definitions and explanations frequently come together and permeate almost all fields of knowledge. This does not exclude mathematics, even when these definitions hold clear links and close connections with our physical world. Here we propose a rather different perspective. Making operational physical assumptions, we show how it is possible to rigorously reconstruct some features of both geometry and topology. Broadly speaking, assuming this operational and more concrete philosophy we not only are capable of defining primitive concepts like points, (...) class='Hi'>straight lines, planes, angles and spaces, but we also go all the way up to more complex constructions. We show that within our method it is also possible to come up with what we call a compass-based topology as well as a normed and a metric space. We hope this operationalist point-of-view can assist other researchers and students to have a clearer understanding of these quite abstract concepts. (shrink)

While the classical account of the linear continuum takes it to be a totality of points, which are its ultimate parts, Aristotle conceives of it as continuous and infinitely divisible, without ultimate parts. A formal account of this conception can be given employing a theory of quantification for nonatomic domains and a theory of region-based topology.

This article is a study of geometric constructions. We consider, as an illustration, the methods used for dividing the straight line into n equal parts (n-section). Architects and practicioners of classical Europe had at their disposal a broad range of geometric constructions: ancient ones were edited and translated, whereas new solutions were constantly published. The wide variety and reasons for selection of these geometric constructions are puzzling: the most widespread construction was not the simplest one. This article wonders (...) why so many constructions were used to solve the same problem and why tedious methods were used. It is argued that the practical conditions and scale at which the problem was tackled could account for such diversity. (shrink)

In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite (...) time line H, which is a hyperfinite initial segment of the hyperintegers. The U-monad topology of H is the quotient topology of the U-topological space H modulo U. In this paper we answer a question of Keisler and Leth about the U-monad topologies by showing that when H is κ-saturated and has cardinality κ, (1) if the coinitiality of U1 is uncountable, then the U1-monad topology and the U2-monad topology are homeomorphic iff both U1 and U2 have the same coinitiality; and (2) H can produce exactly three different U-monad topologies (up to homeomorphism) for those U's with countable coinitiality. As a corollary H can produce exactly four different U-monad topologies if the cardinality of H is ω1. (shrink)

Physical and biological measurements might display range values extending towards infinite. The occurrence of infinity in equations, such as the black hole singularities, is a troublesome issue that causes many theories to break down when assessing extreme events. Different methods, such as re-normalization, have been proposed to avoid detrimental infinity. Here a novel technique is proposed, based on geometrical considerations and the Alexander Horned sphere, that permits to undermine infinity in physical and biophysical equations. In this unconventional approach, a continuous (...) monodimensional line becomes an assembly of countless bidimensional lines that superimpose in quantifiable knots and bifurcations. In other words, we may state that Achilles leaves the straight line and overtakes the turtle. (shrink)

The unified theory of dose and effect, as indicated by the median-effect equation for single and multiple entities and for the first and higher order kinetic/dynamic, has been established by T.C. Chou and it is based on the physical/chemical principle of the massaction law (J. Theor. Biol. 59: 253-276, 1976 (質量作用中效定理) and Pharmacological Rev. 58: 621-681, 2006) (普世中效指數定理). The theory was developed by the principle of mathematical induction and deduction (數學演繹歸納法). Rearrangements of the median-effect equation lead to Michaelis-Menten, Hill, Scatchard, (...) and Henderson-Hasselbalch equations. The “median” serves as the universal reference point and the “common link” for the relationship of all entities and is also the “harmonic mean” of kinetic dissociation constants. Over 300 mechanism-specific equations have been derived and published using the mathematical induction-deduction process. These equations can be deduced into several general equations, including the median-mediated whole/part equation, combination index theorem, isobologram equation, and polygonogram. It is proven that “dose” and “effect” are interchangeable, thus, “substance” and “function” are interchangeable, which leads to “the unity theory” (劑效、心物、知行一元論) in quantitative mathematical philosophy (數學的定量哲學) in functional context. Therefore, a general theory centered on the “median” and based on equilibrium dynamics has evolved. In other words: [「中」的宇宙觀： 以「中」爲基凖的動力學生態平衡]. Based on the median-effect equation of the mass-action law, the fundamental claim is that we can draw “a specific cure” for only two data points, if they are determined accurately. This claim has far reaching consequences since it defies the general held belief that two points can dray only a straight line. Remarkably, the unity theory (一元論) providesscientific/mathematical interpretation in equations and in graphics of Chinese ancient philosophy, including Fu-Si Ba Gua (伏羲八卦), Dao’s Harmony (和諧), the Confucian doctrine of the mean (儒家中庸之道), Chou Dun-Yi’s (周敦頤, 1017-1073) From Wu-ji to Tai-ji and Taiji Tu Sho (無極而太極及太極圖說). The moderntopological analysis for trinity yields an exact correspondence to the Ba-Gua, which was introduced over 4,000 years ago. Furthermore, the median-centered algorithm, promotes modern ecological content (生態學) in the equilibral dynamic state of harmony. It is concluded that Western science and Eastern philosophy are directly linked and complementary to each other. Since the truth in mathematical quantitative philosophy (數學的定量哲學) has no boundaries, East and West philosophies can flourish together for the common goal and ideal in science and in humanity (世界大同). (shrink)

This paper presents the topological arrangements in four geometrical figures of modal propositions and their derivative relations by means of Peirce's gamma graphs and their rules of transformation. The idea of arraying the gamma graphs in a geometric and symmetrical order comes from Peirce himself who in a manuscript drew two cubes in which he presented the derivative relations of some gamma graphs. Therefore, Peirce's insights of a topological order of gamma graphs are extended here backwards from the cube to (...) the line and the square; and then forwards from the cube to the four-dimensional polyhedron. (shrink)

