This book extends the concept of linguistic coordinate geometry using linguistic planes or semi-linguistic planes. In the case of coordinate planes, we are always guaranteed of the distance between any two points in that plane. However, in the case of linguistic and semi-linguistic planes, we can not always determine the linguistic distance between any two points. This is the first limitation of linguistic planes and semi-linguistic planes.

Special and General theories of relativity may be considered as the most significant examples of integrative thinking. From these works we see that Albert Einstein attached great importance to how we understand geometry and dimensions. It is shown that physics powered by the new multidimensional elastic geometry is a reliable basis for science integration. Instead of searching for braneworlds (elastic membranes - EM) in higher dimensions we will start by searching them in our 3+1 dimensional world. The (...) cornerstone of the new philosophy is an idea that lower dimensional EMs are an essential component of the living matter, they are responsible for our perceptions, intellect, pattern recognition and high speed signal propagation. According to this theory each EM has both physical and perceptive (psychological) meanings: it exists as our Universe-like physical reality for its inner objects and at the same time it plays perceptive (psychological) role in the external bulk space-time. This philosophy may help us to build up a science which explains not only inanimate, unconscious phenomena, but consciousness as well. (shrink)

It is widely thought that knowledge is factive – only truths can be known. However, this view has been recently challenged. One challenge appeals to approximate truths. Wesley Buckwalter and John Turri argue that false-but-approximately-true propositions can be known. They provide experimental findings to show that their view enjoys intuitive support. In addition, they argue that we should reject the factive account of knowledge to avoid widespread skepticism. A second challenge, advanced by Nenad Popovic, appeals to multidimensional geometry (...) to build a case where it seems intuitive that a person knows p even though p is false. In addition, Popovic argues that we should reject the factive account of knowledge because most of us would not become widespread skeptics if we discovered that ordinary objects in our world are actually four-dimensional. In this paper, we defend the factive account of knowledge against these arguments by challenging the intuitive appeal of the cases and arguing that there is no real threat of widespread skepticism for the factive account of knowledge. (shrink)

Virtually all epistemologists agree that truth is a necessary condition for knowledge. Assuming that our three-dimensional world is nested within a higher-dimensional space, I use multidimensional geometry to present a type of case which undermines this fundamental principle of epistemology. I further argue that we would be unlikely to revise our epistemic practices in light of new discoveries even if our world turned out to be substantially different and many of our beliefs turned out to be false.

René Descartes included a picture of his "mesolabium" twice in his "Géométrie". The functioning of this device is shown in an interactive simulation and it is further explained. The mesolabium was used by Descartes to compute multidimensional roots. It illustrated Descartes' classification of curves according to their grade of composition. Furthermore, it showed against the opinion of the ancients that composite curves were a legitmate part of geometry. Finally, by its means Descartes established continuous movement as a definitory (...) criterion for geometric constructability. (shrink)

Curved multi-dimensional space-times (5D and higher) are constructed by embedding them in one higher-dimensional flat space. The condition that the embedding coordinates have a separable form, plus the demand of an orthogonal resulting space-time, implies that the curved multi-dimensional space-time has 4D de-Sitter subspaces (for constant extra-dimensions) in which the 3D subspace has an accelerated expansion. A complete determination of the curved multi-dimensional spacetime geometry is obtained provided we impose a new type of “equivalence principle”, meaning that there is (...) a geodesic which from the embedding space has a rectliniar motion. According to this new equivalence principle, we can find the extra-dimensions metric components, each curved multi-dimensional spacetime surface’s equation, the energy-momentum tensors and the extra-dimensions as functions of a scalar field. The generic geodesic in each 5D spacetime are studied: they include solutions where particle’s motion along the extra-dimension is periodic and the 3D expansion factor is inflationary (accelerated expansion). Thus, the 3D subspace has an accelerated expansion. (shrink)

