the view of spinoza as a scion of the mathematico-mechanistic tradition of Galileo and Descartes, albeit perhaps an idiosyncratic one, has been held by many commentators and might be considered standard.1 Although the standard view has a prima facie solid basis in Spinoza's conception of the physical world as extended, law-bound, and deterministic, it has come under sustained criticism of late. Arguing that, for Spinoza, numbers and figures are mere beings of reason and mathematical conceptions of nature belong (...) to the imagination rather than the intellect, recent commentators have cast doubt on the applicability of mathematics to natural science in Spinoza and challenged the association of him with the Galilean and... (shrink)

I argue that life’s meaning is not only a distinct, gradational final value of individual lives, but also an “extradimensional” final value: the realization of meaning in life brings final value along an additional evaluative dimension, much as the realization of depth in solids or width in plane geometric figures brings magnitude along an additional spatial dimension. I go on to consider the extent to which Metz’s (2013) fundamentality theory respects the principle that life’s meaning is an extradimensional (...) final value, and consequently suggest that the theory may stand in need of further refinement and supplementation. (shrink)

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a (...) “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

While some studies suggest cultural differences in visual processing, others do not, possibly because the complexity of their tasks draws upon high-level factors that could obscure such effects. To control for this, we examined cultural differences in visual search for geometric figures, a relatively simple task for which the underlying mechanisms are reasonably well known. We replicated earlier results showing that North Americans had a reliable search asymmetry for line length: Search for long among short lines was faster (...) than vice versa. In contrast, Japanese participants showed no asymmetry. This difference did not appear to be affected by stimulus density. Other kinds of stimuli resulted in other patterns of asymmetry differences, suggesting that these are not due to factors such as analytic/holistic processing but are based instead on the target-detection process. In particular, our results indicate that at least some cultural differences reflect different ways of processing early-level features, possibly in response to environmental factors. (shrink)

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

Leibnizian-Newtonian calculus was a theory that dealt with geometrical objects; the figure continued to play one of the fundamental roles it had played in Greek geometry: it susbstituted a part of reasoning. During the eighteenth century a process of de-geometrization of calculus took place, which consisted in the rejection of the use of diagrams and in considering calculus as an 'intellectual' system where deduction was merely linguistic and mediated. This was achieved by interpreting variables as universal quantities and introducing the (...) notion of function, which replaced the study of curves. However, the emancipation of calculus from its basis in geometry was not comprehensive. In fact, the geometrical properties of curves were attributed de facto to functions and thus eighteenth-century calculus continued implicitly to use principles borrowed from geometry. There was therefore no transition to a purely syntactical theory based on axiomatically introduced terms, a shift which only took place subsequently in modern times. (shrink)

Dans le cadre de sa théorie de la production des figures géométriques par projection des raisons innées, Proclus est le premier à assimiler la phantasia (imagination) au nous pathetikos (intellect passif) évoqué furtivement par Aristote en De Anima III,5. Tout en maintenant cette assimilation, Ammonius (ré)intègrera la notion d’ epinoia dans le processus d’abstraction, statut de la chose abstraite du monde sensible. L’introduction de cette notion provoquera une certaine confusion chez les commentateurs ultérieurs qui, tout en gardant l’assimilation de (...) Proclus, n’hésiteront pas à faire de la phantasia la faculté intellective permettant d’abstraire les formes de la matière en étroite collaboration avec l’intellect actif, de séparer les accidents d’un substrat et de composer des êtres et formes sans substrat. (shrink)

Fundamental to spatial knowledge in all species are the representations underlying object recognition, object search, and navigation through space. But what sets humans apart from other species is our ability to express spatial experience through language. This target article explores the language ofobjectsandplaces, asking what geometric properties are preserved in the representations underlying object nouns and spatial prepositions in English. Evidence from these two aspects of language suggests there are significant differences in the geometric richness with which objects (...) and places are encoded. When an object is named (i.e., with count nouns), detailed geometric properties – principally the object's shape (axes, solid and hollow volumes, surfaces, and parts) – are represented. In contrast, when an object plays the role of either “figure” (located object) or “ground” (reference object) in a locational expression, only very coarse geometric object properties are represented, primarily the main axes. In addition, the spatial functions encoded by spatial prepositions tend to be nonmetric and relatively coarse, for example, “containment,” “contact,” “relative distance,” and “relative direction.” These properties are representative of other languages as well. The striking differences in the way language encodes objects versus places lead us to suggest two explanations: First, there is a tendency for languages to level out geometric detail from both object and place representations. Second, a nonlinguistic disparity between the representations of “what” and “where” underlies how language represents objects and places. The language of objects and places converges with and enriches our understanding of corresponding spatial representations. (shrink)

