In this paper, we are proposing a study of the Vesica Piscis, a symbol of Sacred Geometry, starting from mathematics. This study aims to give a deeper comprehension of the hidden meanings of this icon. The paper is also showing that scientific and philosophical studies can be integrated, leading to very interesting results.

After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the (...) golden ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients. (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)

I argue a case for interpreting Yeats through the metaphysics of The Order of the Golden Dawn and the human/cosmic life cycle of their Rider-Waite tarot deck. In doing so, I will explain how the metaphysics of Indian and Egyptian sacred geometry inform his poetry, and his plays, in particular, ‘A Vision’ (1925) and ‘The Herne’s Egg’ (1938).

Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The (...) spatial content of the formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry. (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.

The notion of linguistic geometry is defined in this book. It is pertinent to keep in the record that linguistic geometry differs from classical geometry. Many basic or fundamental concepts and notions of classical geometry are not true or extendable in the case of linguistic geometry. Hence, for simple illustration, facts like two distinct points in classical geometry always define a line passing through them; this is generally not true in linguistic geometry. Suppose (...) we have two linguistic points as tall and light we cannot connect them, or technically, there is no line between them. However, let's take, for instance, two linguistic points, tall and very short, associated with the linguistic variable height of a person. We have a directed line joining from the linguistic point very short to the linguistic point tall. In this case, it is important to note that the direction is essential when the linguistic variable is a person's height. The other way line, from tall to very short, has no meaning. So in linguistic geometry, in general, we may not have a linguistic line; granted, we have a line, but we may not have it in both directions; the line may be directed. The linguistic distance is very far. So, the linguistic line directed or otherwise exists if and only if they are comparable. Hence the very concept of extending the line infinitely does not exist. Likewise, we cannot say as in classical geometry; three noncollinear points determine the plane in linguistic geometry. Further, we do not have the notion of the linguistic area of well-defined figures like a triangle, quadrilateral or any polygon as in the case of classical geometry. The best part of linguistic geometry is that we can define the new notion of linguistic social information geometric networks analogous to social information networks. This will be a boon to non-mathematics researchers in socio-sciences in other fields where natural languages can replace mathematics. (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.

This essay gives an account of how traditional morality is best understood and also why it is worth defending (even if some reform is needed) and how this might be done. Traditional morality is first contrasted with supposedly more enlightened forms of morality, such as utilitarianism and liberal Kantianism (i.e., autonomy-centered ethics). The focus here is on certain sacred values that are central to traditional morality and which highlight this contrast and bring out the attractions of traditional morality. Next, (...) this essay explores and offers support for the convergence thesis to which traditional morality, understood as common morality, is committed. This thesis states that although there are diverse moral traditions, insofar as they are in good order we should expect them to converge upon a common or universal morality, even if there remain some differences in the details. The defense of this thesis provides justification for the validity of traditional morality as it suggests an objective basis. (shrink)

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)

Some recently popular accounts of perception account for the phenomenal character of perceptual experience in terms of the qualities of objects. My concern in this paper is with naturalistic versions of such a phenomenal externalist view. Focusing on visual spatial perception, I argue that naturalistic phenomenal externalism conflicts with a number of scientific facts about the geometrical characteristics of visual spatial experience.

We cannot imagine two straight lines intersecting at two points even though they may do so. In this case our abilities to imagine depend upon our abilities to visualise.

What, if anything, is fanaticism? Philosophers including Locke, Hume, Shaftesbury, and Kant offered an account of fanaticism, analyzing it as (1) unwavering commitment to an ideal, together with (2) unwillingness to subject the ideal (or its premises) to rational critique and (3) the presumption of a non-rational sanction for the ideal. In the first part of the paper, I explain this account and argue that it does not succeed: among other things, it entails that a paradigmatically peaceful and tolerant individual (...) can be a fanatic. The following sections argue that the fanatic is distinguished by four features: (4) the adoption of sacred values; (5) the need to treat these values as unconditional in order to preserve a particular form of psychic unity; (6) the sense that the status of these values is threatened by lack of widespread acceptance; and (7) the identification with a group, where the group is defined by shared commitment to the sacred values. If the account succeeds, it not only reveals the nature of fanaticism, but also uncovers a distinctive form of ethical critique: we can critique a way of understanding values not on the grounds that it is false, but on the grounds that it promotes a particular form of social pathology. (shrink)

