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.

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)

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

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

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.

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)

Some things look more complex than others. For example, a crenulate and richly organized leaf may seem more complex than a plain stone. What is the nature of this experience—and why do we have it in the first place? Here, we explore how object complexity serves as an efficiently extracted visual signal that the object merits further exploration. We algorithmically generated a library of geometric shapes and determined their complexity by computing the cumulative surprisal of their internal skeletons—essentially (...) quantifying the “amount of information” within each shape—and then used this approach to ask new questions about the perception of complexity. Experiments 1–3 asked what kind of mental process extracts visual complexity: a slow, deliberate, reflective process (as when we decide that an object is expensive or popular) or a fast, effortless, and automatic process (as when we see that an object is big or blue)? We placed simple and complex objects in visual search arrays and discovered that complex objects were easier to find among simple distractors than simple objects are among complex distractors—a classic search asymmetry indicating that complexity is prioritized in visual processing. Next, we explored the function of complexity: Why do we represent object complexity in the first place? Experiments 4–5 asked subjects to study serially presented objects in a self‐paced manner (for a later memory test); subjects dwelled longer on complex objects than simple objects—even when object shape was completely task‐irrelevant—suggesting a connection between visual complexity and exploratory engagement. Finally, Experiment 6 connected these implicit measures of complexity to explicit judgments. Collectively, these findings suggest that visual complexity is extracted efficiently and automatically, and even arouses a kind of “perceptual curiosity” about objects that encourages subsequent attentional engagement. (shrink)

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

Like most, if not all, of his contemporaries, Spinoza never developed a full-fledged philosophy of mathematics. Still, his numerous remarks about mathematics attest not only to his deep interest in the subject (a point which is also confirmed by the significant presence of mathematical books in his library), but also to his quite elaborate and perhaps unique understanding of the nature of mathematics. At the very center of his thought about mathematics stands a paradox (or, at least, an apparent paradox): (...) mathematics provides Spinoza with an epistemic model. Mathematical knowledge is certain (II/138/9 and II/138/9), clear (IV/261/8), and free from teleological thinking (II/79/33), but the objects of mathematical knowledge – i.e., mathematical entities – are nothing but “auxila imaginationis [aids of the imagination],” (IV/57/16 and II/83/15), entities that are not real and merely assist the imagination in carving the world in manner that is suitable to our limited and distortive cognitive capacities. (shrink)

Are mathematical objects affected by their historicity? Do they simply lose their identity and their validity in the course of history? If not, how can they always be accessible in their ideality regardless of their transmission in the course of time? Husserl and Foucault have raised this question and offered accounts, both of which, albeit different in their originality, are equally provocative. Both acknowledge that a scientific object like a geometrical theorem or a chemical equation has a history (...) because it is only constituted in and transmitted through history. But they see that history as a part of its ideality, so that, although historical, a scientific object retains its identity as one and the same object. (shrink)

For Aristotle, the shape of a physical body is perceptible per se (DA II.6, 418a8-9). As I read his position, shape is thus a causal power, as a physical body can affect our sense organs simply in virtue of possessing it. But this invites a challenge. If shape is an intrinsically powerful property, and indeed an intrinsically perceptible one, then why are the objects of geometrical reasoning, as such, inert and imperceptible? I here address Aristotle’s answer to that problem, (...) focusing on the version of it that he presents in De caelo III.8. I argue that if we grant that Aristotle conceived of the shape of a sensible body as some kind of causal power, then the satisfactory resolution of that challenge pushes us to interpret him as having conceived of it as being, more specifically, an impure power—that is, as a property that is not only intrinsically powerful but also, in some way, intrinsically non-powerful as well. This is a notable result not only insofar as it illuminates Aristotle’s conception of shape but also insofar as it contributes to our knowledge of Aristotle’s theory of dunameis and his ontology more broadly. (shrink)