In this paper, an in-depth study of interactive visual communication of network topology through non-line-of-sight congestion control algorithms is conducted to address the real-time routing problem of adapting to dynamic topologies, and a delay-constrained stochastic routing algorithm is proposed to enable packets to reach GB within the delay threshold in the absence of end-to-end delay information while improving network throughput and reducing network resource consumption. The algorithm requires each sending node to select an available relay set based on (...) the location of its neighbor nodes and channel state and computes transfer probabilities for each node in the relay set combining the remaining delay of the packet with the distance from the relay node to GB. Based on the obtained transfer probability and local channel state, the sending node passes the packet to the relay node. The convergence of the algorithm is proved and its performance is verified by simulation. The first part of the algorithm is based on the greedy algorithm to deploy and locate the network flying platform nodes with the goal of efficient coverage of the network flying platform nodes, considering the ground base station services. As the delay on each link varies due to the change of channel state, the source and relay nodes asynchronously update the data generation rate and the pairwise parameters based on the received local information and use the obtained optimal values to pass the packets to GB. (shrink)

Measuring time is expressing temporal relationships between objects in terms of spatial relationships with the aid of geometric points, straight lines and clocks. The concepts, point and line, are abstracted from the concrete substratum of sensory experience. This process of abstraction is integrated with the psychological processes which go on within an observer who is reading a clock. The analysis of clock-reading from a psychological point of view points up the necessity to differentiate between two modalities of time: (...) objective time, or, time relationships between objects and which can be expressed in terms of space with point, line and clock; and subjective time, or, the time structure of the human individual in which region point and line are irrelevant concepts. From this irrelevance of point and line it follows that subjective time cannot be measured. (shrink)

In the topological semantics for modal logic, S4 is well known to be complete for the rational line and for the real line: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete but strongly complete, for the rational line. But no similarly easy amendment is available for the real line. In an earlier paper, we proved (...) a general theorem: S4 is strongly complete for any dense-in-itself metric space. Strong completeness for the real line is a special case. In the current paper, we give a proof of strong completeness tailored to the special case of the real line: the current proof is simpler and more accessible than the proof of the more general result and involves slightly different techniques. We proceed in two steps: first, we show that S4 is strongly complete for the space of finite and infinite binary sequences, equipped with a natural topology; and then we show that there is an interior map from the real line onto this space. (shrink)

And so that there may be less difficulty in understanding what I shall say on this matter, I should like that those not versed in anatomy should take the trouble, before they read this, to have cut up before their eyes the heart of some large animal that has lungs—for it is in all respects sufficiently similar to the heart of man….

This paper is devoted to discussing the topological structure of the arrow of time. In the literature, it is often accepted that its algebraic and topological structures are that of a one-dimensional Euclidean space \, although a critical review on the subject is not easy to be found. Hence, leveraging on an operational approach, we collect evidences to identify it structurally as a normed vector space \\), and take a leap of abstraction to complete it, up to isometries, to the (...) real line. During the development of the paper, the space-time is recognized as a fibration, with the fibers being the sets of simultaneous events. The corresponding topology is also exposed: open sets naturally arise within our construction, showing that the classical space-time is non-Hausdorff. The transition from relativistic to classical regimes is explored too. (shrink)

Deleuze and Guattari's work opens theoretical and political possibilities for us by demonstrating that the boundary between ‘the West’ and the ‘non-West’ is itself a historically unstable epistemological inscription on the surface of the earth, but one that nevertheless retraces itself over and over again. They show us that our political possibilities exist precisely in the ‘non-West’, so long as we understand this term not in the sense of an existing supposedly ‘non-Western’ territory or ‘substance’, but rather as a ‘ (...) class='Hi'>line of flight’ beyond the nation-state, the form in which the ‘original division’ and creation of the social relation called ‘the West’ continues to produce itself. They thus demonstrate that the way out of the Eurocentric epistemological order lies precisely in clarifying how ‘the West’ was created as an effect of the capitalist axiomatic and maintained by the repetition of this paradoxical ‘commencement’ in the form of the ‘anthropological difference’, congealed into the modern nation-state. In the pursuit of this moment, we will investigate the question of how the social surface is territorialised and enclosed for capitalist accumulation, how the purely heterogeneous topology of the social comes to be inscribed with the traces of the apparatus of capture. (shrink)

In a series of recent papers, two of which appeared in this journal, a group of philosophers, physicists, and climate scientists have argued that something they call the `hawkmoth effect' poses insurmountable difficulties for those who would use non-linear models, including climate simulation models, to make quantitative predictions or to produce `decision-relevant probabilites.' Such a claim, if it were true, would undermine much of climate science, among other things. Here, we examine the two lines of argument the group has used (...) to support their claims. The first comes from a set of results in dynamical systems theory associated with the concept of `structural stability.' The second relies on a mathematical demonstration of their own, using the logistic equation, that they present using a hypothetical scenario involving two apprentices of Laplace's omniscient demon. We prove two theorems that are relevant to their claims, and conclude that both of these lines of argument fail. There is nothing out there that comes close to matching the charac. (shrink)

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the (...) choice sequence. But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. "It is almost as if one could hear the two rebels arguing their case in a European café or on a terrace, and coming to a common understanding, with both men taking their hat off to the other, in admiration and gratitude. Dr. van Atten has convincingly applied Husserl's method to Brouwer's program, and has equally convincingly applied Brouwer's intuition to Husserl's program. Both programs have come out the better." Piet Hut, professor of Interdisciplinary Studies, Institute for Advanced Study, Princeton, U.S.A. (shrink)

The Necker cube and the productive class of related stimuli involving multiple depth interpretations driven by corner-like line junctions are often taken to be ambiguous. This idea is normally taken to be as little in need of defense as the claim that the Necker cube gives rise to multiple distinct percepts. In the philosophy of language, it is taken to be a substantive question whether a stimulus that affords multiple interpretations is a case of ambiguity. If we take into (...) account what have been identified as hallmark features of ambiguity and look at the empirical record, it appears that the Necker cube and related stimuli are not ambiguous. I argue that this raises problems for extant models of multistable perception in cognitive neuroscience insofar as they are purported to apply to these stimuli. Helpfully, similar considerations also yield reasons to suggest that the relevant models are well motivated for other instances of multistable perception. However, a different breed of model seems to be required for the Necker cube and related stimuli. I end with a sketch how one may go about designing such a model relying on oscillatory patters in neural firing. I suggest that distinctions normally confined to the philosophy of language are important for the study of perception, a perspective with a growing number of adherents. (shrink)