Logic in Action/Doron Avital Nothing is more difficult, and therefore more precious, than to be able to decide (Napoleon Bonaparte) Introduction -/- This book was born on the battlefield and in nights of secretive special operations all around the Middle East, as well as in the corridors and lecture halls of Western Academia best schools. As a young boy, I was always mesmerized by stories of great men and women of action at fateful cross-roads of decision-making. Then, like as today, (...) I felt the full weight of the moment of truth as it confronts the individual, the man or woman of action, with the imminent necessity to decide. Alongside, as I was climbing up the ladder of education, I found myself just as well captivated by the great intellectual stories of our times, in particular 20th century analytic philosophy and, at its core, the machinery of modern logic. How do then the cool contemplative modes of the philosophical and analytic outlooks, that are better fitting of the philosopher's armchair at the fireplace, hang together with the heat of the battlefield or the angst of the ticking clock of time left to the completion of a daring special operation? Growing up I was pondering exactly that. Not accidently, and in fact as consequence, I have chosen a course in life that challenged me to visit the poles of both theory and practice. I set myself on a mission to unravel the tension between the contemplative inquiry, supposedly theoretical, aiming at the logical structure of things, and the uncompromising test of life and action. The puzzle of the gap between theory and practice has become ever since the guiding question of my life. -/- Not long ago, I met two friends, Avner Shor, a veteran combat soldier of the special forces unit I eventually commanded, and Aviram Halevi, a fellow officer with whom I served. Why will we not put into writing the ideas that guided us during our years of military action, we wondered. At that time, I used to deliver the keynote lectures to the senior officers' course of Israel central intelligence organization, the Mossad. For this I already had to put some of these ideas in writing. Avner was quick to introduce me with a notable publisher and even helped me write the first chapter. With Aviram I sat to many meetings, where he would challenge me with key questions, and we followed them with memories of operations and battles, aiming to draw the required lessons. This served me well when I finally sat down to write the first draft of the book. The result was more extensive than me and my friends had expected. I had to add a particularly long chapter dealing with the story of modern philosophy and logic. This was required, I felt, as no decision-making process lives in an intellectual vacuum. Knowingly or subconsciously, the intellectual climate of the time guides us for better or worse with the decisions we finally take. The initial draft was naturally cumbersome and inaccessible. But here I had the privilege of meeting a particularly talented editor, Rami Rothholtz. I found in Rami both an editor and a critical reader. As an infantry combat soldier at youth and now a senior journalist and editor, Rami insisted on simplifying and refining my ideas. Had it not been for his work, this book would have been literally unreadable. *** I opened my book with a brief description of a single night of a special operation deep at the Lebanon Bekaa valley, far away from Israel northern border. The mission was to kidnap a senior terror leader known to be holding in captivity an Israeli pilot. This was also the last special operation I commanded in my military career. A single special operation night, but what had to precede it? For sure, many months of intense and skillful preparations. What is Planning? This is the question with which I open up chapter (2) of the book. I confront in this chapter two approaches to planning. The first is the one I saw as a mission to myself to cultivate in my many years of military service. The second is the one I saw as representing a deep-rooted fallacy that holds a firm grip on today's conventional wisdom about planning. To further examine the logic of the competing models, I was required in chapter (3) to analyze the concept of "The Standard". I argue that the false model of planning has its roots in an attitude of an uncompromising adherence to the power of literal obedience to standards and rules. The standard, or what is outlined by a rule as prescribing a future course of action, is seen as an answer to the fundamental tension between theory (planning) and practice (execution). I argue in contrast that the standard does not produce an answer but rather what I call: "A Well-Defined Question". In essence, it is a meticulously-well-planned invitation to exercise a fresh judgment. A judgement that has as an anchor the default answer that the standard produces but is open to extension as it confronts the new frame of reference of the test of reality. In chapter (4), I present a new tool for planning and execution: "The Polygon of Risks" or "The Polygon of Execution". The Polygon is a geometric illustration of the underlying tension that exists between all dimensions of the operational project. The construction and conceptualization of the trade-offs that exist between the various dimensions - segments of the polygon - is the first topic on the planner's agenda. I position the concepts of "Daring" and "Taking Proactive and Well-Informed Risks" as necessary logical constituents of the polygon and show how in contrast the false model of planning is responsible for an operational culture that is risk-averse. It encourages, in fact, the piling up of false securities at the segments' level that eventually lead to the collapse of the polygon. The necessary trade-offs relationships that exist between the segments of the polygon express the fundamental tension between partnership and competition: partnership in a joint goal and competition over resources, both material and abstract. When this tension is not managed well, that is, when the players' local optimizations overshadow the global optimization required for the success of the project, the execution polygon collapses. In this scenario, we may find the setting of the bar of "High Standards" in a players' spheres of responsibility to conceal a hidden agenda of personal insurances taken against a possible failure. Once the overall risk is not mitigated by way of collaborative joint executional dialogue, the process that leads to a collapse