This work examines the unique way in which Benedict de Spinoza combines two significant philosophical principles: that real existence requires causal power and that geometrical objects display exceptionally clearly how things have properties in virtue of their essences. Valtteri Viljanen argues that underlying Spinoza's psychology and ethics is a compelling metaphysical theory according to which each and every genuine thing is an entity of power endowed with an internal structure akin to that of geometrical objects. This allows Spinoza to offer (...) a theory of existence and of action - human and non-human alike - as dynamic striving that takes place with the same kind of necessity and intelligibility that pertain to geometry. This fresh and original study will interest a wide range of readers in Spinoza studies and early modern philosophy more generally. (shrink)

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and (...) class='Hi'>geometry of symmetry is controlled by numerical parameters, which act in analogy with a biological organism’s DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional (2D) design parameters, affine geometry and shape symmetry for visual or tactile shape sensation and perception-based decision making in populations of experts and non-experts. It will be shown that 2D fractal symmetry, referred to herein as the “symmetry of things in a thing”, results from principles very similar to those of affine projection. Results from experiments on aesthetic and visual preference judgments in response to 2D fractal trees with varying degrees of asymmetry are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (on a scale from 0 to 10) the perceived beauty of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from “symmetry of things in a thing” significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion. (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)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto (...) intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

Abstract: It is the aim of this paper to establish a temporal and cultural link between entheogen-use1 in Classical mystery cults and their possible use in a segment of the early Christian Gnostic Church. As early Christianity was heavily influenced by the Classical world in which it first developed, it is essential to examine the evidence of entheogen-use within Classical mystery cults, and explore their possible influence on the development of Christian ritual. We will first present textual evidence from the (...) New Testament and Nag Hammadi Library (NHL) that indicate possible use of ritual entheogens. Next we will examine entheogen-use by some of the main religious sects of the ancient Mediterranean world as part of their liturgical practice between c. fifth century BCE and the third century CE. Then we will show that the different schools of philosophical thought examined were quite fluid, and that there was considerable overlap among schools with respect to several key philosophies and practices. Ultimately the subject of entheogen-use in the Classical world necessarily falls into the realm of plausible speculation mainly due to the lack of primary sources. However, should the entheogenic hypothesis2 prove to be correct, it promises to be quite useful in future research and in the translation of original documents. (shrink)

I explore some ways in which one might base an account of the fundamental metaphysics of geometry on the mathematical theory of Linear Structures recently developed by Tim Maudlin (2010). Having considered some of the challenges facing this approach, Idevelop an alternative approach, according to which the fundamental ontology includes concrete entities structurally isomorphic to functions from space-time points to real numbers.

Using as a starting point recent and apparently incompatible conclusions by Saunders and Knox, I revisit the question of the correct spacetime setting for Newtonian physics. I argue that understood correctly, these two versions of Newtonian physics make the same claims both about the background geometry required to define the theory, and about the inertial structure of the theory. In doing so I illustrate and explore in detail the view—espoused by Knox, and also by Brown —that inertial structure is (...) defined by the dynamics governing subsystems of a larger system. This clarifies some interesting features of Newtonian physics, notably the distinction between using the theory to model subsystems of a larger whole and using it to model complete universes, and the scale-relativity of spacetime structure. _1_ Introduction _2_ Newtonian Mechanics and Galilean Spacetime _3_ Vector Relationism and Maxwellian Spacetime _4_ Recovering the Galilei Group: Dynamics of Subsystems _5_ Knox on Inertial Structure _6_ Connections on Maxwellian Spacetime _7_ Knox on Newtonian Gravity _8_ Vector Relationism and Newton–Cartan Theory _9_ Inertial Structure in Newton–Cartan Gravity _10_ Reconciling Knox and Saunders _11_ Conclusions. (shrink)

We discuss the relationship between logic, geometry and probability theory under the light of a novel approach to quantum probabilities which generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories.

This paper examines explanations that turn on non-local geometrical facts about the space of possible configurations a system can occupy. I argue that it makes sense to contrast such explanations from "geometry of motion" with causal explanations. I also explore how my analysis of these explanations cuts across the distinction between kinematics and dynamics.

In the past decade, three philosophers in particular have recently explored the relation between desert and intrinsic value. Fred Feldman argues that consequentialism need not give much weight – or indeed any weight at all – to the happiness of persons who undeservedly experience pleasure. He defends the claim that the intrinsic value of a state of affairs is determined by the “fit” between the amount of well-being that a person receives and the amount of well-being that the person deserves. (...) Shelly Kagan uses a similar claim to motivate the view that equality is not intrinsically valuable. Thomas Hurka argues that desert is a third-order value, which is a function of the relation between the second-order value of having a virtuous or vicious character and the first-order value of experiencing pleasure or pain. In this paper, we sketch a theory of desert as fittingness and defend a general account of the relation between desert, well-being, and intrinsic value. We then discuss various applications of our “geometry of desert,” including a solution to the problem of the Repugnant Conclusion. (shrink)

Looks at the development of the sacred after the Sumerians' named the phallus Supreme Creator in 9000 B.C.E. Lists names invented creating belief in Sacred Phallus and names the part male genitals played in supporting the phallus as sacred.

