In Book II of The Geometry, Descartes distinguishes some special lines, which he calls geometrical curves. From the mathematical perspective, these curves are identified with polynomials of two variables. In this way, curves, which were understood as continuous quantities in Greek mathematics, turned into objects composed of points in The Geome- try. In this article we present assumptions which led Descartes to this radical change of the concept of curve.

We present a formal mechanical analysis using sweeping net methods to approximate surfacing singularities of saddle maps. By constructing densified sweeping subnets for individual vertices and integrating them, we create a comprehensive approximation of singularities. This approach utilizes geometric concepts, analytical methods, and theorems that demonstrate the robustness and stability of the nets under perturbations. Through detailed proofs and visualizations, we provide a new perspective on singularities and their approximations in analytic geometry.

A comprehensive review of diamonds, in the sense of Scholze, is presented. The diamond formulations of the Fargues-Fontaine curve and Bun_G are stated. Principal results centered on the diamond formalism in the global Langlands correspondence and the geometrization of the local Langlands correspondence are given. We conclude with a discussion of future geometrizations, and conjecture a diamond reformulation of quantum computational complexity towards a diamond ER = EPR.

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

In the standard model of cosmology, λCDM, were introduced to explain the anomalies of the orbital velocities of galaxies in clusters highest according estimated by General Relativity the dark matter and the accelerated expansion of the universe the dark energy. The model λCDM is based in the equations of the General Relativity that of the total mass-energy of the universe assigns 4.9% to matter (including only baryonic matter), 26.8%, to dark matter and 68.3% to dark energy adjusted according observed in (...) Planck mission, therefore, excluding bosonic matter (quantum vacuum). However, the composition of dark matter and dark energy are unknown. Due to that it lacks of a correct physical theory of gravity since General Relativity is only their powerful equations, which in their applications, their results are interpreted arbitrarily. Properties as curvature, viscous fluid, dragging frame and gravity action are attributed mistakenly to the spacetime by the materialist substantivalism, the most credible philosophical interpretation that complements the General Relativity, caused by its absence of physical definition of spacetime and static gravitational field as immaterial, but which violates, the conception of gravity as an effect of coordinates of the generalization of the inertial motion to the accelerated motion and, in particular, the description of the metric tensor of gravity as a geometric field. These properties are really of the quantum vacuum, the main existence form of the matter. In this paper we propose that the quantum vacuum is the source of dark matter and dark energy, therefore, the components of the quantum vacuum are of them. Both are opposite effects of the quantum vacuum that when gravitationally interacts with the cosmic structures, the vacuum it curves and when such interaction tends to cease by declination of the formation of these structures, occurring since near five milliard of years ago, vacuum it maintains quasi plane, since it interacts gravitationally very weakly with itself, accelerating expansion of the universe. (shrink)

Gravity is the curvature of spacetime, the structural property of static gravitational field, a geometric field, in curved coordinates, according the functions guv, that express geometric relations between material events. Course, general relativity is a relational theory, however, gravity, a thinking category, has symetric physical effects with matter. We use, analitic and critic method of reread the general relativity, since the perspective of the history of the science and the philosophy of the science. Our goal is driver the debate on (...) gravity, to the arena of the quantum physics, but without the ballast of the general relativity. We find that through of relativist aether was attempted transform spacetime in a substantia without succes, the consequence was return to problematic geometric field. The philosophy of the science intervenes, and according the best philosophical theory of substantivalism, spacetime is a inmaterial, geometric substantia. Then, the metaphysics arrives to a full solution in the super-substantivalism theory, that affirms: matter arises from geometric spacetime. Thus, it explains consistently the symetric physical effects between spacetime and matter. Surely, this solution is a medieval speculation. Our conclusion is that since general relativity do not defined physically spacetime leads necessarily to philosophical definitions of relationism and substantivalism on spacetime that are unacceptable physically. Therefore, gravity is not the curvature of spacetime. (shrink)