A systematic and comprehensive introduction to the study of nonlinear dynamical systems, in both discrete and continuous time, for nonmathematical students and researchers working in applied fields. An understanding of linear systems and the classical theory of stability are essential although basic reviews of the relevant material are provided. Further chapters are devoted to the stability of invariant sets, bifurcation theory, chaotic dynamics and the transition to chaos. In the final two chapters the authors approach the subject from a measure-theoretical (...) point of view and compare results to those given for the geometrical or topological approach of the first eight chapters. Includes about one hundred exercises. A Windows-compatible software programme called DMC, provided free of charge through a website dedicated to the book, allows readers to perform numerical and graphical analysis of dynamical systems. Also available on the website are computer exercises and solutions to selected book exercises. See www.cambridge.org/economics/resources. (shrink)

We are all used to drawing straight lines to represent time, and above them, we plot historical events or physical or economic data. What to us is a self-evident convention, is however of an astonishingly recent date: it emerged only in the second half of the eighteenth century. To us, this late date seems paradoxical and cries out for an explanation. How else did earlier periods measure change, if not as a function of time? it will be argued that (...) since Antiquity, time was taken to measure change, and change to occur in space. ‘Our’ idea of representing time as an independent dimension would have seemed aberrant. But then, a second issue arises. Did not medieval natural philosophers employ timelines, Oresme’s diagram of the mean speed theorem being the most famous case? However, as will be shown, our interpretation of his diagram is probably wrong. This insight, in turn, takes care of a third paradox, namely Galileo’s initial inability to represent the law of free fall correctly. This article will document that the timeline first emerged in the late sixteenth century in works on chronology, made its first appearance in physics in Galileo’s diagrams, and had its general breakthrough in the eighteenth century. (shrink)

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? Mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice (...) sequence. But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. (shrink)

Localised configurations of the free electromagnetic field are constructed, possessing properties of massive, spinning, relativistic particles. In an inertial frame, each configuration travels in a straight line at constant speed, less than the speed of lightc, while slowly spreading. It eventually decays into pulses of radiation travelling at speedc. Each configuration has a definite rest mass and internal angular momentum, or spin. Each can be of “electric” or “magnetic” type, according as the radial component of the magnetic or (...) electric field vanishes in the rest frame, and each has an “antiparticle.” Any such configuration, of electric or magnetic type, is characterized in part by a set of labels (κ, ω0, σ,l, m), where ω0 is the mean of the angular frequencies of the plane waves making up the configuration, σ is the variance of those frequencies, κ is a positive constant with dimensions of action, andl, m are angular momentum “quantum numbers” withl a positive integer andm an integer such that ‖m‖≤l. The rest energy of the “particle” is κω0, its spin is κ ‖m‖, and its lifetime is of the order of 1/σ. Its antiparticle has ω0 replaced by −ω0. (shrink)

Epitome V (1621), and consisted of matching an element of area to an element of time, where each was mathematically determined. His treatment of the area depended solely on the geometry of Euclid's Elements, involving only straight-line and circle propositions – so we have to account for his deliberate avoidance of the sophisticated conic-geometry associated with Apollonius. We show also how his proof could have been made watertight according to modern standards, using methods that lay entirely within his (...) power. The greatest innovation, however, occurred in Kepler's fresh formulation of the measure of time. We trace this concept in relation to early astronomy and conclude that Kepler's treatment unexpectedly entailed the assumption that time varied nonuniformly; meanwhile, a geometrical measure provided the independent variable. Even more surprisingly, this approach turns out to be entirely sound when assessed in present-day terms. Kepler himself attributed the cause of the motion of a single planet around the Sun to a set of `physical' suppositions which represented his religious as well as his Copernican convictions; and we have pared to a minimum – down to four – the number he actually required to achieve this. In the Appendix we use modern mathematics to emphasize the simplicity, both geometrical and kinematical, that objectively characterizes the Sun-focused ellipse as an orbit. Meanwhile we highlight the subjective simplicity of Kepler's own techniques (most of them extremely traditional, some newly created). These two approaches complement each other to account for his success. (shrink)

Der Beitrag beschäftigt sich mit kulturanthropologischen und literarischen Reflexionen auf den Bewegungsmodus des Gehens. Er diskutiert die Frage, inwieweit das Gehen in diesen Diskursen als Linienpraxis aufgefasst wird. Ausgangspunkt ist die Beobachtung, dass die Kulturanthropologie, die dem aufrechten Gang eine Schlüsselfunktion für die Anthropogenese zuweist, diesen zugleich als Produkt eines ›Begradigungsprozesses‹ markiert und an die dichotomische Gegenüberstellung von Natur und Kultur koppelt. In literarischen Texten, aber auch in neueren ökoanthropologischen Ansätzen wird die Natur-Kultur-Opposition und die damit verbundene Privilegierung der geraden (...) Linie kritisch hinterfragt. Setzt die literarische Peripatetik mithin eine alternative Form der Beziehung zwischen Mensch und Umwelt in Szene? Entwirft sie eine Ökologie des Denkens und Wahrnehmens, die sich jenseits der Natur-Kultur-Dichotomie bewegt? Diese Fragen werden an ausgewählten Fallbeispielen beleuchtet. The essay deals with the peripatetic mode of movement as reflected in cultural anthropology and literature. It asks to what extent these discourses view walking as a practice of line-making. The point of departure is the observation that cultural anthropology ascribes a key role to the upright gait within anthropogenesis while associating it with a process of rectilinearization on the one hand and a sharp dichotomy between nature and culture on the other hand. Certain literary texts and recent approaches in ecologically oriented anthropology have, however, challenged the validity of this dichotomy and the concomitant privilege of the straight line. Does literary peripatetics thus propose an alternative way to grasp the relationship between human beings and the environment? Does it outline a new ecology of the mind that supersedes the binary relation between nature and culture? These questions are discussed with reference to selected case studies. (shrink)

Beginning with the situated character of the question concerning the human, this paper argues that the problem of the human is itself inextricably bound to the problem of situation or place. Consequently, any genuine philosophical anthropology must take the form of a philosophical topology. This line of argument is developed through the work Abraham Heschel, Martin Heidegger, Martin Buber, and also Helmut Plessner.