What aspects of spatial relations influence speakers’ choice of locative? This article presents a study of static spatial descriptions from 24 languages. The study reveals two kinds of spatial terms evident cross‐linguistically: specific spatial terms and general spatial terms (GSTs). Whereas specific spatial terms—including English prepositions—occur in a limited range of situations, with concomitant specificity in their meaning, GSTs occur in all spatial descriptions (in languages that employ them). Because of the extreme differences in range of application, the two are (...) considered separately. A multidimensional scaling analysis is used with specific spatial terms to extract statistically valid similarities across the languages sampled. For GSTs, which have not been previously analyzed in the literature, a semantic analysis is proposed and experimentally validated. The results suggest the importance of geometry, function, and qualitative physics to the meanings of both kinds of spatial terms, although the details differ. (shrink)

What does it mean to keep an “open mind”? In casual conversation it’s a popular phrase with enough common sense to negate much need for debate about what the speaker means. Someone with an open mind might be considered considerate, equanimous, empathetic, a good listener, curious, or flexible in opinion. In Western culture an open-minded person might be receptive to new ideas, possibilities, and interpretations, suggesting that they successfully maintain an engaged yet dynamic mental relationship to various subjects or challenges. (...) Yet in science’s nascent study of consciousness, the notion of a mind “opening” is complicated by the field’s inability to clearly articulate what a mind is, let alone how, or into what, it might open. It is the purpose of this research to present the biological significance of “open-mindedness” in order to discuss possible phenomenological implications pertaining to neural correlates of consciousness (NCC). In his seminal book The Open Work, Umberto Eco describes “openness” as a phenomenon of conscious organization that “locates the infinite at the very core of the finite,” and “invites us to conceive, feel, and thus see the world as possibility.” Utilizing the methods of Mechanics, Dynamics, Aesthetics (MDA) analysis, my critical phenomenological inquiry extends Eco’s lens of cultural semiotics into quantum biology to provide key insights for understanding the aesthetic role of field dynamics in qualia physics—as interpretive events (i.e., watching a film, tasting a dessert, or drawing). In particular, pairing Eco’s semiotic analysis of openness with Hameroff and Penrose’s OR Theory concerning cognitive qualia-producing architectures such as microtubules, raises how sensations between organisms and the Zero Point Field (ZPF) are absorbed, integrated, and transmitted through quantum Stochastic Electrodynamic (SED) information states proposed by Joachim Keppler. Therefore, we can say that living systems are those systems that have adapted their material capabilities (including fundamental principles of self-organization and complexity in common cognitive architectures like awareness and attention) to perceive and critically interpret coherently. That is to say—biology has evolved to make meaning through its very aliveness. Our conscious ability to open our minds in order to interpret and communicate our sensations, thoughts, and complete experiences, therefore reveals a radically multidimensional bio-geometry and biolinguistics based on a field dynamics, which Eco’s work constructs. Via an aesthetic information theory, Eco describes this phenomenon as, “a practical level of poetics that acts as programmatic projects for creation.” The writings of Rumi, Husserl, Dewey, Emerson, Merleau-Ponty, Hegel, and others confer depth to the emergent mechanics of perception found throughout biotic systems. Eco’s ideas thereby provoke discourse on the role of openness within recent theoretical works by Jeremy England, Anirban Bandyopadhyay, Robert Lanza, Deepak Chopra, Ervin Laszlo, Giulio Tononi, Stuart Kauffman, Walter J. Freeman, Robin Carhart-Harris, Mark P. Mattson, Robert R McCrae, Selen Atasoy, Katherine Peil, Terence Deacon, and David Chalmers. Unlike models of consciousness that arise from closed, local computation, I argue that the phenomenon of open mindedness unifies cultural and scientific concepts of consciousness as life’s critical integrative force. (shrink)

It is shown in the present paper that the transformation relating a parallel transported vector in a Weyl space to the original one is the product of a multiplicative gauge transformation and a proper orthochronous Lorentz transformation. Such a Lorentz transformation admits a spinor representation, which is obtained and used to deduce the transportation properties of a Weyl spinor, which are then expressed in terms of a composite gauge group defined as the product of a multiplicative gauge group and the (...) spinor group. These properties render a spinor amenable to its treatment as a particle coupled to a multidimensional gauge field in the framework of the Kaluza–Klein formulation extended to multidimensional gauge fields. In this framework, a fiber bundle is constructed with a horizontal, base space and a vertical, gauge space, which is a Lie group manifold, termed its structure group. For the present, the base is the Minkowski spacetime and the vertical space is the composite gauge group mentioned above. The fiber bundle is equipped with a Riemannian structure, which is used to obtain the classical description of motion of a spinor. In its classical picture, a Weyl spinor is found to behave as a spinning charged particle in translational motion. The corresponding quantum description is deduced from the Klein–Gordon equation in the Riemann spaces obtained by the methods of path-integration. This equation in the present fiber bundle reduces to the equation for a Weyl spinor, which is close to but differs somewhat from the squared Dirac equation. (shrink)