In the standard model of cosmology, λCDM, were introduced to explain the anomalies of the orbital velocities of galaxies in clusters highest according estimated by General Relativity the dark matter and the accelerated expansion of the universe the dark energy. The model λCDM is based in the equations of the General Relativity that of the total mass-energy of the universe assigns 4.9% to matter (including only baryonic matter), 26.8%, to dark matter and 68.3% to dark energy adjusted according observed in (...) Planck mission, therefore, excluding bosonic matter (quantum vacuum). However, the composition of dark matter and dark energy are unknown. Due to that it lacks of a correct physical theory of gravity since General Relativity is only their powerful equations, which in their applications, their results are interpreted arbitrarily. Properties as curvature, viscous fluid, dragging frame and gravity action are attributed mistakenly to the spacetime by the materialist substantivalism, the most credible philosophical interpretation that complements the General Relativity, caused by its absence of physical definition of spacetime and static gravitational field as immaterial, but which violates, the conception of gravity as an effect of coordinates of the generalization of the inertial motion to the accelerated motion and, in particular, the description of the metric tensor of gravity as a geometric field. These properties are really of the quantum vacuum, the main existence form of the matter. In this paper we propose that the quantum vacuum is the source of dark matter and dark energy, therefore, the components of the quantum vacuum are of them. Both are opposite effects of the quantum vacuum that when gravitationally interacts with the cosmic structures, the vacuum it curves and when such interaction tends to cease by declination of the formation of these structures, occurring since near five milliard of years ago, vacuum it maintains quasi plane, since it interacts gravitationally very weakly with itself, accelerating expansion of the universe. (shrink)

The General Relativity understands gravity like inertial movement of the free fall of the bodies in curved spacetime of Lorentz. The law of inertia of Newton would be particular case of the inertial movement of the bodies in the spacetime flat of Euclid. But, in the step, from general to particular, breaks the law of inertia of Galilei since recovers apparently the rectilinear uniform movement but not the repose state, unless the bodies have undergone their collapse, although, the curved spacetime (...) becomes flat and the curved geodesies becomes straight lines. For General Relativity is a natural law, within of a gravitational field, the accelerated movement of the bodies, that leads to that a geometric curvature puts out to the bodies in such geodesic movement. In this paper this error of General Relativity, like generalization of the law of inertia of Galilei, is examined and it is found that it is caused by suppression of mass and force that allows conceiving acceleration like property of spacetime. This is a mathematical and non-ontological result. Indeed, mass and force are the fundament that the gravitational acceleration is independent of the magnitude of mass of the bodies but gravity not of the mass and the gravitational force. The action of the gravity force, on inertial and gravitational masses of a body, produces mutual cancellation during its free fallen but too its weight when this cease. By means of the third law of Newton it shows that gravity is a force since weight is caused by gravity. (shrink)

While the philosophers of science discuss the General Relativity, the mathematical physicists do not question it. Therefore, there is a conflict. From the theoretical point view “the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory 's foundations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended? Or is it just a theory that treats gravitation geometrically in the spacetime (...) setting?”. “The voices of dissent proclaim that Einstein was mistaken over the fundamental ideas of his own theory and that their basic principles are simply incompatible with this theory. Many newer texts make no mention of the principles Einstein listed as fundamental to his theory; they appear as neither axiom nor theorem. At best, they are recalled as ideas of purely historical importance in the theory's formation. The very name General Relativity is now routinely condemned as a misnomer and its use often zealously avoided in favour of, say , Einstein's theory of gravitation What has complicated an easy resolution of the debate are the alterations of Einstein's own position on the foundations of his theory”, (Norton, 1993). Of other hand from the mathematical point view the “General Relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance” (Hawking and Penrose, 1996). So, during a time, the declaration of quantum theorists:“I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation.” (Hawking and Penrose, 1996)seemed to solve the problem, but recently achieved with the help of the tightly and collectively synchronized clocks in orbit frontally contradicts fundamental assumptions of the theory of Relativity. These observations are in disagree from predictions of the theory of Relativity. (Hatch, 2004a, 2004b, 2007). The mathematical model was developed first by Grossmann who presented it, in 1913, as the mathematical part of the Entwurf theory, still referred to a curved Minkowski spacetime. Einstein completed the mathematical model, in 1915, formulated for Riemann ́s spacetimes. In this paper, we present as of General Relativity currently remains only the mathematical model, darkened with the results of Hatch and, course, we conclude that a Einstein ́s gravity theory does not exist. (shrink)