Let ξ ≥ 1 be a countable ordinal. We study the Borel subsets of the plane that can be made [Formula: see text] by refining the Polish topology on the real line. These sets are called potentially [Formula: see text]. We give a Hurewicz-like test to recognize potentially [Formula: see text] sets.

It is a commonplace in the analytic geometry of physical space-time that an extended straight line segment, having positive length, is treated as “consisting of” unextended points, each of which has zero length. Analogously, time intervals of positive duration are resolved into instants, each of which has zero duration.

Drawing on work done by Helmholtz, I argue that Reid was in no position to infer that objects appear as if projected on the inner surface of a sphere, or that they have the geometric properties of such projections even though they do not look concave towards the eye. A careful consideration of the phenomena of visual experience, as further illuminated by the practice of visual artists, should have led him to conclude that the sides of visible appearances either look (...) straight, in which case their angles appear to approximate Euclidean measures, or their angles do not appear to approximate Euclidean measures for straight line figures, in which case their sides do not look straight. (shrink)

Reason's capacity to take us where we did not expect to go could also lead to a curious diversion from what one might expect to be the straight line of evolution. We have evolved a capacity to reason because it helps us to survive and reproduce. But if reason is an escalator, then although the first part of the journey may help us to survive and reproduce, we may go further than we needed to go for this purpose (...) alone. We may even end up somewhere that creates tension with other aspects of our nature. In this respect, there may after all be some validity in Kant 's picture of tension between our capacity to reason, and what it may lead us to see as the right thing to do, and our more basic desires. We can live with the contradictions only up to a point. (shrink)

This piece, included in the drift special issue of continent. , was created as one step in a thread of inquiry. While each of the contributions to drift stand on their own, the project was an attempt to follow a line of theoretical inquiry as it passed through time and the postal service(s) from October 2012 until May 2013. This issue hosts two threads: between space & place and between intention & attention . The editors recommend that to experience (...) the drifiting thought that attention be paid to the contributions as they entered into conversation one after another. This particular piece is from the BETWEEN INTENTION & ATTENTION thread: Jeremy Fernando, Sitting in the Dock of the bay, watching... * R.H. Jackson, Reading Eyes * Gina Rae Foster, Nyctoleptic Nomadism: The Drift/Swerve of Knowing * Bronwyn Lay, Driftwood * Patricia Reed, Sentences on Drifitng * David Prater, drift: a way * * * * "… to sleep perchance to dream " 1 To dream: to be not quite asleep, yet not particularly awake. Or, rather: to be awake but not quite know it. For, it is only when we dream, when we are dreaming, that we know that we are not in that final sleep. But we can only know that we are dreaming, that we have dreamt, when we are awake, when we have awoken; after it is too late. When all we know is that the sleep beyond finitude, the sleep that is the step beyond, is not yet upon us, is only to come. To die to sleep … To dream: a sleep that refuses sleep. Perchance to dream: to drift—between sleep and sleep. Aye there's the rub For, can we even know if we have been sleeping? Or, if death has claimed us?—even if a little death. α Ω α Ω α Ω To drift: but from, to, what? For, to drift implies a certain direction that one was headed from, heading to, headed for; without these indications, markers, points in relation with each other, one would just be moving. Can one know—intend—one's drift? Certainly a stunt driver would say so. But even as (s)he is starting her slide, all that (s)he can know is that she is setting the car, herself, the car with herself in it, in motion: after which the drift itself takes over. After which, all (s)he can do is attend to it. At the point of the drift: both (s)he and the car are drifting—here, one might not even be able to separate the movement from those involved in it. Without either of them, there would not be a drift; there is no drifting without the drifter. Both the drifter and the drifting are in a relationality; in which, all that they can know is that they are in relation with each other. Hence, the drift itself is a relationality. A non-essence. But, it is not as if we cannot speak of it. Perhaps though: we can only speak of it as if we can speak of it. Always already an imaginary gesture; where what is being imagined is the relationality between the drift and the ones drifting. Thus, we have a situation where the drifter and drifting are in a relationality; where relationality itself is what is being imagined. Perhaps then, what are we drifting from, to? , is a moot question. As is, what is drifting? Perhaps then, all we can say is drift? To speak of drift is an attempt to speak of the unspeakable. Not that what is speakable and what is unspeakable are antonyms: if that were so, speaking the unspeakable would make no sense, be a contradiction. But that in every act of speaking, something unspeakable is potentially said: something that opens, ruptures, wounds even. And not just that—at the point where it punctures, speaking itself moves out of the way for the unspeakable; speaking itself disappears. "… the whole art is to know how to disappear before dying, and instead of dying. " 2 To disappear; or, to drift out of sight. Where the words themselves slip away. After all: "in the Beginning was the Word. It was only afterwards that Silence came." Perhaps the wish, the hope, is that "the end itself has disappeared …" (Baudrillard, 70) Remaining hidden from us. Perhaps only glimpsed when we dream. Secret. α Ω α Ω α Ω " Bury all your secrets in my skin " (Corey Taylor) Which is the problem: words cling. And they remain. Perhaps not ontologically; but they certainly remain to haunt us. And here, we should not forget Lucretius' lesson that communication occurs in the skin between the parties in communion with each other. Which is not to say that the encounter is determined by atoms—and more precisely atoms that move in straight lines until they collide with each other—that communication is pre-determined. For, one must not forget that will is found, discovered, enacted even, at the moment the atoms swerve. Clinamen . Drift. But even in their movement—drifting—they trace themselves into the skin between; a tangential touching. Perhaps only briefly. But even then, enough … "… there's always texture that betrays the place." (May Ee Wong) Here though, one must not forget that betrayal cannot happen in the absence of love. In fact, betrayal is the very excess of love: where one loves the other so much that one can no longer bear to see the other drift from what (s)he could have been. Whether that idealised other exists or is only in one's head is another question altogether. Perhaps, a fetishised other: keeping in mind that "fetishes are hinged around simulation." After all, "when one is supposed to show up as an oil rig diver no one is expecting actual crude oil" (Amanda Sordes); in fact, actualisation is the perfect way to destroy the fantasy. Perhaps then, the only way to maintain love for another is to maintain a proper distance, as it were, from love: allow the love to constantly alter, change. And here, one must not forget that if love is a relationality between one and another who remains wholly other (otherwise just a mere manifestation of the self), love is a relationality that knows nothing except for the fact that it is in a relation. For, to love one has to attend to—without subsuming another, some other, under oneself. Which means that to love, one has to be willing to risk, to open oneself, to allow oneself to be wounded, torn apart. In new ways, ways that we have yet to understand, come across, ways we do not yet have a name for. Thus, this movement in love is one that occurs in utter blindness; to not only the other, but to what love is. This is love as pure drifting. Perhaps always searching for love itself, without ever knowing what it is that it is looking for. Love: only at the very moment when the word love itself disappears. Perhaps all we can do is sit, and attend: watching the tides flowing away—as if they were having their "last swim of the summer." (Hendrik Speck) Like a butterfly. α Ω α Ω α Ω Isn't it quite amazing how the appearance of a butterfly can inject a stutter or pause into any conversation? Words and words pour out of the animals in assembly, before they are all of a sudden arrested by the passing flight. Heads turn to trace a lilting poetics, attempting to close the distance with this seemingly awkward beauty. There are no straight lines here, only a relative arrival and departure to bracket a brilliant and bewildering trajectory, surging and lurching in a vibrating and nomadic line avant la lettre. (Sean Smith, 'I Seek You: Countdown to Stereoscopic Tear') Before the letter. Before the possibility of naming. Before being sayable. Quite possibly also before language. And yet, a "surging and lurching," a movement with an effect—"vibrating and nomadic"—tracing itself before there is even anything to trace. Leaving something, even if that thing remains unknowable, for us to attend to. Drifting into us. I had some dreams they were clouds in my coffee, clouds in my coffee..." (Carly Simon) NOTES William Shakespeare, Hamlet , Act 3 Scene 1. Jean Baudrillard, Why hasn't everything already disappeared? , 25.  . (shrink)