of the polygon is literally unstoppable. *** In chapter (5), I discuss "(Fateful) Decisions". I examine the way in which we should translate the polygon into sequential critical decisions on the timeline. The execution polygon is not a rigid scaffolding but a dynamic conceptual illustration of the -/- project. It will need now to be converted into successive decision junctions. We will find the content of a decision and its timing on the timeline to be inseparable. The right decision is the one that is taken at the right time. The moment of decision is in fact the last moment of deliberation, the first moment available for action, i.e. the first possible "Can" is also the latest possible "Must". Deciding too soon is analog to the "Jumping of the Gun". The decision maker is forcing a pre-conceived answer to a state of affairs not ripe for an answer, e.g. the target in the shooting range is not yet erected. Late to decide and the decision maker may envision what he or she should have done but reality is no longer available for them to harness it right, e.g. consider a tennis player's frustration when reviewing a too-late-to-react move in the video replay. I review in this chapter many fateful decisions, both historical, from Napoleon's Waterloo, to a Bridge Too Far of WWII, to the failed rescue attempt of American hostages at the time of the Iranian Revolution, as well as fateful decisions I had to take in the course of my military life, both in combat and in special operations, and show the sense in which they all answer to our analysis. The theme of chapter (6) is "The Moment of Truth". This is the moment when everything is on the line and from which there is no turning back. I discuss the formative repetitive structure of the schooling and simulation phase and contrast it with the "One Shot, One Opportunity" nature of the moment of truth. This is extremely helpful when we analyze the logical structure of failures as unwarranted repetitions of past lessons. The schooling curriculum indeed consists of important abstractions drawn from lessons of the past. However, it is when we project them in a fashion that is literal or mechanical that we cross path with the possibility of a failure. The paradox of education is that what we must repeat in school and simulation must be extended and not repeated in real life. What is required of us so we do not fall into the fallacy of repetition in face of the moment of truth is the core subject of the discussion of this chapter. In chapter (7), I took a lengthy philosophical detour surveying the intellectual drama of modern logic, and in general, the story of 20th century analytic philosophy. Readers who choose to bypass the somehow demanding narrative of this drama can do so and still keep the argumentative thread of the book. However, even by way of skipping the more technical parts, I would still urge the readers to delve into the discussion in this chapter as I believe this would grant them a deeper understanding of the overall conceptual framework. I explore here the work of two key figures: the logician Kurt Gödel and the philosopher Ludwig Wittgenstein, both were crowned in 2000 by TIME Magazine, as the leading Logician and Philosopher, respectively, of the 20th Century. I review the fascinating logical result of Gödel's Incompleteness Theorem and examine its relevance as setting a theoretical limit to the power of formalization. This amazing logical result - a result that shakes the logical foundations of 20th century thought in as much as the discovery of irrational numbers shook the foundations of the Pythagorean program of the ancient Greeks - bears significantly on the question of action and the blind trust we place in the power of formalization in the form of protocols and rules. It is here that Wittgenstein's philosophy enters in full force. We follow Wittgenstein as he dismantles our conception of rules as if there were "Railroads to Infinite", i.e. as prescribing for us in-advance their proper employments in all possible -/- future eventualities. Instead we propose that the dilemma of action demands of us an extension of rules rather than a mechanical or blind repetition of their literal imperative. In turn, this extension can serve as a new lesson added to the curriculum of school and simulation. The theme of chapter (8), the concluding chapter, is Strategy, Logic, and Ethics. The chapter comes to place the ethical dilemma in the conceptual map we presented in the book. I discuss here the tension between strategy as a process of setting goals and deriving backward as it were the steps that are necessary for achieving these goals, to the inescapable dictation of the ethical imperative. This unavoidable tension is particularly important against the backdrop of contemporary culture that is guided by a vision of personal success and achievement that may threaten our ability to do what is right. Most of all, I am troubled in this chapter, with the role logic may play, first as a liberating force enabling a change that is tuned to the ethical imperative, and second, as a force designed to justify on logical grounds, as it were, the existing power structure. It is in this junction that brave individuals will be called to challenge the governing logic of the zeitgeist - logic that operates now as an oppressive force that serves the vested interest of beneficials of the existing order. When success is favored over doing what is right, as we must admit is the prevailing cultural mood of our times, that the words of Emanuel Kant carry their full weight: "Do the Right thing and leave the Consequences to God". For this we need heroes and heroines that challenge the time by bearing the full weight of responsibility that comes with the doing of what is Right. *** This book was written more than anything from the perspective of the heroes and heroines of action. Sometimes they may be missing the exact words or the full insight into the solid logical complexity that stands between them and the doing of what is right. They feel what is right to do on the battlefield of life, whether in real battle, in political or economic life or whether in the life of creativity and the quest of the intellect. They do need, in analogy, proper logical and intellectual artillery. I hope they will find the logical cannons they require as they travel through the pages of this book. The rest is in their hands at the Moment of Truth. -/- Doron Avital, Tel Aviv 2011 -/- . (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)