The General Relativity understands gravity like inertial movement of the free fall of the bodies in curved spacetime of Lorentz. The law of inertia of Newton would be particular case of the inertial movement of the bodies in the spacetime flat of Euclid. But, in the step, of the particular to the general, breaks the law of inertia of Galilei since recovers the rectilinear uniform movement but not the repose state, unless the bodies have undergone their union, although, the curved (...) spacetime becomes flat and the curved geodesies becomes straight lines. For General Relativity is a natural law, within of a gravitational field, the uniform accelerated movement of the bodies, that leads to that a geometric curvature puts out to the bodies of the repose state for animate them of the movement of free fallen. In this paper this error of General Relativity, like generalization of the law of inertia of Galilei, is examined and it is found that it is caused by suppression of mass and force that allows conceiving acceleration like property of spacetime. This is a mathematical and non-ontological result. Indeed, mass and force are the fundament that the gravitational acceleration is a constant value for all the bodies, independently of the magnitude of mass but not of the mass and the gravitational force. The action of the gravity force, on inertial and gravitational masses of a body, produces mutual cancellation during its free fallen. In addition, by means of the third law of Newton it demonstrates that gravity is a force since weight is caused by gravity force. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this (...) “crisis” as follows: Mathematicians had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof... (p. 67) Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called “logicization”, “when the discipline requires nothing but purely logical fundamental concepts and propositions for its development.” On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals to visual representations. This would be a radical reformation of past practice, necessary, according to its advocates, for avoiding “unclarities and mistakes” like the one exposed by Peano. (shrink)

Bayesians often assume, suppose, or conjecture that for any reasonable explication of the notion of simplicity a prior can be designed that will enforce a preference for hypotheses simpler in just that sense. But it is shown here that there are simplicity-driven approaches to curve-fitting problems that cannot be captured within the orthodox Bayesian framework.

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)

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.

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

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)

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

How is knowledge of geometry developed and acquired? This central question in the philosophy of mathematics has received very different answers. Spelke and colleagues argue for a “core cognitivist”, nativist, view according to which geometric cognition is in an important way shaped by genetically determined abilities for shape recognition and orientation. Against the nativist position, Ferreirós and García-Pérez have argued for a “culturalist” account that takes geometric cognition to be fundamentally a culturally developed phenomenon. In this paper, I argue that (...) when understood as moderate versions supported by the state-of-the-art research, the nativist and culturalist views are in fact possible to reconcile. While Ferreirós and García-Pérez present the work of Spelke and colleagues as implying that geometric cognition is genetically determined, I argue that they fail to appreciate the role that Spelke and colleagues see for cultural factors. On this basis, I provide theoretical and terminological clarifications and show that moderate versions of the nativist and culturalist view are in fact consistent with each other. I then propose a unifying theoretical framework for future study that can integrate the two accounts in ontogeny by moving beyond the crude nature (nativism) vs. nurture (culturalism) dichotomy. (shrink)

This paper examines Aristotle's discussion of the priority of actuality to potentiality in geometry at Metaphysics Θ9, 1051a21–33. Many scholars have assumed what I call the "geometrical construction" interpretation, according to which his point here concerns the relation between an inquirer's thinking and a geometrical figure. In contrast, I defend what I call the "geometrical analysis" interpretation, according to which it concerns the asymmetrical relation between geometrical propositions in which one is proved by means of the (...) other. His argument as so construed is ultimately based on the asymmetrical relation between the corresponding geometrical facts. Then I explore this ontological priority in geometry by drawing attention to a parallel passage, Posterior Analytics II.11, 94a24–35, where Aristotle explains the relation between the same geometrical propositions in connection to material causation. (shrink)