What Mamardašvili meant by “process of knowledge” is not an all-embracing vision of reality accomplished “once-and-for-all”; it is not a step by step procedure of deduction; rather it is an anti-dialectical reconstruction of a constellation of signs put together over and over again by the subject by an act of non-premeditated genius. It is a kind of aesthetic act that makes the sense appear, like a vertical cut in the sequential line of space and time.

In [5], Metakides and Nerode introduced the study of recursively enumerable substructures of a recursively presented structure. The main line of study presented in [5] is to examine the effective content of certain algebraic structures. In [6], Metakides and Nerode studied the lattice of r.e. subspaces of a recursively presented vector space. This lattice was later studied by Kalantari, Remmel, Retzlaff and Shore. Similar studies have been done by Metakides and Nerode [7] for algebraically closed fields, by Remmel [10] (...) for Boolean algebras and by Metakides and Remmel [8] and [9] for orderings. Kalantari and Retzlaff [4] introduced and studied the lattice of r.e. subsets of a recursively presented topological space. Kalantari and Retzlaff consideredX, a topological space with ⊿, a countable basis. This basis is coded into integers and with the help of this coding, r.e. subsets ofωgive rise to r.e. subsets ofX. The notion of “recursiveness” of a topological space is the natural next step which gives rise to the question of what should be the “degree” of an r.e. open subset ofX? It turns out that any r.e. open set partitions ⊿; into four sets whose Turing degrees become central in answering the question raised above.In this paper we show that the degrees of the elements of the partition of ⊿ imposed by an r.e. open set can be “controlled independently” in a sense to be made precise in the body of the paper. In [4], Kalantari and Retzlaff showed that givenAany r.e. set andany r.e. open subset ofX, there exists an r.e. open set ℋ which is a subset ofand is dense in and in whichAis coded. This shows that modulo a nowhere dense set, an r.e. open set can become as complicated as desired. After giving the general technical and notational machinery in §1, and giving the particulars of our needs in §2, in §3 we prove that the set ℋ described above could be made to be precisely of degree ofA. We then go on and establish various results on the mentioned partitioning of ⊿. One of the surprising results is that there are r.e. open sets such that every element of partitioning of ⊿ is of a different degree. Since the exact wording of the results uses the technical definitions of these partitioning elements, we do not summarize the results here and ask the reader to examine §3 after browsing through §§1 and 2. (shrink)

In this essay Elizabeth de Freitas follows Tim Ingold's groundbreaking anthropological work on lines and their cultural and material significance to argue that the line is the engine of theory, be it the drawn line of inscription or mathematical measure, the exclusionary line of delineation, or the undulating generative line of flight. De Freitas focuses on contemporary theories of perception that deploy the line — and mobilize the force of theory — so as to encode (...) and reconfigure the student's body. She draws on the work of Gilles Deleuze to develop a topological approach to the line, and argues that such an approach aligns with current philosophical approaches to social interaction that might be termed “topo-philosophy.”. (shrink)

With the rising influence of digitalization and its immense penetration intoeven everyday life, the last decade of the 20th Century addressed to a critical threshold in the successive transformation process of the spatiality in its long-term run. The advanced digital technologies of ubiquitous computing and generative design, as well as the invention of smart materials in late 90’s have all provoked the fluid characteristics of spatiality, and strengthen the transformative capacities of the architectural space through the emergence of computer-augmented territories. (...) Additionally, while they are becoming as the body extensions, the advent of novel apparatuses and gadgets further enhanced the inte- gration of the corporal and incorporal bodies with the spatio-temporal multiplicities, where the hyperdimensionality of the space has been triggered to its outmost range, in relation to the “soft and smart technologically augmented immanent millieu”, in Spuybroek terms. Thus, like Spuybroek points out as the “haptonomist” presence of the body merges itself with these diverse bodily extensions on one hand; and on the other hand, as the rising influence of nomadic view of the world further stimulates the unboundedness and endless fluidity of space, so that the spatiality becomes a landscape of successive transformations, a topology of emergence or a plane of becoming, which is merely defined by lines of forces, and occures as an alive territory rather than a limited space of predefined boarders. Therefore, this evolvable territory which is affectable and being affected by the lines of forces –inner and outer forces–, emerges as an animated existence, an interactive organism. So, by interacting with the Deleuzian Philosophy and their notions like lines of forces, folding, becoming, smooth space, territory, spatium, this article aims to reveal the relevance of these notions in architectural discourse, as well as the emergence of the smooth space in the contemporary architectural practice, by magnifiying one of the very initial examples of its kind; in terms of unfolding the Fresh Water Pavilion of NOX Architecture by Lars Spuybroek into question to reveal the essences of thecontemporary transformable-evolvable architectural spatiality. (shrink)