The aim of this paper is to compare two approaches to semantics, namely the standard Tarskian theory and Wittgenstein’s picture theory of meaning. I will compare them with respect to an unusual subject matter, namely to geometrical pictures. The choice of geometry rather than arithmetic or set theory as the basis, on which this comparison will be made has two reasons. One reason is related to Wittgenstein’s picture theory of meaning. This theory was developed more or less as a metaphor, (...) comparing the language to a picture. Nevertheless, if we take pictures themselves in the role of the language to which we apply Wittgenstein’s picture theory of meaning, this theory stops being a metaphor and starts to work in a technical way. I believe that in this way we can create a new approach to semantics in geometry. The other reason for taking geometry as the basis for our investigation is that the Tarskian approach to semantics does not work in geometry as we would wish. (shrink)

ArgumentIn his Philosophia practica universalis, Christian Wolff proposes a “mathematical” theory of moral action that includes his statements on the Aesopian fable. As a sort of moral example, Wolff claims, the fable is an appropriate means to influence human conduct because it conveys general truths to intuition. This didactic concept is modeled on the geometrical figure: Just as students intuit mathematical demonstrations by looking at figures on a blackboard, one can learn how to execute complex actions by listening to (...) a fable. Wolff's “scientific” fable theory met with an ambivalent reception. Lessing, who in his fable treatises re-translates Wolff's suggestions into the conceptual framework of poetics, interprets the geometric model as a stylistic ideal. The famous passage on Homer's successive descriptions in Lessing's Laokoon can be seen as another attempt to apply the representational model of geometry to literature. Herder, reading Wolff in a way that might be called deconstructive, replaces Wolff's geometric theory of poetry by a poetic anthropology of geometry. (shrink)

The fragmentary nature ofGSmakes it difficult to read as it stands, and for this reason, I have rearranged the material slightly so that it falls into four primary, reasonably coherent, parts. Their titles are: ‘The nature of mathematical objects’, ‘Thirteen propositions of Euclid 1’, ‘The philosophy of parallel lines’ and ‘On the algebra of geometrical figures’.GSactually starts with ‘Thirteen propositions of Euclid 1’. The justification for the reversal of order in the translation is to have Hegel's philosophical basis for (...) geometry available first, to provide a context for his detailed discussion of geometrical propositions. The reader can easily reconstruct the original order ofGSas follows. Interchange the first and second parts and also the third and fourth parts. Next, move the passages marked a …. a', b …. b',c….c'to the places in the text markeda-a',b-b'andc-c'. For the reader wishing to locate the original German passages inDok., the page and paragraph for each of these passages are given at the start of that passage.In addition, subtitles for sections in these parts have been added which do not appear in the original. They are included in the translation to help orient the reader. I have inserted in some places words not in the original in order to clarify the meaning of the text. These insertions as well as my comments on the text are also enclosed within square brackets. Further clarifications can be found in the appended endnotes. (shrink)

There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential compass-and-straightedge drawings qua quantitative and (...) continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous. (shrink)

If the dominant ethico-philosophical thinking of responsibility in the West is founded upon, or tied to a certain figure of death, it is because this ethical notion of responsibility is also a certain econo-onto-thanatology. Here the notion of the gift to the other is always already inscribed within a certain economic equivalence of value, or an economic determination of temporality as the geometric figure of the circle, or a certain economy of the experiences of abandonment and mourning, through which (...) the event-character of the gift, its excess and its infinite surplus is economised, reduced, repressed, or even annulled. Reading Jacques Derrida's deconstruction of this econo-onto-thanatology, and relating him to Schelling, Heidegger, Levinas and Kierkegaard, this article attempts to reveal this very complex relationship of the ethical notion of responsibility and the gift with death, in order to think anew – in the spirit of Derrida – a responsibility in relation to mourning and abandonment, and in relation to a death that does not figure in any figuration of self-figuration and self-presence, but – to speak with Maurice Blanchot – as interminable, incessant worklessness, as endless ruination and abandonment of itself. This impossible aporia of the notion of responsibility is itself a dis-figuring of death, which is also an aporia of an instant which escapes, in its event character, the geometric figure of time as circle. (shrink)

This paper begins with an assessment of the differences between two states of Jacopo Caraglio's engraved Diogenes after Parmigianino, and between each of those states and Parmigianino's preparatory drawing of the composition. What follows is an attempt to trace both the textual sources and the creative development of this unusual iconographie subject, culminating in a hypothesis about the chronological sequence of the earliest prints of Parmigianino's Diogenes. It is argued that, originally, the artist devised the composition in collaboration with a (...) learned advisor. His own interest in geometry then led him, incongruously, to introduce a geometric solid—an error corrected in Ugo da Carpi's chiaroscuro woodcut version. (shrink)

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

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1–32, 1996, and (...) Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, 2002). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, 2009). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the “conjugate rule” or a “completion of the square,” but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore 2007b, Ch. 11 and App. 1). (shrink)

This book concerns Spinoza's theory of knowledge and closely related issues: Spinoza's conceptions of geometrical figure or shape, number, and observational sci.