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

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

This study analyses the predictions of the General Theory of Relativity (GTR) against a slightly modified version of the standard central mass solution (Schwarzschild solution). It is applied to central gravity in the solar system, the Pioneer spacecraft anomalies (which GTR fails to predict correctly), and planetary orbit distances and times, etc (where GTR is thought consistent.) -/- The modified gravity equation was motivated by a theory originally called ‘TFP’ (Time Flow Physics, 2004). This is now replaced by the ‘Geometric (...) Model’, 2014 [20], which retains the same theory of gravity. This analysis is offered partially as supporting detail for the claim in [20] that the theory is realistic in the solar system and explains the Pioneer anomalies. The overall conclusion is that the model can claim to explain the Pioneer anomalies, contingent on the analysis being independently verified and duplicated of course. -/- However the interest lies beyond testing this theory. To start with, it gives us a realistic scale on which gravity might vary from the accepted theory, remain consistent with most solar-scale astronomical observations. It is found here that the modified gravity equation would appear consistent with GTR for most phenomena, but it would retard the Pioneer spacecraft by about the observed amount (15 seconds or so at time). Hence it is a possible explanation of this anomaly, which as far as I know remains unexplained now for 20 years. -/- It also shows what many philosophers of science have emphasized: the pivotal role of counterfactual reasoning. By putting forward an exact alternative solution, and working through the full explanation, we discover a surprising ‘counterfactual paradox’: the modified theory slightly weakens GTR gravity – and yet the effect is to slow down the Pioneer trajectory, making it appear as if gravity is stronger than GTR. The inference that “there must be some tiny extra force…” (Musser, 1998 [1]) is wrong: there is a second option: “…or there may be a slightly weaker form of gravity than GTR.” . (shrink)

This study presents a new type of foundational model unifying quantum theory, relativity theory and gravitational physics, with a novel cosmology. It proposes a six-dimensional geometric manifold as the foundational ontology for our universe. The theoretical unification is simple and powerful, and there are a number of novel empirical predictions and theoretical reductions that are strikingly accurate. It subsequently addresses a variety of current anomalies in physics. It shows how incomplete modern physics is by giving an example of a theory (...) that is genuinely unified. In doing this, it radically alters the metaphysical interpretation of the nature of time, space and matter currently interpreted from modern physics. It also profoundly challenges materialist expectations about a naturalistic account our own existence. I contend here that there is sufficient evidence to support this theory as a leading paradigm for a unified foundational theory. (shrink)

Mechanisms for QoS provisioning in communication networks range from flow-based resource reservation schemes, providing QoS guarantees, through QoS differentiation based on reservation aggregation techniques to adaptation of applications, compensating for incomplete reservations. Scalable, aggregation-based reservations can also be combined with adaptations for a more flexible and robust overall QoS provisioning. Adaptation is particularly important in wireless networks, where reservations schemes are more difficult to realize. It is widely accepted that usability of Cellular or Mobile IP can be largely improved if (...) adaptation is incorporated. Adaptation, in turn, depends on coding techniques involved and on user requirements. Those impose certain preferences and ordering relations on QoS hierarchies or degradation paths to be followed along when adaptation is performed. With the advent of MPEG4 coding techniques, more sophisticated adaptation schemes are more likely to be supported in the future. (shrink)

All acknowledged proofs of the Four Colour Theorem (4CT) are computerdependent. They appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, putatively minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient (...) in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of the single, ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. We shall further show that although Kempe fallaciously concluded that a 5-sided configuration was also in the ‘unavoidable’ set, and appealed to unproven properties of ‘Kempe’ chains in a graphical representation to then argue for its ‘reducibility’, neither flaw in his ‘proof’ is fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