Undoubtedly the Penrose-Hameroff Orch OR model may be considered as a good theory for describing information processing mechanisms and holistic phenomena in the human brain, but it doesn’t give us satisfactory explanation of human perception. In this work a new approach explaining our perception is introduced, which is in good agreement with Orch OR model and other mainstream science theories such as string theory, loop quantum gravity and holographic principle. It is shown that human perception cannot be explained in the (...) terms of elementary particles and we should introduce new indivisible holistic objects with geometry based on smooth infinitesimal analysis - elastic membranes. The example of such a membrane is our Universe which is an indivisible whole. It is shown that our perception may be considered as the result of elastic oscillations of two dimensional (2D) elastic membranes with closed topology embedded in our bodies. Only one elastic membrane responsible for its perceptions will correspond to the selected organism, but there may be other membranes, even at the cell level. In other words, reality may be considered as the process of time evolution of holistic energetically very weak macro objects - elastic membranes with the geometry based on smooth infinitesimal analysis. An embedded membrane in this multidimensional world will look different for the external and internal observers: from the outside it will look like a material object with smooth infinitesimal geometry, while from the inside our Universe-like space-time fabric. When interacting with elementary particles and other membranes, a membrane will transform their energy into its elastic energy (a new form of energy) - the energy of stretching of the infinitesimal segments. The theory postulates that these elastic deformations will not be observable from the point of view of the internal observer. Heisenberg’s uncertainty principle will work in this physics only from the point of view of the internal observer. For the external observer each embedded elastic membrane may be stretched and even a very small region will become observable. For example, living organisms play the role of internal observers of the Universe, and at the same time they serve as external observers for 2D membranes embedded into our Universe. We can observe our 2D self-membranes through our perceptions, which are encoded in elastic oscillations of the elastic membrane. According to the theory, elastic membranes occupy energetically favorable positions around microtubules involved into Orch OR. The theory not only gives us a really multidimensional holistic picture of reality, but it also provides us with a new method for understanding such phenomena as perception, self-awareness and will. (shrink)

We study properties of definable sets and functions in power-bounded T -convex fields, proving that the latter have the multidimensional Jacobian property and that the theory of T -convex fields is b -minimal with centers. Through these results and work of I. Halupczok we ensure that a certain kind of geometrical stratifications exist for definable objects in said fields. We then discuss a number of applications of those stratifications, including applications to Archimedean o-minimal geometry.

Multidimensional adjectives are ubiquitous in natural language. An adjective F is multidimensional just in case whether F applies to an object or pair of objects depends on how those objects stand with respect to multiple underlying dimensions of F-ness. Developing a semantics for multidimensional adjectives requires us to address the problem of dimensional aggregation: how do the application conditions of an adjective F in its positive and comparative forms depend on its underlying dimensions? Here we develop a (...) semantics for multidimensional adjectives that incorporates aggregation functions. We then explore an analogy between dimensional aggregation and preference aggregation, bringing results from social choice theory to bear on the number and kind of aggregation functions that are admissible in a context. These results suggest that, for any given adjective, there will often be multiple aggregation functions admissible, meaning that multidimensional comparatives are often vague. (shrink)