I provide a mereological analysis of Zeno of Sidon’s objection that in Euclid’s Elements we need to supplement the principle that there are no common segments of straight lines and circumferences. The objection is based on the claim that such a principle is presupposed in the proof that the diameter cuts the circle in half. Against Zeno, Posidonius attempts to prove against Zeno the bisection of the circle without resorting to Zeno’s principle. I show that Posidonius’ proof is flawed (...) as it fails to account for the case in which one of the two circumferences cut by the diameter is a proper part of the other. When such a case is considered, then either the bisection of the circle is false or it presupposes Zeno’s principle, as claimed by Zeno. (shrink)

As a result of the development of new technologies such as satellite communication, digitalization, and multimedia computer networks, new media such as blogs, online magazines, and wireless network media have sparked a lot of interest. This study uses 3D clothing display technologies to improve the customer experience of online clothing marketing, aid in the improvement of online clothing marketing efficacy, and extensively discuss the digital clothing anthropometric model. Furthermore, this study employs the convex hull approach and NURBS fitting technology to (...) address concerns with digital clothing anthropometric measurement and provides a practical solution for measuring straight line length, circumference length, and curve length. Furthermore, this research incorporates 3D intelligent technology to develop an online fitting system as well as an intelligent display system for garments. The simulation experiment investigation revealed that the smart clothing 3D display system proposed in this work has a substantial impact on growing online e-commerce clothing marketing as well as improving clothing marketing effectiveness. (shrink)

Let us agree, to begin with, that we are not shown [in Last Judgment], as Life Magazine long ago phrased it, a Saint Bartholomew who "holds his own mortal skin, in which Michelangelo whimsically painted a distorted portrait of himself.”1 The face was sloughed with the rest of the skin and goes with it. What we see is a Saint Bartholomew with another's integument in his hand. We next consider an aspect of the self-portrait which even La Cava left out (...) of account - its relative siting. This has to matter since the portrait lies in the path of Christ's imminent action. More than that, it lies on a diagonal that traverses the fresco like a heraldic bend chief to base - from left top to right bottom. The twofold competence thus assumed by the self-portrait - in its concrete location and in the range of its influence - is something to marvel at. A hangdog face flops to one side, helpless and limp. But the tilt of its axis projected upward across the field strikes the apex of the left-hand lunette, the uppermost point of the fresco. And if, departing once again from the skin's facial axis, we project its course netherward, we discover the line produced to aim straight at the fresco's lower right corner. Such results do not come by chance. To put it literally, letting metaphor fall where it may: it is the extension of the self's axis that strings the continuum of heaven and hell. · 1. Life Magazine, 6 December 1949, p. 45. So also Redig de Campos speaks of the lifeless Apostle's own skin, "dove il Buonarroti ha nascoto un singolare autoritratto..in caricatura tragica della sua 'infinita miseria'" . Tolnay sees the matter correctly: "It is the artist's empty skin which the saint holds in his hand" Leo Steinberg is Benjamin Franklin Professor and University Professor of the History of Art at the University of Pennsylvania. His publications include Other Criteria: Confrontations with Twentieth-Century Art, Michelangelo's Last Paintings, Borromini's San Carlo alle Quattro Fontane, as well as studies of Leonardo, Pontormo, Velazquez, and Picasso. (shrink)

This book offers a philosophical exploration of lines in art and culture, and traces their history from Antiquity onwards. Lines can be physical phenomena, cognitive responses to observed processes, or both at the same time. Based on this assumption, the book describes the “philosophy of lines” in art, architecture, and science. The book compares Western and Eastern traditions. It examines lines in the works of Paul Klee, Wassily Kandinsky, and Henri Michaux, as well as in Chinese and Japanese art and (...) calligraphy. Lines are not merely a matter of aesthetics but also reflect the psychological states of entire cultures. In the nineteenth century, non-Euclidean geometry sparked the phenomenon of the “self-negating line,” which influenced modern art; it also prepared the ground for virtual reality. Straight lines, distorted lines, blurred lines, hot and cold lines, dynamic lines, lines of force, virtual lines, and on and on, lines narrate the development of human civilization. (shrink)

Despite the various perceptual-motor deficits documented in strabismus, there is a paucity of studies evaluating visual illusions in patients with strabismus. The aim of this study was to examine how the illusionary perception occurs in children/adolescents with strabismus with referral for surgery to correct ocular deviations. A controlled cross-sectional study was carried out in which 45 participants with strabismus and 62 healthy volunteers aged 10–15 years were evaluated. The behavioral response to three geometric illusions [Vertical-Horizontal illusion, Müller-Lyer illusion and Ponzo (...) illusion] and respective neutral stimuli regarding the estimation of image size and response time were measured using the Method of Adjustment. To analyze the influence of secondary factors: type of ocular deviation ; amount of eye deviation; presence of amblyopia and stereopsis, a one-way ANOVA was performed. Among the tested illusions, children with strabismus showed greater susceptibility and response time to Ponzo’s illusory images. Children with strabismus and preserved stereopsis, on the other hand, showed similar susceptibility and response time to control group patients to the Ponzo illusion. Patients with amblyopia showed overcorrection in the estimate of non-illusory Ponzo images. Children with horizontal ocular deviation associated with vertical deviation showed higher susceptibility to vertical adjustment images for the Müller-Lyer illusion. Individuals with strabismus tended to overcorrect the length of the straight-line segment adjusted for non-illusory images when testing non-illusory images in the Müller-Lyer test, as well as for the neutral images in the Vertical-Horizontal test. The findings indicated impairment in the perception of geometric illusions and neutral figures, especially for the Ponzo illusion test by children with strabismus. As the behavioral response to illusory images may indirectly reflect the visual and morphofunctional alterations present in these individuals, we suggest that the investigation of visual illusory perception can be used as a new research strategy in the field of investigating the visual function in strabismus. (shrink)