Can green growth policies help protect the environment while keeping the industry growing and infrastructure expanding? The City of Kitakyushu, Japan, has actively implemented eco-friendly policies since 1967 and recently inspired the pursuit of sustainable development around the world, especially in the Global South region. However, empirical studies on the effects of green growth policies are still lacking. This study explores the relationship between road infrastructure development and average industrial firm size with air pollution in the city through the Environmental (...) Kuznets Curve (EKC) hypothesis. Auto-Regressive Distributed Lag (ARDL) and Non-linear Auto-Regressive Distributed Lag (NARDL) methods were applied on nearly 50-years’ time series data, from 1967 to 2015. The results show that the shape of the EKC of industrial growth, measured by average firm size, depends on the type of air pollution: inverted N-shaped relationships with NO2 and CO, and the U-shaped relationships with falling dust particle and Ox. Regarding infrastructure development, on the one hand, our analysis shows a positive effect of road construction on alleviating the amount of falling dust and CO concentration. On the other hand, the emissions of NO2 and Ox are shown to rise when plotted against road construction. The decline of CO emission, when plotted against both industrial growth and road development, indicates that the ruthlessness of the local government in pursuing green growth policies is effective in this case. However, the story is not straightforward when it comes to other air pollutants, which hint at limits in the current policies. The case of Kitakyushu illustrates the complex dynamics of the interaction among policy, industry, infrastructure, and air pollution. It can serve as an important reference point for other cities in the Global South when policies are formed, and progress is measured in the pursuit of a green economy. Finally, as an OECD SDGs pilot city and the leading Asian green-growth city, policymakers in Kitakyushu city are recommended to revise the data policy to enhance the findability and interoperability of data as well as to invest in the application of big data. (shrink)

The upshot of this article is that dance functions in Irigaray’s work in the following three ways: as (1) a symbol of a more positive comportment for heterosexual relationships; (2) an indication that the ambivalence in Irigaray’s work is self-consciously strategic; and (3) an example that teases apart the concepts of negative and positive mimesis, specifically by fleshing out the latter. More concisely, dance constitutes a figure of positive ambivalence (whether between heterosexual lovers, participants in a philosophical dialogue, or aspects (...) of a concept) as self-mimetic, curved silvering in the pursuit of social justice. (shrink)

Luminance and color are strong and self-sufficient cues to pictorial depth in visual scenes and images. The present study investigates the conditions Under which luminance or color either strengthens or overrides geometric depth cues. We investigated how luminance contrasts associated with color contrast interact with relative height in the visual field, partial occlusion, and interposition in determining the probability that a given figure is perceived as ‘‘nearer’’ than another. Latencies of ‘‘near’’ responses were analyzed to test for effects of attentional (...) selection. Figures in a pair were supported by luminance contrast or isoluminant color contrast and combined with one of the three geometric cues. The results of Experiment 1 show that luminance contrasts associated with hue, when it does not interact with other hues, produces the same effects as achromatic luminance contrasts: The probability of‘‘near’’ increases with luminance contrast while the latencies for ‘‘near’’ responses decrease. Partial occlusion is found to be a strong enough pictorial cue to support a weaker red luminance contrast. Interposition cues lose out against cues of spatial position and partial occlusion. The results of Experiment 2, with isoluminant displays of varying color contrast, reveal that red color contrast on a light background supported by any of the three geometric cues wins over green or white supported by any of the three geometric cues. On a dark background, red color contrast supported by the interposition cue loses out against green or white color contrast supported by partial occlusion. These findings reveal that color is not an independent depth cue, but is strongly influenced by luminance contrast and stimulus geometry. Systematically shorter response latencies for stronger ‘‘near’’ percepts demonstrate that selective visual attention reliably detects the most likely depth cue combination in a given configuration. (shrink)

This essay offers an in-depth reading of the geometrical illustration of Ethics IIP8S and shows how it can be used to explicate the whole architecture of Spinoza’s system by specifying the way in which all the key structural features of his basic ontology find their analogies in the example. The illustration can also throw light on Spinoza’s ontology of finite things and inform us about what is at stake when we form universal ideas. In general, my reading of IIP8S (...) thus elucidates what it means, for Spinoza, to think geometrically or to consider geometry as a model: fundamentally, geometricity is not a form of exposition but the way in which reality itself is structured. (shrink)