This essay explores theories of place, or lived-space, as regards the role of objectivity and the problem of relativism. As will be argued, the neglect of mathematics and geometry by the lived-space theorists, which can be traced to the influence of the early phenomenologists, principally the later Husserl and Heidegger, has been a major contributing factor in the relativist dilemma that afflicts the lived-space movement. By incorporating various geometrical concepts within the analysis of place, it is demonstrated that the (...) lived-space theorists can gain a better insight into the objective spatial relationships among individuals and within groups—and, more importantly, this appeal to mathematical content need not be construed as undermining the basic tenants of the lived-space approach. (shrink)

Spinoza’s geometrical approach to his masterwork, the Ethics, can be represented by a digraph, a mathematical object whose properties have been extensively studied. The paper describes a project that developed a series of computer programs to analyze the Ethics as a digraph. The paper presents a statistical analysis of the distribution of the elements of the Ethics. It applies a network statistic, betweenness, to analyze the relative importance to Spinoza’s argument of the individual propositions. The paper finds that (...) a small percentage of the propositions greatly exceed the majority in this importance. The paper then describes two logical structures that appear respectively in the Ethics and argues that they result in redundancy in the sense that about ten percent of the propositions could have been eliminated. The appendices list these structures and describes how the resources of the study can be made available to readers. (shrink)

As soon as you believe an imagination to be nonfictional, this imagination becomes your ontological theory of the reality. Your ontological theory (of the reality) can describe a system as the reality. However, actually this system is only a theory/conceptual-space/imagination/visual-imagery of yours, not the actual reality (i.e., the thing-in-itself). An ontological theory (of the reality) actually only describes your (subjective/mental) imagination/visual-imagery/conceptual-space. An ontological theory of the reality, is being described as a situation model (SM). There is no way to prove/disprove (...) that there is only one reality, or there are two realities (i.e., “subjective reality” and “objective reality”). So, every ontology talk/theory/imagination about the two realities is only a talk/theory/imagination – we will never know whether it is true or not. The conventionally-called “physical/objective reality/world” around my conventionally-called “physical/objective body” is actually a geometric mathematical model (being generated/mathematically-modeled by my brain) – it's actually a subset/component/part/element of my brain’s mind/consciousness/manifest-image. Our cosmos is an autonomous objective parallel computing automaton (aka state machine) which evolves by itself automatically/unintentionally – wave-particle duality and Heisenberg’s uncertainty principle can be explained under this SM of my brain. Each elementary particle (as a building block of our cosmos) is an autonomous mathematical entity itself (i.e., a thing in itself). Our cosmos has the same nature as a Game of Life system – both are autonomous objective parallel-computing automata. Cosmos (as a state machine) is indistinguishable from a digital simulation – my consciousness (as something nonphysical) is not cosmos (as a state machine). If we are happy to accept randomness/stochasticity, then it is obviously possible that all other worlds in the many-worlds interpretation (MWI) actually do not exist (objectively). As one metaphysical option, we can treat all other worlds as subjective only (even if they are actually objective). Under the context of this metaphysical option, we are only living in one world (i.e., the world we are currently living in; this world) – we are not living in many worlds at the same time parallelly. Under the context of this metaphysical option, there is only one possible future. The relationship among any number of elementary particles, is governed/described by Schrodinger equation. If (in theory) Schrodinger equation can’t be used to reliably forecast whether I will go to McDonald for dinner in this world (based on the current state of all elementary particles of the cosmos), then what can Schrodinger equation do? The conventionally-called “space” does not exist objectively. “Time” and “matter” are not physical. Consciousness is the subjective-form (aka quale) of the mathematical models (of the objective cosmos) which are intracorporeally/subjectively used by the control logic of a Turing machine’s program fatedly. A Turing machine’s mind/consciousness/manifest-image or deliberate decisions/choices should not be able to actually/objectively change/control/drive the (autonomous or fated) worldline of any elementary particle within this world (i.e., the world we are currently living in, under the context of MWI). Besides the Schrodinger equation (or another mathematical equation/function which is yet to be discovered) which is a valid/correct/factual causality of our cosmos/state-machine, every other causality (of our cosmos/state-machine) is either invalid/incorrect/counterfactual or can be proved by deductive inference based on the Schrodinger equation (or the aforementioned yet-to-be-discovered mathematical equation/function) only. Closed causality entails no causality. Consciousness plays no causal role (“epiphenomenalism”), or in other words, any cognitive/behavioural activity can in principle be carried out without consciousness (“conscious inessentialism”). If the “loop quantum gravity” theory is correct, then time/space does not actually/objectively exist in the objective-evolution of the objective cosmos, or in other words, we should not use the subjective/mental concept of “time”, “state” or “space” to describe/imagine the objective-evolution of our cosmos. (shrink)