Noetherian type and Noetherian π-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the cellularity, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian π-type of κ-Suslin Lines, and we are able to determine it for every κ up to the first (...) singular cardinal. We then prove a consequence of Changʼs Conjecture for ℵω regarding the Noetherian type of countably supported box products which generalizes a result of Lajos Soukup. We finish with a connection between PCF theory and the Noetherian type of certain Pixley–Roy hyperspaces. (shrink)

The completeness of the modal logic S4 for all topological spaces as well as for the real line , the n-dimensional Euclidean space and the segment etc. was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvili with a further simplification. The proof strategy is to embed a finite rooted Kripke structure for S4 into a (...) subspace of the Cantor space which in turn encodes . This provides an open and continuous map from onto the topological space corresponding to . The completeness follows as S4 is complete with respect to the class of all finite rooted Kripke structures. (shrink)

The mathematical structure of realist quantum theories has given rise to a debate about how our ordinary 3-dimensional space is related to the 3N-dimensional configuration space on which the wave function is defined. Which of the two spaces is our (more) fundamental physical space? I review the debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based on three criteria. I argue that when we consider which view leads to a deeper understanding of the physical world, especially given the deeper topological (...) explanation from the unordered configurations to the Symmetrization Postulate, we have strong reasons in favor of 3D-Fundamentalism. I conclude that our evidence favors the view that our fundamental physical space in a quantum world is 3-dimensional rather than 3N-dimensional. I outline lines of future research where the evidential balance can be restored or reversed. Finally, I draw lessons from this case study to the debate about theoretical equivalence. (shrink)

ArgumentThis article presents ancient documents on the subject of homeomeric lines. On the basis of such documents, the article reconstructs a definition of the notion as well as a proof of the result, which is left unproved in extant sources, that there are only three homeomeric lines: the straight line, the circumference, and the cylindrical helix. A point of particular historiographic interest is that homeomeric lines were the only class of lines defined directly as the extension of a (...) mathematical property, a move that is unparalleled in Greek mathematics. The far-reaching connections between mathematical homeomery and key issues in the ancient cosmological debate are extensively discussed here. An analysis of its relevance as a foundational theme will be presented in a companion paper in a future issue of Science in Context. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Kant and the Capacity to Judge; Sensibility and Discursivity in the Transcendental Analytic of the Critique of Pure Reason by Beatrice LonguenesseMichelle GreerBeatrice Longuenesse. Kant and the Capacity to Judge; Sensibility and Discursivity in the Transcendental Analytic of the Critique of Pure Reason. Translation by Charles T. Wolfe. Princeton, NJ: Princeton University Press, 1998. Pp. xv + 420. Cloth, $59.50.Kant and the Capacity to Judge is a translation (...) of Longuenesse’s Kant et Pouvoir de juger, although Longuenesse has revised and added to the original version. The central argument seeks to demonstrate the essential role played by the logical forms of judgment in the arguments of Kant’s first Critique. Longuenesse emphasizes straight away Kant’s efforts (in the Metaphysical Deduction) to deduce a set of pure concepts from the logical forms of judgment. Against many standard readings, she aims to reveal the very “tight” connection between this attempt and subsequent arguments in the Transcendental [End Page 372] Analytic. That Kant takes the functions of judgment to be relevant to these arguments is hardly a novel claim. The originality of Longuenesse’s work, however, stems both from the central and pervasive role she assigns to the logical forms of judgment, and from the impressive, painstaking effort she makes to sustain the close connection between these forms and Kant’s subsequent arguments. According to Longuenesse, any proper understanding of the arguments of the Transcendental Analytic absolutely depends upon relating these “down to the minutest details of their proofs,” to the role of the logical forms (5).These minute details do not escape Longuenesse in her subsequent elaboration on this original thesis. Given the length and detail of her work, I can only highlight a few of the central points and strategies. One should note from the outset, however, that the detail and the scholarship evinced to support her claims constitute much of what is of value in the book. Many of these “supporting arguments” and discussions, which include efforts to trace Kantian conceptions back to their historical/philosophical origins, stand in their own right as independent contributions to Kant scholarship and interpretation.The work is divided into three parts. The first part is essentially propaedeutic. Longuenesse begins by drawing a distinction between the potential or capacity to judge and judgement itself (i.e., the “actualization” of this judgmental capacity under [externally grounded] sensory stimulation) (7). Taking this distinction as her cue, Longuenesse seeks to reveal the way in which the line of argumentation in the transcendental analytic presents an extended investigation into the formal capacity (ibid). The centrality of the logical forms of judgment is secured early on by assigning them a substantive role in the very presentation of something in intuition “as” an “object.” Insofar as she assigns this role to the logical forms themselves, her account clearly undercuts the primary function of the categories in this regard. Relevant here is Longuenesse’s construal of “appearances” as the intentional correlates of our representational capacities and activities (cf. 20–21). Given this, the problem now motivating the transcendental deduction is how to make sense of the relation between this “internalized” object of (indeterminate) empirical intuition (the appearance) and the object corresponding to intuition which concepts allow one to think in a determinate way (the phenomenon). This ability to think the object in a determinate fashion appears to be the essential contribution made by judgement.These preliminary considerations ground Longuenesse’s efforts (in Part Two) to isolate and examine the discursive activity of judgment. Longuenesse stresses the reflective (and not simply the determinative) role of judgment and its forms. Indeed, she suggests that concepts (not merely empirical concepts, but also the pure concepts of the understanding themselves) are formed by reflecting (through acts of comparison, reflection, and abstraction) on the “sensible given.” Drawing extensively from the Logic, she interestingly argues that the form of judgment itself carries with it some inherent claim to objectivity. Thus, objectivity is folded into the formal activity of judging by being identified as a “normative propensity” inherent in every judgment by virtue of its form alone (87–88). Longuenesse accordingly eschews efforts to downplay the role of the logical use of the... (shrink)