This paper examines the second geometrical problem in the Meno. Its purpose is to explore the implication of Cook Wilson’s interpretation, which has been most widely accepted by scholars, in relation to the nature of hypothesis. I argue that (a) the geometrical hypothesis in question is a tentative answer to a more basic problem, which could not be solved by available methods at that time, and that (b) despite the temporary nature of a hypothesis, there is a rational (...) process for formulating it. The paper also contains discussion of the method of analysis, problem reduction and a diorism, which have often been ambiguously explained in relation to the geometrical problem in question. (shrink)

General relativity (GR) describes gravity through the curvature of spacetime. However, there are two equivalents of GR that describe flat spacetimes with gravitational effects attributed to torison or non-metricity. These theories, together with GR, are known as the geometrical trinity of gravity and are said to present a case of underdetermination by Wolf et al. (2024). In this article, I argue against this stance by examining the empirical equivalence and possible interpretations of the trinity. I propose a unifying framework (...) where the trinity emerge as different facets of the same underlying ontology, thereby breaking down the underdetermination. (shrink)

In this paper I challenge the pernicious aspects of Milton Friedman's methodological outlook that continues to hold sway over mainstream neoclassical economists. I do this by showing how Friedman's own methodological dicta could have been used against him when he famously advanced the expectations critique of the Phillips curve at his presidential address to the American Economic Association. I use this case study to further suggest that psychological and neurophysiological data should not be deemed irrelevant to economic science.

Focused attention is needed to perceive change (Rensink et al., 1997; Psychological Science, 8: 368-373) . But how much attentional processing is given to an item? And does this depend on the nature of the task? To answer these questions, "flicker" displays were created, where an original and a modified image continually alternated, with brief blanks between them. Each image was an array of simple figures, half being horizontal and the other half vertical. In half the trials, one of the (...) items changed orientation; in the remainder, all items remained the same. Observers were asked to detect whether a change was occurring in each trial. -/- Within-observer comparisons for different shapes of items yielded two classes of speed: relatively fast search (50 ms/item) for simple lines, and slower search (90 ms/item) for compound figures made of two or more lines. Evidently, more processing was given to compound figures, even though shape was irrelevant to the task. A similar set of experiments asked observers to detect changes in contrast polarity. Here, no differences were found for different shapes, indicating that geometric information is not processed as much when the task involves only nongeometric properties. (shrink)

Albert Einstein’s theory of General Relativity was once the leading theory in theoretical physics. Unfortunately the theory describes macroscopic reality without a clear link with the the microcosm in respect to the properties of spacetime. However the theory of General Relativity has proved to predict macroscopic phenomena in a very accurate way. Nowadays most theoretical physicists use the conceptual framework of quantum theory. So it is not surprisingly that the question about the “true nature” of spacetime becomes very intrigue.

The cognitive basis of geometry is still poorly understood, even the ‘simpler’ issue of what kind of representation of geometric objects we have. In this work, we set forward a tentative model of the neural representation of geometric objects for the case of the pure geometry of Euclid. To arrive at a coherent model, we found it necessary to consider earlier forms of geometry. We start by developing models of the neural representation of the geometric figures of ancient Greek practical (...) geometry. Then, we propose a related model for the earliest form of pure geometry – that of Hippocrates of Chios. Finally, we develop the model of the neural representation of the geometric objects of Euclidean geometry. The models are based on the hub-and-spoke theory. In our view, the existence of specific models opens the possibility of addressing the relationship between geometric figures and geometric objects, in a novel way, in terms of their neural representation. (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)

All accepted proofs of the Four Colour Theorem (4CT) are computer-dependent; and appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient in (...) their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of a ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. Although Kempe appealed to putative properties of ‘Kempe’ chains in a graphical representation to fallaciously argue that a 5-sided configuration was also in the ‘unavoidable’ set, and ‘reducible’, we shall show why the flaw in his ‘proof’ is not fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

The developing body of empirical work on the "Gettier effect" indicates that, in general, the presence of a Gettier-type structure in a case makes participants less likely to attribute knowledge in that case. But is that a sufficient reason to diverge from a JTB theory of knowledge? I argue that considerations of good model selection, and worries about noise and overfitting, should lead us to consider that a live, open question. The Gettier effect is perhaps so transient, and so sensitive (...) to other, epistemologically-inappropriate factors, that it raises the question of whether it ought to be counted as something to include in our theories -- or as a piece of noise to be excluded from them. (shrink)