In the brain the relations between free neurons and the conditioned ones establish the constraints for the informational neural processes. These constraints reflect the systemenvironment state, i.e. the dynamics of homeocognitive activities. The constraints allow us to define the cost function in the phase space of free neurons so as to trace the trajectories of the possible configurations at minimal cost while respecting the constraints imposed. Since the space of the free states is a manifold or a non orthogonal space, (...) the minimum distance is not a straight line but a geodesic. The minimum condition is expressed by a set of ordinary differential equation ( ODE ) that in general are not linear. In the brain there is not an algorithm or a physical field that regulates the computation, then we must consider an emergent process coming out of the neural collective behavior triggered by synaptic variability. We define the neural computation as the study of the classes of trajectories on a manifold geometry defined under suitable constraints. The cost function supervises pseudo equilibrium thermodynamics effects that manage the computational activities from beginning to end and realizes an optimal control through constraints and geodetics. The task of this work is to establish a connection between the geometry of neural computation and cost functions. To illustrate the essential mathematical aspects we will use as toy model a Network Resistor with Adaptive Memory (Memristors).The information geometry here defined is an analog computation, therefore it does not suffer the limits of the Turing computation and it seems to respond to the demand for a greater biological plausibility. The model of brain optimal control proposed here can be a good foundation for implementing the concept of "intentionality",according to the suggestion of W. Freeman. Indeed, the geodesic in the brain states can produce suitable behavior to realize wanted functions and invariants as neural expressionsof cognitive intentions. (shrink)

In this chapter, I explore how experiences of social belonging might emerge and be sustained in online communities, drawing from the work on rituals by Randall Collins. I argue that rather than viewing mediated interactions in terms of whether they are suitable substitutes for face-to-face interactions, we should consider mediated encounters in their own right. This allows us to recognize the creative ways that people can create rituals in a mediated setting and thus support and create a sense of belonging (...) with others online. I also highlight how some online platforms not only support online rituals but that they are well-positioned to create and circulate collective symbols (or sacred objects) of groups and, thus, can continue to scaffold and sustain on-going feelings of belonging. (shrink)

The Bugis-Makassar indigenous people who live around Mount Bawakaraeng perform a ritual pilgrimage (hajj) to the top of Mount Bawakaraeng (as a sacred space). This ritual is often considered heretical and deviant. These negative assumptions are the result of the monopoly definition of “sacred place” by the world religion paradigm which is only limited to the doctrine of the holy book and is hierarchical-exclusive. Meanwhile, in the indigenous religion paradigm, “sacred place” is closely related tothe surrounding environment (...) (nature) which also gives life to indigenous peoples. The Bugis-Makassar indigenous people who live around Mount Bawakaraeng construct the sacredness of the mountain, not only as a place for religious rituals but also as a guarantor of their life. There were lacking previous researches discussing “sacred place” through the indigenous religion paradigm approach. This research contributes to that lack. This study examines how the indigenous religion paradigm interprets “sacred place”. The research method used in this research is qualitative. This study argues that there is no better way to understand why indigenous people perform rituals on Mount Bawakaraeng than using the indigenous religion paradigm. This study also shows that the “sacred place” associated with Mount Bawakaraeng is a way for the indigenous people who live around the mountain to preserve the nature around them which has enabled them to live and make a living such as accessing water, gathering medicines from nature, and so on. Eventually, with research that provides a better explanation of what a “sacred place” is in the indigenous religion paradigm, negative assumptions about indigenous people who regard a mountain as sacredplace can be better understood. (shrink)

This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding of (...) natural philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience with causal principles borrowed from geometry. Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the _Dialogus Physicus, sive De natura aeris_, should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in _Dialogus Physicus_ be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation. (shrink)