This book interrogates the ontology of mathematical entities in Spinoza as a basis for addressing a wide range of interpretive issues in Spinoza’s epistemology—from his antiskepticism and philosophy of science to the nature and scope of reason and intuitive knowledge and the intellectual love of God. Going against recent trends in Spinoza scholarship, and drawing on various sources, including Spinoza’s engagements with optical theory and physics, Matthew Homan argues for a realist interpretation of geometrical figures in Spinoza; illustrates their (...) role in a Spinozan hypothetico-deductive scientific method; and develops Spinoza’s mathematical examples to better illuminate the three kinds of knowledge. The result is a portrait of Spinoza’s epistemology as sanguine and distinctive yet at home in the new Cartesian and Galilean scientific-philosophical paradigm. (shrink)

The paper addresses some rhetorical uses of the figure of water management from the perspective of an affirmative approach to contamination which Derrida saw as constitutive of affirmation itself. Contaminated water and its discontents discussed in the text frequently appears in various kinds of writings as a frightening figure of contamination which simultaneously brings in the figure of water management as a way of controlling the purity of cultural exchanges and transmissions in which, as Caroline Petronius puts it, contagion journeys (...) out of medicine into culture. The paper also addresses water and its management from the perspective of Astrida Neimanis’s liquidizing of the border between the solid and the fluid as a border between the human and the inhuman. This perspective opens up the sphere of mutual contamination of the human and the inhuman and translates posthumanist theoretical positions into spheres of affirmative exchanges managed not by masters, but by Donna Haraway’s companion species whom, or which, we all are. (shrink)

This paper has two specific goals. The first is to demonstrate that the theorem in MetaphysicsΘ 9, 1051a24-27 is not equiva-lent to Euclid’s Proposition 32 of book I (which contradicts some Aristotelian commentators, such as W. D. Ross, J. L. Heiberg, and T. L. Heith). Agreeing with Henry Mendell’s analysis, I ar-gue that the two theorems are not equivalent, but I offer different reasons for such divergence: I propose a pedagogical-philosoph-ical reason for the Aristotelian theorem being shorter than the Euclidean (...) one (and the previous Aristotelian versions). Aristotle wants to emphasize the deductive procedure as a satisfactory method to discover scientific knowledge. The second objective, opposing some consensus about geometrical deductions/theo-rems in Aristotle, is to briefly propose that the theorem, exactly as we found it in Metaphysics and without any emendation to the text (therefore opposing Henry Mendell’s suggested amend-ments), allows the ancient philosopher to demonstrate that universal mathematical knowledge is in potence in geometrical figures. This tentatively proves that Aristotle emphasizes that geometrical deduction is sufficient to actualize mathematical knowledge. (shrink)

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 our discussions around the theme of solid fluids, we often resort to everyday words, many of them of ancient derivation and rich in association. We have decided to make a list of some of the words that come up most often – barring those that already figure as the principal characters of individual contributions – and to distribute among ourselves the task of writing a sort of mini-biography for each. The resulting lexicon with 19 entries, ranging from ‘cloud’ (...) and ‘concrete’ to ‘wave’ and ‘wood’, serves as a conclusion to the collection as a whole. (shrink)

Spherical geometry studies the sphere not simply as a solid object in itself, but chiefly as the spatial context of the elements which interact on it in a complex three-dimensional arrangement. This compels to establish graphical conventions appropriate for rendering on the same plane—the plane of the diagram itself—the spatial arrangement of the objects under consideration. We will investigate such “graphical choices” made in the Theodosius’ Spherics from antiquity to the Renaissance. Rather than undertaking a minute analysis of every (...) particular element or single variant, we will try to uncover the more general message each author attempted to convey through his particular graphical choices. From this analysis, it emerges that the different kinds of representation are not the result of merely formal requirements but mirror substantial geometrical requirements expressing different ways of interpreting the sphere and testify to different ways of reasoning about the elements that interact on it. (shrink)