We consider momentum operators on intrinsically curved manifolds. Given that the momentum operators are Killing vector fields whose integral curves are geodesics, it is shown that the corresponding manifold is either flat, or otherwise of compact type with positive constant sectional curvature and dimension equal to 1, 3 or 7. Explicit representations of momentum operators and the associated Casimir element will be discussed for the 3-sphere. It will be verified that the structure constants of the underlying Lie algebra are proportional (...) to 2ħ/R, where R is the curvature radius of the sphere. This results in a countable energy and momentum spectrum of freely moving particles. It is shown that the maximum resolution of the possible momenta is given by the de-Broglie wave length λ=πR, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates and the associated commutator relations of position and momentum are established. (shrink)

During 17th century a scientific controversy existed on the derivation of Snell’s laws of reflection and refraction. Descartes gave a derivation of the laws, independent of the minimality of travel time of a ray of light between two given points. Fermat and Leibniz gave a derivation of the laws, based on the minimality of travel time of a ray of light between two given points. Leibniz’s calculus method became the standard method of derivation of the two laws. We demonstrate in (...) this article that Snell’s law of reflection follows from simple results of geometry. We do not use the concept of motion or the time of travel in our demonstration. That is, time plays no role in our demonstration. (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)

We give a brief overview of the evolution of mathematics, starting from antiquity, through Renaissance, to the 19th century, and the culmination of the train of thought of history’s greatest thinkers that lead to the grand unification of geometry and algebra. The goal of this paper is not a complete formal description of any particular theoretical framework, but to show how extremisation of mathematical rigor in requiring everything be drivable directly from first principles without any arbitrary assumptions actually leads to (...) relaxing the computational difficulty along with maximal conceptual clarity. With this, we consider a revision of the foundations of elementary geometry and algebra based on the work of Grassmann and Clifford and apply it to conceptual and practical problems of past and present modern mathematics and mathematical physics. (shrink)

This paper explores the physical and metaphysical implications of time travel, focusing on the possibility of changing the past, through a comparative analysis of David Lewis' modal realism and Everett's many-worlds interpretation of quantum mechanics. The existence of closed timelike curves (CTCs) in certain solutions to Einstein's field equations provides a theoretical basis for the possibility of backwards time travel, but this leads to a range of paradoxes, most notably the grandfather paradox. David Lewis argues that time travel must maintain (...) consistency and that the past cannot be changed, even if time travel were possible. His view is supported by his modal realism, which posits the existence of a plurality of possible worlds. In contrast, Everett's many-worlds interpretation, which holds that quantum mechanics describes a multiverse of branching realities, might seem to allow for the possibility of multiple, inconsistent histories. By examining these two ontological frameworks, this paper aims to shed light on the paradoxes of time travel and to explore whether the many-worlds interpretation provides a more satisfactory resolution to these paradoxes than Lewis' modal realism. It is argued that while Lewis' view provides a coherent model for maintaining consistency in time travel, Everett's interpretation may offer a more physically grounded and philosophically compelling account of the metaphysics of time travel. The paper concludes by suggesting that the many-worlds interpretation, by locating the branching of timelines within the structure of quantum mechanics itself, may provide a more promising framework for understanding the possibility and implications of time travel than Lewis' modal realism. However, significant questions remain about the nature of personal identity and free will in a branching multiverse. This comparative analysis aims to contribute to the ongoing debate about the metaphysical possibility of time travel and to highlight the value of engaging with interpretations of quantum mechanics in this context. Keywords: time travel, closed timelike curves, modal realism, many-worlds interpretation, grandfather paradox, metaphysics of time. (shrink)