The overreliance on verbal models and theories in psychology has been criticized for hindering the development of reliable research programs (Harris, 1976; Yarkoni, 2020). We demonstrate how the conceptual space framework can be used to formalize verbal theories and improve their precision and testability. In the framework, scientific concepts are represented by means of geometric objects. As a case study, we present a formalization of an existing three-dimensional theory of emotion which was developed with a spatial metaphor in mind. Wundt (...) posits that the range of human emotion can be represented along three axes of basic emotions (pleasure/displeasure, excitement/inhibition, tension/relaxation), just as color vision can be represented using three basic colors. We use dimensions to represent basic emotions, points to represent emotional states, and regions to represent broader emotional concepts. We then compare our formalization to an existing structuralist formalization of Wundt’s theory. Further, we discuss the empirical predictions that our formalization generates, such as comparisons of similarity and intensity. We conclude by demonstrating how the tools developed in the conceptual space framework can be used to formulate a theory of emotion based on empirical observation. (shrink)

The overreliance on verbal models and theories in psychology has been criticized for hindering the development of reliable research programs (Harris, 1976; Yarkoni, 2020). We demonstrate how the conceptual space framework can be used to formalize verbal theories and improve their precision and testability. In the framework, scientific concepts are represented by means of geometric objects. As a case study, we present a formalization of an existing three-dimensional theory of emotion which was developed with a spatial metaphor in mind. Wundt (...) posits that the range of human emotion can be represented along three axes of basic emotions (pleasure/displeasure, excitement/inhibition, tension/relaxation), just as color vision can be represented using three basic colors. We use dimensions to represent basic emotions, points to represent emotional states, and regions to represent broader emotional concepts. We then compare our formalization to an existing structuralist formalization of Wundt’s theory. Further, we discuss the empirical predictions that our formalization generates, such as comparisons of similarity and intensity. We conclude by demonstrating how the tools developed in the conceptual space framework can be used to formulate a theory of emotion based on empirical observation. (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)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at (...) Göttingen. Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)

The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity (...) conditions required by the modern version of the theory of proportions. Secondly, Grassmann’s conception of mathematical knowledge will be investigated. Parting from the traditional definition of mathematics as a science of magnitudes, Grassmann considered mathematical forms as particulars rather than universals: the classification of the branches of mathematics was thus based on different operational rules, rather than on empirical criteria of abstraction or on the distinction of different species belonging to a common genus. It will be argued that a different notion of generalization is thus involved, and that the knowledge of mathematical forms relies on the understanding of the rules of generation of the forms themselves. Finally, the paper will analyse if Grassmann’s approach in the first edition of the Ausdehnungslehre should be explained in terms of the notion of purity of method, and if it clashes with Grassmann’s later conventionalism. Although in the second edition the features of the operations are chosen by convention, as it is the case for the anti-commutative property of the multiplication, the choice is oriented by our understanding of the resulting forms: a simplification in the algebraic calculus need not correspond to a simplification in the ‘dimensional’ interpretation of the result of the multiplicative operation. (shrink)