This paper describes Alfred Clebsch’s 1871 article that gave a geometrical interpretation of elements of the theory of the general algebraic equation of degree 5. Clebsch’s approach is used here to illuminate the relations between geometry, intuition, figures, and visualization at the time. In this paper, we try to delineate clearly what he perceived as geometric in his approach, and to show that Clebsch’s use of geometrical objects and techniques is not intended to aid visualization matters, but rather (...) is a way of directing algebraic calculations. We also discuss the possible reasons why the article of Clebsch has been eventually completely forgotten by the historiography. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:61. HUME'S GEOMETRIC "OBJECTS" Arithmetic and algebra allow of precision and certainty. The science of geometry is not likewise a perfect and infallible science. At any rate, this is Hume's teaching in the Treatise. When two numbers are so combin ' d, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and 'tis for want of such a (...) standard of equality in extension, that geometry can scarce be esteem'd a perfect and infallible science. 1 The ideas of, say, equality and right line shade into· inequality and curve. It must not be supposed, however, that the vagueness of these ideas renders all judgments uncertain and fallible. 'Tie evident, that the eye, or rather the mind is often able at one view to determine the proportions of bodies....Such judgments are not only common, but in many cases certain and infallible. (T47) Nevertheless, geometry is a fallible science because its essential ideas are vague. Vagueness, due to lack of boundaries between ideas, is in the ideas themselves. For this reason, 'tis impossible to produce any definition of them, which will fix the precise boundaries betwixt them. (T49) (Yet, some forty pages earlier Hume argued at length that the mind cannot form any notion of quantity or quality without forming a precise notion of the degrees of each [T18]) As far as the Treatise goes, geometry is the science of measurement and as such it must allow for error. Even if precise geometric standards could be had, they would not make the propositions of geometry certain. Thus, we might precisely define a right line by a rule of order of indivisible points. But geometry, construed as the science of measurement, would not be perfected. Nor, if it were (the standard), is there any such firmness in our senses or imagination, as to determine when such an order is violated or preserv ' d. The original standard of a right line is in reality nothing but a certain general appearance. (T52) 62. In the Enquiries, Hume continues to hold that notions essential to geometry, for example, equality, right line, and so on, are sensibles. But they are no longer generic appearances. They are determinate. The great advantage of the mathematical sciences above the moral consists in this, that the ideas of the former, being sensible, are always clear and determinate... even when no definition is employed, the object itself may be presented to the senses, and by that means be steadily and clearly apprehended. 2 In addition, geometry is no longer merely the science of measurement. The truths of Euclid, Hume holds, would remain truths even if there were no figures in nature. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. (EHU25) Professor Zabeeh, in his commentary on Hume, holds that Hume did not mean to imply that Euclid could dispense with the observation of figures. However, whenever possible Hume should be taken at his word: observation of figures is not a truth condition for propositions of geometry. A distinction must be drawn between discovery conditions and what Hume calls the "truth or evidence" of a geometric proposition. We may suppose that observation of figures is a discovery condition. But it is hardly a truth condition. The certainty, as opposed to the discovery, of geometric propositions does not depend on the existence in nature of any figure. I am mainly concerned, however, with Professor Zabeeh' s account of how, according to Hume, geometric propositions retain their certainty and evidence in the face of recalcitrant experience. Noting that Hume does not "fully explain," he gives the real reason for Hume's belief in the apodeictic certainty of mathematical truths. He appears to assume we never allow such truths to be controverted by empirical evidence. That is to say, if perchance we find, by measurement, that the sum of the angles of a Euclidean triangle 63. does not equal 180 degrees, either we say that we measured wrongly or we say that the triangle we have been measuring is not Euclidean.4 This formulation of Hume's reason avoids... (shrink)

Owing to intensive development of the theory of self-organization of complex systems called also synergetics, profound changes in our notions of time occur. Whereas at the beginning of the 20th century, natural sciences, by picking up the general spirit of Einstein's theory of relativity, consider a geometrization as an ideal, i.e. try to represent time and force interactions through space and the changes of its properties, nowadays, at the beginning of the 21st century, time turns to be in the focus (...) of attention. It turns to be possible to represent space through time, because synergetics shows that historical and evolutionary stages of development of a complex structure can be found now, in its present spatial configuration. A whole series of paradoxical notions, such as "the influence of the future upon the present", a "possibility of touching of a rather remote future today", "availability of the past and the future now, in praesenti", "irreversibility and elements of reversibility in the course of evolutionary processes in time", "discrete unites, quanta of time", appear in synergetics. (shrink)

Solid-state processing of metal material is a very complex physical and chemical process, which is coupled by a series of variations including heat transfer, momentum transfer, mass transfer, and phase change. Applying three-dimensional finite element numerical method to the simulation of solid-state processing can perform analysis of metal material’s forging processes before production trial production, can obtain their relevant information such as material flow law, temperature field, and strain field under the minimum physical test conditions, thereby predicting metal (...) material’s forming defects and improving their forging quality. On the basis of summarizing and analyzing previous research works, this paper expounded the current status and significance of solid-state processing of metal materials, elaborated the development background, current status, and future challenges of 3D finite element numerical simulation, introduced the discrimination method and free surface solution method of numerical simulation calculation, conducted finite element model’s geometric assumptions, material selection, element division, model establishment, parameter selection, and initial and boundary condition determination, and simulated and analyzed rheological casting, remelting heating, thixoforming, and rotary piercing processes of metal materials. The results show that the 3D finite element numerical method can not only simulate various processes of flow field, temperature field, stress field, and microstructure in solid-state processing but also can provide a reliable basis for effectively obtaining a reasonable description and finding a more optimized design plan for metal material processing in a short time, which plays an important role in understanding and analyzing solid metal forming process, controlling and optimizing process parameters, guiding and mastering rheological casting, and secondary heating and rotary piercing of metal materials. (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)