Here we propose a discussion about the "mandorla" or "vesica piscis". It is a type of 2-dimensional lens, that is, a geometric shape formed by the intersection of two circles with the same radius, intersecting in such a way that the centre of each circle lies on the perimeter of the other. The aim of the discussion is that of understanding when such a geometric shape became a symbol and when this symbol received a specific name. We will find that (...) the name "mandorla" was used long before the term "vesica piscis", which is the Latin translation of the German "fischblosen" used by Albrecht Dürer in his book on geometry. Therefore, the name invented by Dürer was not used by the painter for a sacred form. However, after the middle of the nineteenth century the term "vesica piscis" exploded in literature. Its use was criticized and, at the same time, it was stressed that the proper term for the symbol is "mandorla". Nonetheless, the "vesica piscis" continues to be largely used in the sacred geometry, which ascribes symbolic and sacred meanings to certain geometric shapes and proportions. In the proposed discussion, we will also show that, recently, the 2-dimensional lenticular symbol has been related to the Pythagorean philosophy. It is told that the followers of this philosophy had the habit of using an apple for symbolic communications. Sliced across, the core of the apple is displaying a pentagram, but sliced lengthwise it forms two intersecting circles, that is a "mandorla". Then, the investigation about terms and related uses of this specific geometric shape is the core of the matter of our discussion or, let us tell, the "core of the apple", in a Pythagorean-like approach to find how a German "fischblosen" evolved into a sacred symbol, thanks to a translation into Latin. (shrink)

Many species rely on the three-dimensional surface layout of an environment to find a desired goal following disorientation. They generally do so to the exclusion of other important spatial cues. Two influential frameworks for explaining that phenomenon are provided by geometric-module theories and view-matching theories of reorientation respectively. The former posit a module that operates only on representations of the global geo- metry of three-dimensional surfaces to guide behavior. The latter place snapshots, stored representations of the subject’s two-dimensional retinal stimulation (...) at specific locations, at the heart of their accounts. In this paper, I take a fresh look at the debate between them. I begin by making a case that the empirical evidence we currently have does not clearly favor one framework over the other, and that the debate has reached something of an impasse. Then, I present a new explanatory problem—the representation selection problem—that offers the pro- spect of breaking the impasse by introducing a new type of explanatory consideration that both frameworks must address. The representation selection problem requires explaining how subjects can reliably select the relevant representation with which they initiate the reorientation process. I argue that the view-matching framework does not have the resources to address this problem, while a certain type of theory within the geometric-module framework can provide a natural response to it. In showing this, I develop a new geometric-module theory. (shrink)

Notoriously, the Einstein equations of general relativity have solutions in which closed timelike curves occur. On these curves time loops back onto itself, which has exotic consequences: for example, traveling back into one's own past becomes possible. However, in order to make time travel stories consistent constraints have to be satisfied, which prevents seemingly ordinary and plausible processes from occurring. This, and several other "unphysical" features, have motivated many authors to exclude solutions with CTCs from consideration, e.g. by conjecturing a (...) chronology protection law. In this contribution we shall investigate the nature of one particular class of exotic consequences of CTCs, namely those involving unexpected cases of indeterminism or determinism. Indeterminism arises even against the backdrop of the usual deterministic physical theories when CTCs do not cross spacelike hypersurfaces outside of a limited CTC-region|such hypersurfaces fail to be Cauchy surfaces. We shall compare this CTC-indeterminism with four other types of indeterminism that have been discussed in the philosophy of physics literature: quantum indeterminism, the indeterminism of the hole argument, non-uniqueness of solutions of differential equations and lack of predictability due to insuffcient data. By contrast, a certain kind of determinism appears to arise when an indeterministic theory is applied on a CTC: things cannot be different from what they already were. Again we shall make comparisons, this time with other cases of determination in physics. We shall argue that on further consideration both this indeterminism and determinism on CTCs turn out to possess analogues in other, familiar areas of physics. CTC- indeterminism is close to the epistemological indeterminism we know from statistical physics, while the "fixedness" typical of CTC-determinism is pervasive in physics. CTC- determinism and CTC-indeterminism therefore do not provide incontrovertible grounds for rejecting CTCs as conceptually inadmissible. (shrink)

Review of Geometric Possibility (2011), by Gordon Belot. Oxford and New York: Oxford University Press. x + 219 pp.

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.