Art among Iranian and Islamic philosophers has always been associated with moral, so that many philosophers have considered art to be synonymous with virtue. By examining Farabi's opinions, it is possible to extract his special ideas about art and artist. In Farabi's theory of Virtuous Art, the artist is on the second floor of utopia and carries religious truths and reasonable happiness. Also, the theory of Virtuous Art has all the aesthetic features and artistic creativity, and in fact, all artistic (...) values are the presupposition of being an art of Virtuous Art. By reflecting on Suhrawardi’s numerous works and intellectual foundations, we can also see that he also paid special attention to the virtuous art and the supreme aspects of art; although he did not directly theorize about the basics of art, but by examining his theories, we can infer the basics of Ishraghi's Virtuous Art. Moreover, Suhrawardi has written some of his works in allegorical and artistic language, and this is a manifestation of Suhrawardi’s Illuminating Art. Among contemporary thinkers, Seyyed Hossein Nasr and Titus Burckhardt have presented an approach with the title of Sacred Art, considering the sacred foundations of art and Traditionalism. (shrink)

Explications of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in 'The Fold: Leibniz and the Baroque' focus predominantly on the role of the infinitesimal calculus developed by Leibniz.1 While not underestimat- ing the importance of the infinitesimal calculus and the law of continuity as reflected in the calculus of infinite series to any understanding of Leibniz’s metaphysics and to Deleuze’s reconstruction of it in The Fold, what I propose to examine in this paper is the role played by other (...) developments in mathematics that Deleuze draws upon, including those made by a number of Leibniz’s near contemporaries – the projective geometry that has its roots in the work of Desargues (1591–1661) and the ‘proto-topology’ that appears in the work of Du ̈rer (1471–1528) – and a number of the subsequent developments in these fields of mathematics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. What is provided in this paper is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

ABSTRACT In order to explore what it means to pursue philosophical investigations for theological reasons, this paper argues that Ludwig Wittgenstein continues and corrects Pseudo-Denys’ project in The Divine Names. I first argue that The Divine Names should be interpreted as attempting to render human thought transparent to the divine by relativizing our concepts. The success of this project is compromised because the concept of ‘unity’ is not relativized. I then develop the claim that Wittgenstein does relativize unity in a (...) similar way and for similar religious reasons to Pseudo-Denys. As such, he can be read as continuing and correcting the Pseudo-Dionysian project. I conclude by reflecting on several of this argument’s implications for the relationship between philosophy and Christian systematic theology. (shrink)

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as “natural”. We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry – i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3–34 years), and 25 (...) participants from the Amazon (age 5–67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in “sense”). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans’ knowledge in Euclidean geometry could possibly be grounded. (shrink)

Planar geometry was exploited for the computation of symmetric visual curves in the image plane, with consistent variations in local parameters such as sagitta, chordlength, and the curves’ height-to-width ratio, an indicator of the visual area covered by the curve, also called aspect ratio. Image representations of single curves (no local image context) were presented to human observers to measure their visual sensation of curvature magnitude elicited by a given curve. Nonlinear regression analysis was performed on both the individual (...) and the average data using two types of model: (1) a power function where. (shrink)

Kant was engaged in a lifelong struggle to achieve what he calls in the 1756 Physical Monadology (PM) a “marriage” of metaphysics and geometry (1:475). On one hand, this involved showing that metaphysics and geometry are complementary, despite the seemingly irreconcilable conflicts between these disciplines and between their respective advocates, the Leibnizian-Wolffians and the Newtonians. On the other hand, this involved defining the terms of their union, which meant among other things, articulating their respective roles in grounding Newtonian (...) natural science. In this paper, consider how Kant’s project of marrying metaphysics and geometry evolves from the pre-Critical to the Critical period and how key discussions in the Prolegomena are related to the lifelong marriage project. (shrink)

Suffering can make sacred, so it may partly be nature, and not culture alone, that leads us to apprehend a sacred aspect in victims of oppression. Those who recognize this sacredness show piety—a special form of respect—toward members of oppressed groups. The result is a system of social constructions often dismissed as “identity politics.” This essay starts with an analysis of the intentionality of piety and sacredness and how they relate to suffering, sacrifice, sanctions, pollution, and purification. It (...) then argues that the sacralization of oppressed groups is an expression of the perennial human disposition to acknowledge sacredness and to respond piously. The essay then analyzes this sacred status as socially constructed. Based on the sacred-making power of suffering, the sacred status elicits piety, gives its bearers special authority, surrounds them with sanctions, and calls for symbolic sacrificial punishments of the impious. By dissecting sacrificial politics as a system of social constructions, we see that, although the oppressed groups are made sacred, certain people in the oppressor groups—“the Pious”—continue to exercise fundamental power. This essay, by displaying the inner logic of this cultural phenomenon, helps us both to sympathize with and to critique the system and then to pose questions about what good or bad the system might be doing. (shrink)