Multidimensional concepts are everywhere, and they are important. Examples include moral value, welfare, scientific confirmation, democracy, and biodiversity. How, if at all, can we aggregate the underlying dimensions of a multidimensional concept F to yield verdicts about which things are Fer than which overall? Social choice theory can be used to model and investigate this aggregation problem. Here, we focus on a particularly thorny problem made salient by this social choice-theoretic framework: the underlying dimensions of a given concept (...) might be measurable on different types of scales—e.g., some ordinal and some cardinal. An underappreciated impossibility theorem due to Anna Khmelnitskaya shows that seemingly plausible constraints on aggregation across scale types are inconsistent. This impossibility threatens to render the notion of overall Fness incoherent. We attempt to defuse this threat, arguing that the impossibility depends on an overly restrictive conception of measurement and of how measurement constrains aggregation. Adopting a more flexible—and, we think, more perspicuous—conception of measurement opens an array of possibilities for aggregation across disparate scale types. (shrink)

This high-level study discusses Newtonian principles and 19th-century views on electrodynamics and the aether. Additional topics include Einstein's electrodynamics of moving bodies, Minkowski spacetime, gravitational geometry, time and causality, and other subjects. Highlights include a rich exposition of the elements of the special and general theories of relativity.

Die Geometrie ist seit Euklids Elementen nicht nur Vorbild fur Theorieform und Wissenschaftlichkeit. Sie hat auch seit der Antike die erkenntnistheoretische Diskussion uber das Verhaltnis von Wirklichkeit und Erkenntnis beeinflusst. In ihrer formalaxiomatischen Auffassung der physikalischen Geometrie durch A. Einstein wird die Wissenschaftstheorie des 20. Jahrhunderts in ihren Mehrheitspositionen auf einen Logischen Empirismus festgelegt. Vollstandig ausgeklammert bleibt dabei das philosophische Problem der Gegenstandskonstitution. Wovon ist Geometrie eine Wissenschaft, und wodurch erhalt sie ihre Passung auf die Anwendungen in Handwerk, Technik und (...) messender Naturwissenschaft? (shrink)

This study investigates professional tax practitioners' ethical judgments and behavioral intentions in cases involving client pressure to adopt aggressive reporting positions, an issue that has been identified as the most difficult ethical/moral problem facing public accounting practitioners. The multidimensional ethics scale (MES) was used to measure the extent to which a hypothetical behavior was consistent with five ethical philosophies (moral equity, contractualism, utilitarianism, relativism, and egoism). Responses from a sample of 67 tax professionals supported the existence of all dimensions (...) of the MES other than egoism. Regressions of ethical judgments and behavioral intentions on the MES dimensions indicate that ethical decision making is most heavily influenced by the moral equity dimension, followed by the contractualism dimension. In contrast, the utilitarianism and relativism dimensions were only related to ethical judgments and behavioral intentions in isolated instances. (shrink)

Why does ‘ethnic cleansing’ occur? Why does the rise of nationalist feeling in Europe and of Black separatist movements in the United States often go hand in hand with an upsurge of anti-Semitism? Why do some mixings of distinct religious and ethnic groups succeed, where others (for example in Northern Ireland, or in Bosnia) fail so catastrophically? Why do phrases like ‘balkanisation’, ‘dismemberment’, ‘mutilation’, ‘violation of the motherland’ occur so often in warmongering rhetoric? All of these questions are, it will (...) turn out, connected. To understand how they are connected we we will need to examine how human beings acquire a relationship to specific chunks of land, a relationship that is emotionally so strong that they are prepared to die – or kill – to protect that land for themselves or to win it back from others. Territoriality, the biologically rooted predisposition to defend core areas of home ranges against intruders, is a near-universal phenomenon amongst animals of all species. But the ways in which defended territories are conceived and demarcated differ widely from species to species and from group to group. By coming to an understanding of the geometry of these differences we can come to understand also some of the factors which give rise to interethnic conflict. (shrink)

This paper analyses managers'' moral decision-making, and studies the role of ethical theories in it by following the research tradition using the multidimensional ethics scale. The research question is: what kinds of ethical dimensions do Finnish business managers reveal when they are making moral decisions, and how have these dimensions changed in the 1990s? This question is answered by examining what kinds of factors emerge when the multidimensional ethics scale is used to analyse Finnish managers'' attitudes toward moral (...) dilemmas. The results show that Finnish managers'' decision-making reflects a variety of ethical theories. Teleological thinking is strongly emphasised by Finnish managers, and relativist thinking is used as well, but often combined with either deontology or justice thinking. In addition, a strong moralistic dimension emerged in Finnish managers'' decision-making. The analysis was carried out in two different surveys in years 1994 and 1999, and the results show that the ways of decision-making were more complex at the end of the 1990s than almost six years earlier. (shrink)