At the turn of the seventeenth century, Bruno and Cavalieri independently developed two theories, central to which was the concept of the geometrical indivisible. The introduction of indivisibles had significant implications for geometry – especially in the case of Cavalieri, for whom indivisibles provided a forerunner of the calculus. But how did this event occur? What can we learn from the fact that two theories of indivisibles arose at about the same time? These are the questions addressed in this (...) paper. Relying on the methodology of “historical epistemology”, this paper asserts that the similarities and differences between the theories of Bruno and Cavalieri can be explained in terms of “shared knowledge”. The paper shows that the idea – on which both Bruno and Cavalieri build – that geometrical objects are generated by motion was part of the mathematical culture of the time. Tracing this idea back to its Pythagorean origins thus sheds light on the relationship between motion and continuum in mathematics. (shrink)

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

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)

The Simulation Hypothesis proposes that all of reality, including the earth and the universe, is in fact an artificial simulation, analogous to a computer simulation, and as such our reality is an illusion. In this essay I describe a method for programming mass, length, time and charge (MLTA) as geometrical objects derived from the formula for a virtual electron; $f_e = 4\pi^2r^3$ ($r = 2^6 3 \pi^2 \alpha \Omega^5$) where the fine structure constant $\alpha$ = 137.03599... and $\Omega$ = (...) 2.00713494... are mathematical constants and the MLTA geometries are; M = (1), T = ($2\pi$), L = ($2\pi^2\Omega^2$), A = ($4\pi \Omega)^3/\alpha$. As objects they are independent of any set of units and also of any numbering system, terrestrial or alien. As the geometries are interrelated according to $f_e$, we can replace designations such as ($kg, m, s, A$) with a rule set; mass = $u^{15}$, length = $u^{-13}$, time = $u^{-30}$, ampere = $u^{3}$. The formula $f_e$ is unit-less ($u^0$) and combines these geometries in the following ratio M$^9$T$^{11}$/L$^{15}$ and (AL)$^3$/T, as such these ratio are unit-less. To translate MLTA to their respective SI Planck units requires an additional 2 unit-dependent scalars. We may thereby derive the CODATA 2014 physical constants via the 2 (fixed) mathematical constants ($\alpha, \Omega$), 2 dimensioned scalars and the rule set $u$. As all constants can be defined geometrically, the least precise constants ($G, h, e, m_e, k_B$...) can also be solved via the most precise ($c, \mu_0, R_\infty, \alpha$), numerical precision then limited by the precision of the fine structure constant $\alpha$. (shrink)

This paper discuss the problem of motion within sense-data concept. Using the sense of speed as starting-point, we debate how it is possible to find a conceptual formulation that combines the idea of mental states with its physicalist criticism. The answer lies in the field of quantum mechanics and its concept of tensor, a geometric object that has a mathematical matrix representation. Thinking about examples taken from the car racing world, where the sense of speed is preponderant, we see (...) how the mental condition of speed is represented matrix-like, tensor-like, rather than ephemerally as the more traditional sense-data formulation advocates. (shrink)

The idea that there could be spatially extended mereological simples has recently been defended by a number of metaphysicians (Markosian 1998, 2004; Simons 2004; Parsons (2000) also takes the idea seriously). Peter Simons (2004) goes further, arguing not only that spatially extended mereological simples (henceforth just extended simples) are possible, but that it is more plausible that our world is composed of such simples, than that it is composed of either point-sized simples, or of atomless gunk. The difficulty for these (...) views lies in explaining why it is that the various sub-volumes of space occupied by such simples, are not occupied by proper parts of those simples. Intuitively at least, many of us find compelling the idea that spatially extended objects have proper parts at every sub-volume of the region they occupy. It seems that the defender of extended simples must reject a seemingly plausible claim, what Simons calls the geometric correspondence principle (GCP): that any (spatially) extended object has parts that correspond to the parts of the region that it occupies (Simons 2004: 371). We disagree. We think that GCP is a plausible principle. We also think it is plausible that our world is composed of extended simples. We reconcile these two notions by two means. On the one hand we pay closer attention to the physics of our world. On the other hand, we consider what happens when our concept of something—in this case space—contains elements not all of which are realized in anything, but instead key components are realized in different features of the world. (shrink)

A detailed axiomatisation of diagrams (in affine geometry) is presented, which supports typing of geometric objects, calculation of geometric quantities and automated proof of theorems.