Knowing someone personally centrally involves engaging in various patterns of affective response. Inasmuch as humans can know God personally, this basic insight about the relationship between personal knowledge and affective response also applies to God: knowing God involves responding to him, and to the world, in various affectively toned ways. In light of this insight, I explore how one particular practice might contribute to human knowledge of God: namely, engaging with sacred music – in particular, sacred music in (...) the Western, classical tradition. In investigating this topic, I tackle an issue at the interface between the philosophy of religion and aesthetics: sacred music’s capacity to arouse religious emotions. I use as a springboard Peter Kivy’s view that ‘music alone’ can only arouse emotions about itself. This view, I argue, has a counterintuitive consequence: the music in sacred works would play no part in arousing emotions with religious objects. In contrast, I develop an account of how music combines with religious factors to arouse religious emotions. I then draw out two consequences for how music can facilitate understanding between religious and non-religious outlooks. The upshot of all this, I suggest, is that by expanding the listener’s religious-emotional landscape, sacred music can enrich her knowledge of God – and that this is possible whether or not the listener subscribes propositionally to God’s reality or to any other religious doctrines. (shrink)

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

The logic of soku-hi is presented as an articulation of a post-Kantian view of reality that embraces the truths of science with the assumption of the transcendental subject. As such, soku-hi represents the philosophical posture of both the secular Zen of the Kyoto School and the new materialists of contemporary continental philosophy. It describes how material reality is not all even though there is nothing else.

Sacred value attitudes involve a distinctive profile of norm psychology: an absolutist prohibition on transgressing the value, combined with outrage at even hypothetical transgressions. This article considers three mechanisms by which such attitudes may be adaptive, and relates them to central theories regarding the evolution of religion. The first, “deterrence” mechanism functions to dissuade coercive expropriation of valuable resources. This mechanism explains the existence of sacred value attitudes prior to the development of religion and also explains analogues of (...) sacred value in non-human species. The two remaining mechanisms, “assurance” and “coalitional,” are more likely to have been involved in the cultural evolution of specifically religious behavior. In the assurance mechanism, sacred value attitudes increase the cost of ideological commitments, making them more reliable as signals of a cooperative disposition. In the coalitional mechanism, sacred value attitudes make it dangerous for third parties to dissent from a social norm, and thus discourage competitor ideologies. While both these mechanisms are compatible with major accounts of the evolution of religion, different theories suggest a greater or lesser emphasis on one mechanism or the other. (shrink)

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in (...) this debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond. (shrink)

The role of conventions in the formulation of Thomas Reid’s theory of the geometry of vision, which he calls the “geometry of visibles”, is the subject of this investigation. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid’s “geometry of visibles” and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a (...) choice of conventions regarding the construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reid’s direct realist theory of perception with his geometry of visibles. (shrink)

The latest draft (posted 05/14/22) of this short, concise work of proof, theory, and metatheory provides summary meta-proofs and verification of the work and results presented in the Theory and Metatheory of Atemporal Primacy and Riemann, Metatheory, and Proof. In this version, several new and revised definitions of terms were added to subsection SS.1; and many corrected equations, theorems, metatheorems, proofs, and explanations are included in the main text. The body of the text is approximately 18 pages, with 3 sections; (...) 1, an Introduction (with a 108 page listing of key terms & definitions), sect. 2, the Results, sect. 3, Discussion (commentary & predictions), and sect. 4, Works Cited. As much as possible, the style is intended for readability and understanding by very bright children (with some interest & knowledge of maths, etc.) and very interested nonprofessionals. The results of this project also enable upgrades of number theory, set theory, proof theory, metamathematics, the foundations of science, and quantum mechanics theory (etc.). (shrink)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows from the (...) premises: there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...) recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical (...) geometry, some of which are basically explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

Against Thomas Mormann's argument that differential topology does not support Carnap's conventionalism in geometry we show their compatibility. However, Mormann's emphasis on the entanglement that characterizes topology and its associated metrics is not misplaced. It poses questions about limits of empirical inquiry. For Carnap, to pose a question is to give a statement with the task of deciding its truth. Mormann's point forces us to introduce more clarity to what it means to specify the task that decides between competing (...) hypotheses and in what way such a task may be both in practice and/or in principle impossible to carry out. (shrink)

In this paper I present critical evaluations of Ayer and Putnam's views on the analyticity of geometry. By drawing on the historico-philosophical work of Michael Friedman on the relativized apriori; and Roberto Torretti on the foundations of geometry, I show how we can make sense of the assertion that pure geometry is analytic in Carnap's sense.