Pain is often used as the paradigmatic example of a phenomenal kind with a phenomenal quality common and unique to its instantiations. Philosophers have intensely discussed the relation between the subjective feeling, which unites pains and distinguishes them from other experiences, and the phenomenal properties of sensory, affective, and evaluative character along which pains typically vary. At the center of this discussion is the question whether the phenomenal properties prove necessary and/or sufficient for pain. In the empirical literature, sensory, affective, (...) and evaluative properties have played a decisive role in the investigation of psychophysical correspondence and clinical diagnostics. This paper addresses the outlined philosophical and empirical issues from a new perspective by constructing amultidimensional phenomenal spacefor pain. First, the paper will construe the phenomenal properties of pains in terms of a property space whosestructurereflects phenomenal similarities and dissimilarities by means of spatial distance. Second, philosophical debates on necessary and sufficient properties are reconsidered in terms of whether there is a phenomenal space formed of dimensions along which all and only pains vary. It is concluded that there is no space of this kind and, thus, that pain constitutes aprimitivephenomenal kind that cannot be analyzed entirely in terms of its varying phenomenal properties. Third, the paper addresses theutilityof continued reference to pain and its phenomenal properties in philosophical and scientific discourses. It is argued that numerous insights into the phenomenal structure of pain can be gained that have thus far received insufficient attention. (shrink)

Adams’s Thesis, the claim that the probabilities of indicative conditionals equal the conditional probabilities of their consequents given their antecedents, has proven impossible to accommodate within orthodox possible-world semantics. This essay proposes a modification to the orthodoxy that removes this impossibility. The starting point is a proposal by Jeffrey and Stalnaker that conditionals take semantic values in the unit interval, interpreting these (à la McGee) as their expected truth-values at a world. Their theories imply a false principle, namely, that the (...) probability of a conditional is independent of any proposition inconsistent with its antecedent. But they also point to something important, namely, that our uncertainty about conditionals is not confined to uncertainty about the facts (what the actual world is like) but also expresses uncertainty about the counterfacts (what the world would be like if one or another supposition were true). To capture this observation, this essay proposes that the semantic contents of conditionals be treated as sets of vectors of possible worlds, not singleton worlds, with the coordinates of each specifying the world that is or would be true under the supposition that it represents. The probabilities of truth for conditionals will then depend on the joint probabilities of the facts and counterfacts, the latter in turn depending on the mode of supposition. The implication of this treatment is that the probabilities of conditionals are conditional probabilities whenever the mode of supposition is evidential. (shrink)

Computers have transformed science and help to extend the boundaries of human knowledge. However, does the validation and diffusion of results of computational inquiries and computer simulations call for a novel epistemological analysis? I discuss how the notion of novelty should be cashed out to investigate this issue meaningfully and argue that a consequentialist framework similar to the one used by Goldman to develop social epistemologySocial epistemology can be helpful at this point. I highlight computational, mathematical, representational, and social stages (...) on which the validity of simulation-based belief-generating processes hinges, and emphasize that their epistemic impact depends on the scientific practices that scientists adopt at these different stages. I further argue that epistemologists cannot ignore these partially novel issues and conclude that the epistemology of computational inquiries needs to go beyond that of models and scientific representations and has cognitive, social, and in the present case computational, dimensions. (shrink)

Pain is often used as the paradigmatic example of a phenomenal kind with a phenomenal quality common and unique to its instantiations. Philosophers have intensely discussed the relation between the subjective feeling, which unites pains and distinguishes them from other experiences, and the phenomenal properties of sensory, affective, and evaluative character along which pains typically vary. At the center of this discussion is the question whether the phenomenal properties prove necessary and/or sufficient for pain. In the empirical literature, sensory, affective, (...) and evaluative properties have played a decisive role in the investigation of psychophysical correspondence and clinical diagnostics. This paper addresses the outlined philosophical and empirical issues from a new perspective by constructing a multidimensional phenomenal space for pain. First, the paper will construe the phenomenal properties of pains in terms of a property space whose structure reflects phenomenal similarities and dissimilarities by means of spatial distance. Second, philosophical debates on necessary and sufficient properties are reconsidered in terms of whether there is a phenomenal space formed of dimensions along which all and only pains vary. It is concluded that there is no space of this kind and, thus, that pain constitutes a primitive phenomenal kind that cannot be analyzed entirely in terms of its varying phenomenal properties. Third, the paper addresses the utility of continued reference to pain and its phenomenal properties in philosophical and scientific discourses. It is argued that numerous insights into the phenomenal structure of pain can be gained that have thus far received insufficient attention. (shrink)