Following the 26th General Conference on Weights and Measures are fixed the numerical values of the 4 physical constants ($h, c, e, k_B$). This is premised on the independence of these constants. This article discusses a model of a mathematical electron from which can be defined the Planck units as geometrical objects (mass M=1, time T=2$\pi$ ...). In this model these objects are interrelated via this electron geometry such that once we have assigned values to 2 Planck units then (...) we have fixed the values for all Planck units. As all constants can then be defined using geometrical forms (in terms of 2 fixed mathematical constants, 2 unit-specific scalars and a defined relationship between the units $kg, m, s, A$), the least precise CODATA 2014 constants ($G, h, e, m_e, k_B$...) can then be solved via the most precise ($c, \mu_0, \alpha, R_\infty$), with numerical precision limited by the precision of the fine structure constant $\alpha$. In terms of this model we now for example have 2 separate values for elementary charge, calculated from ($c, \alpha, R_\infty$) and the 2017 revision. (shrink)

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

I present a discussion of some issues in the ontology of spacetime. After a characterisation of the controversies among relationists, substantivalists, eternalists, and presentists, I offer a new argument for rejecting presentism, the doctrine that only present objects exist. Then, I outline and defend a form of spacetime realism that I call event substantivalism. I propose an ontological theory for the emergence of spacetime from more basic entities. Finally, I argue that a relational theory of pre-geometric entities can give rise (...) to substantival spacetime in such a way that relationism and substantivalism are not necessarily opposed positions, but rather complementary. In an appendix I give axiomatic formulations of my ontological views. (shrink)

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

In 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.

Is vision merely a state of the beholder’s sensory organ which can be explained as an immediate effect caused by external sensible objects? Or is it rather a successive process in which the observer actively scanning the surrounding environment plays a major part? These two general attitudes towards visual perception were both developed already by ancient thinkers. The former is embraced by natural philosophers (e.g., atomists and Aristotelians) and is often labelled “intromissionist”, based on their assumption that vision is an (...) outcome of the causal influence exerted by an external object upon a sensory organ receiving an entity from the object. The latter attitude to vision as a successive process is rather linked to the “extramissionist” theories of the proponents of geometrical optics (such as Euclid or Ptolemy) who suggest that an entity – a visual ray – is sent forth from the eyes to the object. The present paper focuses on the contributions to this ancient controversy proposed by some 13th-century Latin thinkers. [...]. (shrink)

Conscious experiences are characterized by mental qualities, such as those involved in seeing red, feeling pain, or smelling cinnamon. The standard framework for modeling mental qualities represents them via points in geometrical spaces, where distances between points inversely correspond to degrees of phenomenal similarity. This paper argues that the standard framework is structurally inadequate and develops a new framework that is more powerful and flexible. The core problem for the standard framework is that it cannot capture precision structure: for (...) example, consider the phenomenal contrast between seeing an object as crimson in foveal vision versus merely as red in peripheral vision. The solution I favor is to model mental qualities using regions, rather than points. I explain how this seemingly simple formal innovation not only provides a natural way of modeling precision, but also yields a variety of further theoretical fruits: it enables us to formulate novel hypotheses about the space and structures of mental qualities, formally differentiate two dimensions of phenomenal similarity, generate a quantitative model of the phenomenal sorites, and define a measure of discriminatory grain. A noteworthy consequence is that the structure of the mental qualities of conscious experiences is fundamentally different from the structure of the perceptible qualities of external objects. (shrink)