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

Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely (...) homotopy theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models. (shrink)

In this paper, we study dialogue as a game, but not only in the sense in which there would exist winning strategies and a priori rules. Dialogue is not governed by game rules like for chess or other games, since even if we start from a priori rules, it is always possible to play with them, provided that some invariant properties are preserved. An important discovery of Ludics is that such properties may be expressed in geometrical terms. The main feature (...) of a dialogue is “convergence”. Intuitively, a dialogue “diverges” when it stops prematurely by some disruption, or a violation of the tacit agreed upon conditions of the discourse. It converges when the two speakers go together towards a situation where they agree at least on some points. As we shall see, convergence may be thought of through the geometrical concept of orthogonality . Utterances in a dialogue have as their content, not only the processes (similar to proofs) which lead to them from a monologic view, but also their interactions with other utterances. Finally, any utterance must be seen as co-constructed in an interaction between two processes. That is to say that it not only contains one speaker’s intentions but also his or her expectations from the other interlocutor. From our viewpoint, discursive strategies like narration , elaboration , topicalization may derive from such interactions, as well as speech acts like assertion, question and denegation. (shrink)

Topology has been characterized as an unsuitable mathematical framework for the investigation of geometric-perceptual phenomena. This has been attributed to the highly abstract nature of topology leading to failures in tasks such as making distinctions between geometrical figures (e.g., a cube versus a sphere) in which the human perceptual system succeeds easily. An alternative thesis is proposed on both philosophical and empirical grounds. The present analysis applies the Müller-Lyer (ML) illusion as a method of investigation to examine the (...) suitability of topology in raising questions about geometric-perceptual phenomena and formulating potential answers to these questions. Not only will it be concluded that topology is a suitable mathematical framework in which one can formulate relevant questions and potential answers with respect to perceptual phenomena such as the ML illusion, it will also be discussed that topology will not be replaced by other means of spatial representation consistent with the findings that indicate its contributions to our daily spatial behavior along with other means such as Euclidean geometry. In addition, the analysis will be discussed with respect to two relevant themes: 1) the implications of the proposed view in relation to Husserl’s phenomenology and 2) the effect of developmental stages and education that may change the involvement of topology in such perceptual responses over time. (shrink)

RésuméEn offrant une méthode de plus en plus féconde, les transformations géométriques conduisent les mathématiciens, entre les ixe et xie siècles, à modifier leurs modes d’appréhension des figures géométriques. Le présent article voudrait mettre en évidence la contribution d’al-Sijzī à cette mutation en se fixant deux tâches: en premier lieu, comprendre précisément ce qu’al-Sijzī entend par transformation ; et en second lieu, rendre compte de ses recherches sur les invariants géométriques, obtenus en faisant varier certains éléments d’une figure. L’usage (...) des transformations et la recherche d’invariants apparaissent comme les deux volets d’une même tendance, celle de rompre avec une manière euclidienne de considérer les figures de façon isolée et statique pour mieux exploiter les propriétés communes qui peuvent les unir. L’article s’achève par l’édition et la traduction d’un petit traité consacré à la mise en évidence d’invariants. (shrink)

The purpose of the paper is to examine the potential of participatory budgeting for the formation of citizen identity and attachment to the place in terms of individual, territorial and thematic focus. In the theoretical discussion, the authors analyse the concepts of place attachment, social identity and their influence on civic participation. The authors propose a conceptual framework for the analysis of relationships between PB, place attachment, and social identity. In the case of the community development model of PB, place (...) attachment should lead to the citizens’ increased inclination to participate. In the case of participatory democracy model of PB citizen participation can lead to a stronger place attachment. The con­ceptual framework presented in the paper requires empirical confirmation. Further research on the subject should revolve around the influence of place attachment on the formation of social identity and vice versa by application of the discussed models of PB. The placemakers should take into account literature-based evidence that advisory models of PB do not reinforce place identity. The places that apply one of the transition models should consider the evolution of their PB policies towards either community development model or participatory democracy model. By review of diverse theoretical approaches on place identity and local participation, the paper creates a solid foundation for further analysis of the relationships between the applica­tion of PB and the development of civic identity and place attachment. (shrink)

In his account of visual perception, Thomas Reid describes visible figure as both ‘real and external’ to the eye and as the ‘immediate object of sight’. These claims appear to conflict with Reid's direct realism, since if the ‘immediate’ object of vision is also its direct object, then sight would be perceptually indirect due to the role of visible figure as a perceptual intermediary. I argue that this apparent threat to Reid's direct realism may be resolved by understanding visible figure (...) as the set of geometrical properties that holds between an object's visible surfaces and some particular perspective or point of view. On this relational interpretation of visible figure, and once an ambiguity over the use of the term ‘object’ is resolved, Reid's account of vision is both epistemically and perceptually direct, as well as consistent with his account of the other senses and doctrine of signs. (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)

Students of geometry do not prove Euclid's first theorem by examining an accompanying diagram, or by visualizing the construction of a figure. The original proof of Euclid's first theorem is incomplete, and this gap in logic is undetected by visual imagination. While cognition involves truth values, vision does not: the notions of inference and proof are foreign to vision.

