithmetical proposition. -/- Moreover, we offer two scenarios as to why/how Fermat could have laconically concluded in his recorded marginal noting that FLT is a true arithmetical proposition---even though he either did not (or could not to his own satisfaction) succeed in cogently evidencing, and recording, why FLT can be treated as an evidence-based, pre-formal, arithmetical truth (presumably without appeal to properties of real and complex numbers). It is primarily such a putative, unrecorded, evidence-based reasoning underlying Fermat's laconic assertion which this investigation seeks to reconstruct; and to justify by appeal to a plausible resolution of some philosophical ambiguities concerning the relation between evidence-based, pre-formal, truth and formal provability. (shrink)

Andrew Wiles' analytic proof of Fermat's Last Theorem FLT, which appeals to geometrical properties of real and complex numbers, leaves two questions unanswered: (i) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? (ii) Why is x^n+y^n=z^n solvable only for n<3? In this inter-disciplinary perspective, we offer insight into, and answers to, both queries; yielding a pre-formal proof of why FLT can be treated as a (...) true arithmetical proposition (one which, moreover, might not be provable formally in the first-order Peano Arithmetic PA), where we admit only elementary (i.e., number-theoretic) reasoning, without appeal to analytic properties of real and complex numbers. We cogently argue, further, that any formal proof of FLT needs---as is implicitly suggested by Wiles' proof---to appeal essentially to formal geometrical properties of formal arithmetical propositions. (shrink)

The author presents the history of gravitational waves according to Einstein, linking it to his biography and his time in order to understand it in his connection with the history of the Semites, the personality of Einstein in the handling of his conflict-generating circumstances in his relationships competition with his colleagues and in the formulation of the so-called general theory of relativity. We will fall back on the vicissitudes that Einstein experienced in the transition from his scientific work to (...) normal science as a pillar of theoretical physics. We will deal with how Einstein introduced the relativistic ether, conferring an "odor of materiality" to his geometric explanation of gravity, where undoubtedly it does not fit, but that he had to give in to the pressure that was justified by his most renowned colleagues, led by Lorentz. Einstein had to do it to stay in the queue that would lead him to the Nobel. It was thus, as developing the relativistic ether thread, in June 1916, he introduced the gravitational waves of which, in an act of personal liberation and scientific honesty, when he could, in 1938, he demonstrated how they could not exist, within the scenario of his relativity, to immediately also put an end to the relativistic ether. (shrink)

Special relativity has changed the fundamental view on space and time since Einstein introduced it in 1905. It substitutes four dimensional spacetime for the absolute space and time of Newtonian mechanics. It is believed that the validities of Lorentz invariants are fully confirmed empirically for the last one hundred years and therefore its status are canonical underlying all physical principles. However, spacetime metric is a geometric approach on nature when we interpret the natural phenomenon. A geometric flaw on this will (...) be exhibited and the alternative is suggested. The reasonable geometric model of space and time is a three dimensional space which is translating along the time direction. This model legitimately represents the true characteristic of nature. (shrink)

The scientific study of socio-cultural phenomenon requires a translocation of topics elaborated from the social perspective of the individual to a rationally ordered rendition of processes suitable for comprehension from a scientific perspective. Scholarly curiosity seeded from exposure in the natural setting to economic, political, socio-cultural, evolutionary, processes dictates that study of the self, should be a science with a necessary place in the body of world literatures; yet it has proven difficult to find a perspective to contain discussions of (...) topics in a coherent manner for scientific approach: for example, anthropology, the study of mankind, finds difficulty elaborating definition for the orientation of study; it is a member of the same set that contains it. In this presentation, based on features indigenous to a supposed distant perspective that is exposed employing experiences of history and criteria of common sensory perception, it is conjectured that a civilization lifetime pathology is contemporarily active. Example is taken from philosophical and sociological discourses, modern science theory, medicine in pursuit of international health issues, to capture conceptually a role of motions of external agents occurred within the interval of observation, elaboration of concepts, choice of directions, as a source of paradox and confusion. In supposition that does not escape simple logic, ubiquitously appealing to the experience bound senses for understanding, hidden motions, common to both observer and observed, are hypothesized to render from a sense of familiarity, a continued frustration in attaining an understanding of the self and nature. A psychical seduction is proposed to exist, related to historical behaviors associated with centrism and asceticism, produces eccentric interpretations that are bound modernly to logical circular, centric geometrical reasonings; world conceptualizations are conjectured to acquire an avoidance of a state of ‘motionless’ rather than death within selection processes. Projection by the imagination upon the unknown is conjectured to result in a seduction by an active ‘live-wire” embodied to motions occurred to a distant surface. (shrink)

We indicate a new way in the solution of the problem of the quantum measurement . In past papers we used the well-known formalism of the density matrix using an algebraic approach in a two states quantum spin system S, considering the particular case of three anticommuting elements. We demonstrated that, during the wave collapse, we have a transition from the standard Clifford algebra, structured in its space and metrics, to the new spatial structure of the Clifford dihedral algebra. This (...) structured geometric transition, which occurs during the interaction of the S system with the macroscopic measurement system M, causes the destruction of the interferential factors. In the present paper we construct a detailed model of the (S+M) interaction evidencing the particular role of the Time Ordering in the (S+M) coupling since we have a time asymmetric interaction . We demonstrate that , during the measurement , the physical circumstance that the fermion creation and annihilation operators of the S system must be destroyed during such interaction has a fundamental role . (shrink)

This paper is intended to examine the distinction made by Leibniz in Generales Inquisitiones de Analysi Notionum et Veritatum between essential and existential propositions, in order to shed some light on the nature of the analogies that Leibniz proposes between the logical problem of contingency and the geometrical problem of the continuum.

Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: the (...) parallel postulate. -/- Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. -/- There are other differences that are or might become important. -/- Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. -/- Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. -/- The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content. (shrink)

In this work we present the function and we determine the nature of conventions and hypotheses for the scientific foundations according with the conventionalist doctrine that arose in France during the turning of the XIX century to the XX. The doctrine was composed by Henri Poincaré, Pierre Duhem and Édouard Le Roy. Moreover, we analyze the relation that conventions and hypotheses can establish with metaphysical thesis through criteria used by scientists in order to determine the preference for certain theories. (...) Thereunto, we promote an immanent interpretation of published works between 1891 and 1905. As result, we reveal that the authors, though being classified as belonging to the same doctrine, don't have only common grounds, but also divergences. Poincaré and Le Roy agree that geometrical conventions are chosen in accordance with convenience criteria. However, they disagree about the value convenience aggregate to scientific knowledge. In regards to natural phenomena, the three authors agree that reality can't be described univocally by the same set of conventions and hypotheses. Yet, Poincaré and Duhem both believe that there are experimental, rational and axiological criteria that justify scientist's satisfaction with certain theories and we indicate how those criteria are related with metaphysics. We conclude that conventionalists, even if warily and implicitly, searched to approach metaphysics in order to justify scientific activity. (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.

Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be nontrivially Bayes-compatible. We show by contrast that geometric pooling can be nontrivially Bayes-compatible. Indeed, we show that, under certain assumptions, geometric and (...) Bayes-compatible pooling are equivalent. Granting supra-Bayesianism its usual normative status, one upshot of our study is thus that, in a certain class of epistemic contexts, geometric pooling enjoys a normative advantage over linear pooling as a social learning mechanism. We discuss the philosophical ramifications of this advantage, which we show to be robust to variations in our statement of the Bayes-compatibility problem. (shrink)