A semantics of pictorial representation should provide an account of how pictorial signs are associated with the contents they express. Unlike the familiar semantics of spoken languages, this problem has a distinctively spatial cast for depiction. Pictures themselves are two-dimensional artifacts, and their contents take the form of pictorial spaces, perspectival arrangements of objects and properties in three dimensions. A basic challenge is to explain how pictures are associated with the particular pictorial spaces they express. Inspiration here comes from recent (...) proposals that analyze depiction in terms of geometrical projection. In this essay, I will argue that, for a central class of pictures, the projection-based theory of depiction provides the best explanation for how pictures express pictorial spaces, while rival perceptual and resemblance theories fall short. Since the composition of pictorial space is itself the basis for all other aspects of pictorial content, the proposal provides a natural foundation for further pictorial semantics. (shrink)

The chapter deals with misconceptions in perception theory that are based on the idea of slicing the nature of perception along the joints of physics and on corresponding ill-conceived ʹpurposesʹ and ʹgoalsʹ of the perceptual system. It argues that the conceptual structure underlying the percept cannot be inferentially attained from the sensory input. The output of the perceptual system, namely meaningful categories, is evidently vastly underdetermined by the sensory input, namely physico-geometric energy patterns. Thus, the core task of perception theory (...) is to understand the internal conceptual structure with which our perceptual system is endowed. The conceptual structure underlying the semantic distinctions that characterise the output of the perceptual system can, by conceptual necessity, only be expressed by a logical language that is strictly more powerful than the logical language by which sensory notions can be expressed. Consequently, the internal structure underlying perceptual meaning – including core notions such as ‘Gestalt’ or ‘perceptual object’ – cannot be derived, by whatever kind of general inductive machinery, from the sensory input, or, more generally, from experience. (shrink)

Ordinary reasoning about space—we argue—is first and foremost reasoning about things or events located in space. Accordingly, any theory concerned with the construction of a general model of our spatial competence must be grounded on a general account of the sort of entities that may enter into the scope of the theory. Moreover, on the methodological side the emphasis on spatial entities (as opposed to purely geometrical items such as points or regions) calls for a reexamination of the conceptual (...) categories required for this task. Building on material presented in an earlier paper, in this work we offer some examples of what this amounts to, of the difficulties involved, and of the main directions along which spatial theories should be developed so as to combine formal sophistication with some affinity with common sense. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (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.

In the Fifth Meditation, Descartes makes a remarkable claim about the ontological status of geometrical figures. He asserts that an object such as a triangle has a 'true and immutable nature' that does not depend on the mind, yet has being even if there are no triangles existing in the world. This statement has led many commentators to assume that Descartes is a Platonist regarding essences and in the philosophy of mathematics. One problem with this seemingly natural reading (...) is that it contradicts the conceptualist account of universals that one finds in the Principles of Philosophy and elsewhere. In this paper, I offer a novel interpretation of the notion of a true and immutable nature which reconciles the Fifth Meditation with the conceptualism of Descartes' other work. Specifically, I argue that Descartes takes natures to be innate ideas considered in terms of their so-called 'objective being'. (shrink)

There have recently been various empirical attempts to answer Molyneux’s question, for example, the experiments undertaken by the Held group. These studies, though intricate, have encountered some objections, for instance, from Schwenkler, who proposes two ways of improving the experiments. One is “to re-run [the] experiment with the stimulus objects made to move, and/or the subjects moved or permitted to move with respect to them” (p. 94), which would promote three dimensional or otherwise viewpoint-invariant representations. The other is “to use (...) geometrically simpler shapes, such as the cube and sphere in Molyneux’s original proposal, or planar figures instead of three-dimensional solids” (p. 188). Connolly argues against the first modification but agrees with the second. In this article, I argue that the second modification is also problematic (though still surmountable), and that both Schwenkler and Connolly are too optimistic about the prospect of addressing Molyneux’s question empirically. (shrink)