A focused review of the literature on reasoning suggests that mechanisms based upon contraries are of fundamental importance in various abilities. At the same time, the importance of contraries in the human perceptual experience of space has been recently demonstrated in experimental studies. Solving geometry problems represents an interesting case as both reasoning abilities and the manipulation of perceptual–figural aspects are involved.In this study we focus on perceptual changes in geometrical problem solving processes in order to understand whether a mental (...) manipulation in terms of opposites might help. Four conditions were studied, two of which concerned the search for contraries as an implicit or explicit strategy.Results demonstrated that contraries, when used explicitly in solution processes, constitute an effective heuristic: The number of correct solutions increased, less time was needed to find a solution and participants were oriented towards the use of perception-based solutions—not only wer.. (shrink)

This paper argues that Socrates’ second definition of figure in Plato’s Meno (76a5–7) is deliberately insufficient: It states only a necessary condition for something’s being a figure, not a condition that is necessary as well as sufficient. For although it is true that every figure (in plane geometry) is (or corresponds to) a limit of a solid, not every limit of solid is a figure, i.e. not if the solid has a curved surface. It is argued that (...) this mistake is one Meno was meant to detect, since one of the three concepts he has been asked to agree to beforehand (75e1–76a3), namely the concept ‘plane’, has not been used in the definition. If this concept is put to work, we get a proper definition of figure, i. e. figure is the plane limit of a solid. Meno’s failure shows that he did not grasp the import of Socrates’ first definition of figure (at 75b9–1), and hence not what this example was meant to show him on a definition generally. Socrates’ first account of figure did indeed state a proper definition of figure, taking the definiens to be a necessary as well as a sufficient condition for the definiendum, as shown by the two words standing in for quantifiers, ‘only’ and ‘always’. Some general lessons about the importance of the dramatic elements in a Platonic dialogue are drawn from this interpretation. (shrink)

Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passage we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a philosophical (...) context: not a demonstrative tool, but a purely analogical model. In the case of the geometrical examples discussed in this paper, the diagrams are not conceived as part of a formalized proof, but as a work in progress. Aristotle is not interested in the final diagram but in the construction viewed in its process of development; namely in the figure a geometer draws, and gradually modifies, when he tries to solve a problem. The way in which the geometer makes use of the elements of his diagram, and the relation between these elements and his inner state of knowledge is the real feature which interests Aristotle. His goal is to use analogy in order to give the reader an idea of the states of mind involved in a more general process of knowing. (shrink)

In this article, we assess the heuristic power of Greek geometrical analysis by trying to reconstruct some analyses of extant propositions of which only the demonstration is found in the text. We have reconstructed the analysis of the trisection of an angle, the property of the tangent to the parabola, to the hyperbola/ellipse, and to the spiral line. In all of these cases, the results and the demonstrations can be found by the analysis alone, without arguments by analogy with other (...) figures or kinematic considerations. We have thus concluded that analysis is a much more powerful tool than is usually assumed to be. We have also examined a part of Apollonius’ Cutting off of a ratio and confirmed that the number of possible cases to be distinguished in analysis tended to increase because the geometric objects and the magnitudes related to them were always considered in the geometric arrangement in which they appear. We suggest that the absence of an algebraic approach that would have detached the magnitudes from their geometrical context may have been one of the major obstacles to further development of Greek geometry. (shrink)

This article attempts to study cost minimizing multi-objective fractional solid transportation problem with fuzzy cost coefficients c ˜ i j k r {\tilde{c}}_{ijk}^{r}, fuzzy supply quantities a ˜ i {\tilde{a}}_{i}, fuzzy demands b ˜ j {\tilde{b}}_{j}, and/or fuzzy conveyances e ˜ k {\tilde{e}}_{k}. The fuzzy efficient concept is introduced in which the crisp efficient solution is extended. A necessary and sufficient condition for the solution is established. Fuzzy geometric programming approach is applied to solve the crisp problem by (...) defining membership function so as to obtain the optimal compromise solution of a multi-objective two-stage problem. A linear membership function for the objective function is defined. The stability set of the first kind is defined and determined. A numerical example is given for illustration and to check the validity of the proposed approach. (shrink)

ABSTRACTThis paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on ‘algebraical geometry’ and ‘geometrical algebra’ in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to (...) initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the ‘abstract algebra’ and ‘abstract geometry’ of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to ‘algebraical geometry’ and ‘geometrical algebra’ of the second generation of ‘scientific’ symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s–1840s. (shrink)