This research concentrates on the design and analysis of an algorithm referred to as Virtual Network Configuration (VNC) which uses predicted future states of a system for faster network configuration and management. VNC is applied to the configuration of a wireless mobile ATM network. VNC is built on techniques from parallel discrete event simulation merged with constraints from real-time systems and applied to mobile ATM configuration and handoff. Configuration in a mobile network is a dynamic and continuous process. Factors such (...) as load, distance, capacity and topology are all constantly changing in a mobile environment. The VNC algorithm anticipates configuration changes and speeds the reconfiguration process by pre-computing and caching results. VNC propagates local prediction results throughout the VNC enhanced system. The Global Positioning System is an enabling technology for the use of VNC in mobile networks because it provides location information and accurate time for each node. This research has resulted in well defined structures for the encapsulation of physical processes within Logical Processes and a generic library for enhancing a system with VNC. Enhancing an existing system with VNC is straight forward assuming the existing physical processes do not have side effects. The benefit of prediction is gained at the cost of additional traffic and processing. This research includes an analysis of VNC and suggestions for optimization of the VNC algorithm and its parameters. (shrink)

Charles Darwin argued that human beings are what happen whenphysical laws act upon a planet with the characteristics that earthhad five billion years ago. Similarly, I have argued that theprimacy of individual will is what eventually happens when asociety allocates and limits coercion based upon rights. From timeto time particular visions of the good or the right dominate publicbehavior, but they are eventually enframed by rights — the authoritative claim of each person to respect.I have argued that the propositional structure (...) of American law—the laws themselves — can be seen to be a logically consistentsystem of propositions stemming from the axiom that the will ofeach person is worthy of respect. This is an explanatory, not anormative, proposition. The axiom was not put there by anyoneand the law derived from it, any more than the human brian wasput there and the theory of relativity derived from it. The axiomcame to be embodied in k because of a fact — the single universalcharacteristic of human beings that is relevant to the question ofarranging coercion is individual will — and a process — the right ofeach person to demand a justification for coercion used upon him.Since will is universal to human beings, this would suggest thatany rights-based legal system would evince a general structuresimilar to our own. Particularities of national culture, naturalresources, population density, and so on would produce a verydifferent liberty frontier from the one facing this country andhence, different laws. But the general structure of law — the relationship between principle and policy decision, the role of thebasic rights, and so on — should be similar. This similarity shouldprovide a common basis for cooperation between states, transcending particularities of economic structure, political structure and ideology. We have seen that a very broad range ofeconomic and political institutions may be justified. The essential difference between states lies not in the different ways that theyarrange institutions but in the different ways that they justifythem. Those that justify them to people as persons are similar.Those that justify them by conformity to a design are different.The theory set out here is not a design. It is an explanation. Onevirtue of explanations is that they draw forth other explanations.More importantly, they offer perspective — they tell us what weare “up to.” As the social relations which law must rationalizebecome ever more complex, perspective becomes ever more necessary. The simple laws have already been written. The connectionbetween the doctrine of consideration and the first principle isobvious. The connection between the “hard look” doctrine ofreviewing administrative agencies and the second principle isnowhere near so obvious (though it is a lovely example of thejudicial process enframing the realm of uncertainty). The morecomplex and artificial the institution, the poorer the guidance ofintuition and the more necessary are conscious guides to decision.Justification comes easy to printers. Most of them don't knowwhy a page of print that has straight margines left and right is“justified.” They don't need to know, for the idea has immediateintuitive appeal; it is easy to accept and to remember, and, onceremembered, it is an effective guide to behavior. It is easy to seethat this line of print is not justified and to do somethingabout it. It is not so easy to tell whether the “hard look” doctrine orthe enforcement of a surrogate motherhood contract sits fairly on itspage. Justification of law requires an understanding of thecriterion against which it is being done. There is an intuitive core— a “sense” — to any act of judgment, but that core can be illuminated and developed by an understanding of the framework withinwhich it operates. (shrink)

This paper deals with the very different attitudes that Descartes and Pascal had to the cycloid—the curve traced by the motion of a point on the periphery of a circle as the circle rolls across a right line. Descartes insisted that such a curve was merely mechanical and not truly geometric, and so was of no real mathematical interest. He nevertheless responded to enquiries from Mersenne, who posed the problems of determining its area and constructing its tangent. Pascal, in (...) contrast, saw the cycloid as a paradigm of geometric intelligibility, and he made it the focus of a series of challenge problems he posed to the mathematical world in 1658. After dealing with some of the history of the cycloid , I trace this difference in attitude to an underlying difference in the mathematical epistemologies of Descartes and Pascal. For Descartes, the truly geometric is that which can be expressed in terms of finite ratios between right lines, which in turn are expressible as closed polynomial equations. As Descartes pointed out, this means that ratios between straight and curved lines are not geometrically admissible, and curves that require them must be banished from geometry. Pascal, in contrast, thought that the scope of geometry included curves such as the cycloid, which are to be studied by employing infinitesimal methods and ratios between curved and straight lines. (shrink)

There are many network topology designs that have emerged to fulfill the growing need for networks to provide a robust platform for a wide range of applications like running businesses and managing emergencies. Amongst the most famous network topology designs are star network, mesh network, hexagonal network, honeycomb network, etc. In a star network, a central computer is linked with various terminals and other computers over point-to-point lines. The other computers and terminals are directly connected to the central (...) computer but not to one another. As such, any failure in the central computer will result in a failure of the entire network and computers in star network will not be able to communicate. The star topology design can be represented by a graph where vertices represent the computer nodes and edges represent the links between the computer nodes. In this paper, we study the vertex-edge-based topological descriptor for a newly designed hexagon star network. (shrink)