In the not-too-distant past, vision was often said to involve three levels of processing: a low level concerned with descriptions of the geometric and photometric properties of the image, a high level concerned with abstract knowledge of the physical and semantic properties of the world, and a middle level concerned with anything not handled by the other two. The negative definition of mid-level vision contained in this description reflected a rather large gap in our understanding of visual processing: How could (...) the here-and-now descriptions of the low levels combine with the enduring knowledge of the high levels to produce our perception of the surrounding world? A number of experimental and theoretical efforts have been made over the past few decades to solve this "mid-level crisis". One of the more recent of these is based on the phenomenon of change blindness—the difficulty in seeing a large change in a scene when the transients accompanying that change no longer convey information about its location (Rensink, O'Regan, & Clark, 1997; Rensink, 2000a). Phenomenologically, this effect is quite striking: the change typically is not seen for several seconds, after which it suddenly snaps into awareness. During the time the change remains "invisible", there is an apparent disconnection of the low-level descriptions (which respond to the change) from subjective visual experience (which does not). As such, this effect would seem to have the potential to help us understand how mid-level mechanisms might knit low- and high-level processes into a coherent representation of our surroundings. -/- It is argued here that this potential can indeed be realized, and that change blindness can teach us much about the nature of mid-level vision.3 A number of studies are first reviewed showing that the perception of a scene does not involve a steady buildup of detailed representation: rather, it is a dynamic process, with focused attention playing one of the main roles, viz., forming coherent object representations whenever needed. It is then argued that change blindness can also shed considerable light on the nature of focused attention itself, such as its speed, capacity, selectivity, and ability to bind together visual properties into coherent structures. (shrink)

A gas in a box is perhaps the most important model system studied in thermodynamics and statistical mechanics. Usually, studies focus on the gas, whereas the box merely serves as an idealized confinement. The present article focuses on the box as the central object and develops a thermodynamic theory by treating the geometric degrees of freedom of the box as the degrees of freedom of a thermodynamic system. Applying standard mathematical methods to the thermody- namics of an empty box (...) allows equations with the same structure as those of cosmology and classical and quantum mechanics to be derived. The simple model system of an empty box is shown to have interesting connections to classical mechanics, special relativity, and quantum field theory. (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)

It is a received view that Kant’s formal logic (or what he calls “pure general logic”) is thoroughly intensional. On this view, even the notion of logical extension must be understood solely in terms of the concepts that are subordinate to a given concept. I grant that the subordination relation among concepts is an important theme in Kant’s logical doctrine of concepts. But I argue that it is both possible and important to ascribe to Kant an objectual notion of logical (...) extension according to which the extension of a concept is the multitude of objects falling under it. I begin by defending this ascription in response to three reasons that are commonly invoked against it. First, I explain that this ascription is compatible with Kant’s philosophical reflections on the nature and boundary of a formal logic. Second, I show that the objectual notion of extension I ascribe to Kant can be traced back to many of the early modern works of logic with which he was more or less familiar. Third, I argue that such a notion of extension makes perfect sense of a pivotal principle in Kant’s logic, namely the principle that the quantity of a concept’s extension is inversely proportional to that of its intension. In the process, I tease out two important features of the Kantian objectual notion of logical extension in terms of which it markedly differs from the modern one. First, on the modern notion the extension of a concept is the sum of the objects actually falling under it; on the Kantian notion, by contrast, the extension of a concept consists of the multitude of possible objects—not in the metaphysical sense of possibility, though—to which a concept applies in virtue of being a general representation. While the quantity of the former extension is finite, that of the latter is infinite—as is reflected in Kant’s use of a plane-geometrical figure (e.g., circle, square), which is continuum as opposed to discretum, to represent the extension in question. Second, on the modern notion of extension, a concept that signifies exactly one object has a one-member extension; on the Kantian notion, however, such a concept has no extension at all—for a concept is taken to have extension only if it signifies a multitude of things. This feature of logical extension is manifested in Kant’s claim that a singular concept (or a concept in its singular use) can, for lack of extension, be figuratively represented only by a point—as opposed to an extended figure like circle, which is reserved for a general concept (or a concept in its general use). Precisely on account of these two features, the Kantian objectual extension proves vital to Kant’s theory of logical quantification (in universal, particular and singular judgments, respectively) and to his view regarding the formal truth of analytic judgments. (shrink)