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

In the context of multidimensional measures of well-being, a key question for policy is whether particular groups have differing priorities and are therefore likely to react differently to given economic or social shocks. We explore this issue by presenting the results of two related analyses that suggest positive answers on both counts. First, we apply reference class weights to unique data on adult capabilities in the UK and show that relative weights vary across some groupings. Furthermore, in some cases, (...) deprivation rankings of groups vary depending on which weights are used. Second, we explore possible behavioural consequences of different weights by examining the extent to which groups respond differently to three economic and social shocks (unemployment, widowhood and ill health). In this case, we find that weights and responses vary noticeably with age and region and sometimes with gender. We conclude that whilst equal weighting may be practically unavoidable when constructing indices of welfare in the absence of information on weights, their estimation from survey or experimental data is likely to be justified and may change views about policy needs or efficacy. (shrink)

We introduce a ranking of multidimensional alternatives, including uncertain prospects as a particular case, when these objects can be given a matrix form. This ranking is separable in terms of rows and columns, and continuous and monotonic in the basic quantities. Owing to the theory of additive separability developed here, we derive very precise numerical representations over a large class of domains (i.e., typically notof the Cartesian product form). We apply these representationsto (1)streams of commodity baskets through time, (2)uncertain (...) social prospects, (3)uncertain individual prospects. Concerning(1), we propose a finite horizon variant of Koopmans’s (1960) axiomatization of infinite discounted utility sums. The main results concern(2). We push the classic comparison between the exanteand expostsocial welfare criteria one step further by avoiding any expected utility assumptions, and as a consequence obtain what appears to be the strongest existing form of Harsanyi’s (1955) Aggregation Theorem. Concerning(3), we derive a subjective probability for Anscombe and Aumann’s (1963) finite case by merely assuming that there are two epistemically independent sources of uncertainty. (shrink)

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, (...) but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom, which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions. (shrink)

This paper analyses the possibility of granting legitimacy to democratic decisionmaking procedures in a context of deep pluralism. We defend a multidimensional account according to which a legitimate system needs to grant, on the one hand, that citizens should be included on an equal footing and acknowledged as reflexive political agents rather than mere beneficiaries of policies, and, on the other hand, that their decisions have an epistemic quality. While Estlund’s account of imperfect epistemic proceduralism might seem to embody (...) a dualistic conception of democratic legitimacy, we point out that it is not able to recognize citizens as reflexive political agents and is grounded in an idealized model of the circumstances of deliberation. To overcome these ambiguities, we develop an account of democratic legitimacy according to which disagreement is the proper expression of citizens’ reflexive agency and the attribution of epistemic authority does not stem from a major expertise or specific ability, but it comes through the public confrontation among disagreeing agents. Consequently, the epistemic value of deliberation should be derived from the reasons-giving process rather than from the reference to the alleged quality of its outcomes. In this way, we demonstrate the validity of the multidimensional perspective of legitimacy, yet abstain from introducing any outcome-oriented criterion. Finally, we argue that this account of legitimacy is well suited for modeling deliberative democracy as a decision-making procedure that respects the agency of every citizen and grants her opportunity to influence public choices. (shrink)

In the article, an argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world.

The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then (...) introduce two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

We axiomatize, in the multidimensional case, a social evaluation function that can accommodate a natural Pigou–Dalton principle and correlation increasing majorization. This is performed by building upon a simple class of inframodular functions proposed by Müller and Scarsini under risk.

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion (...) that was first conceived by ancient Greek mathematicians. (shrink)

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