The parceling of land into real estate is more than a simple geometrical affair. Real estate is a historical product of interaction between human beings, political, legal and economic institutions, and the physical environment. And while many authors, from Jeremy Bentham to Hernando de Soto, have drawn attention to the ontological (metaphysical) aspect of property in general, no comprehensive analysis of landed property has been attempted. The paper presents such an analysis and shows how landed property differs from other (...) types of property in a way which implies a special role for political and economic philosophy of property rights in land. This is the Chinese translation of "The Metaphysics of Real Estate", Topoi, 20: 2 (September 2001), 161–172. (shrink)

We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and (...) we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula, which can be interpreted by introducing a suitable distributive property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions. The first condition gives a geometrical characterization for the coherence of prevision assessments on a family F constituted by n conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $F\cup K$, where $K$ is the set of conditional constituents associated with the conditional events in $F$. Then, we give some further theoretical results and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications. (shrink)

Cross-linguistic strategies for mapping lexical and spatial relations from body partonym systems to external object meronymies (as in English ‘table leg’, ‘mountain face’) have attracted substantial research and debate over the past three decades. Due to the systematic mappings, lexical productivity and geometric complexities of body-based meronymies found in many Mesoamerican languages, the region has become focal for these discussions, prominently including contrastive accounts of the phenomenon in Zapotec and Tzeltal, leading researchers to question whether such systems should be (...) explained as global metaphorical mappings from bodily source to target holonym or as vector mappings of shape and axis generated “algorithmically”. I propose a synthesis of these accounts in this paper by drawing on the species-specific cognitive affordances of human upright posture grounded in the reorganization of the anatomical planes, with a special emphasis on antisymmetrical relations that emerge between arm-leg and face-groin antinomies cross-culturally. Whereas Levinson argues that the internal geometry of objects “stripped of their bodily associations” (1994: 821) is sufficient to account for Tzeltal meronymy, making metaphorical explanations entirely unnecessary, I propose a more powerful, elegant explanation of Tzeltal meronymic mapping that affirms both the geometric-analytic and the global-metaphorical nature of Tzeltal meaning construal. I do this by demonstrating that the “algorithm” in question arises from the phenomenology of movement and correlative body memories—an experiential ground which generates a culturally selected pair of inverse contrastive paradigm sets with marked and unmarked membership emerging antithetically relative to the transverse anatomical plane. These relations are then selected diagrammatically for the classification of object orientations according to systematic geometric iconicities. Results not only serve to clarify the case in question but also point to the relatively untapped potential that upright posture holds for theorizing the emergence of human cognition, highlighting in the process the nature, origins and theoretical validity of markedness and double scope conceptual integration. (shrink)

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...) of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

This paper aims to address the difficulties of high school students in bridging their computational understanding with their visualization skills in understanding the notion of the limits in their calculus class. This research used a pre-experimental one-group pretest-posttest design research on 62 grade 10 students enrolled in the Science, Technology, and Engineering Program (STEP) in one of the public high schools in Zamboanga del Sur, Philippines. A series of remedial sessions were given to help them understand the function values, one-sided (...) limits, limits of a function, and its discontinuities with the aid of the GeoGebra. Results revealed that there is a significant improvement in their mathematics performance before and after the remediation, t(61) = -19.99, p<.05. The real-time feedback provided by the drawing interface of GeoGebra has provided the students the insights into how objects behave in geometrical space, bridging their understanding in locating the values shown by the computational algebraic processes. Additionally, the significant difference in the achievement reveals a large effect size (Cohen’s =2.54) which could be accounted for by the remediation using interactive activities in the GeoGebra. (shrink)