This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a "pre" form of this thesis that every proof can be presented in everyday statements-only form.

This paper presents examples of infinite diagrams whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a “pre” form of this thesis that every proof can be presented in everyday statements-only form.

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point but “in” the point. We are interested in our research in the (...) diagrams which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role, and an unveiling role. In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. (shrink)

The infinite regress of Carroll’s ‘What the Tortoise said to Achilles’ is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry.

Some cardinal invariants from Cichon's diagram can be characterized using the notion of cut-and-choose games on cardinals. In this paper we give another way to characterize those cardinals in terms of infinite games. We also show that some properties for forcing, such as the Sacks Property, the Laver Property and ω ω -boundingness, are characterized by cut-and-choose games on complete Boolean algebras.

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point (as in the standard limit theory) but “in” the point. We are (...) interested in our research in the diagrams which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role (to externalize rough mental models), and an unveiling role (to help create new and interesting mathematical concepts, theories, and structures). In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. (shrink)

Introduction: contemporary conditions and diagrammatic trajectory -- From joy to the gap: the accessing of the infinite by the finite (Spinoza, Nietzsche, Bergson) -- The care of the self versus the ethics of desire: two diagrams of the production of subjectivity (and of the subject's relation to truth) (Foucault versus Lacan) -- The aesthetic paradigm: from the folding of the finite-infinite relation to schizoanalytic metamodelisation (to biopolitics) (Guattari) -- The strange temporality of the subject: life in-between the (...) infinite and the finite (Deleuze contra Badiou) -- Desiring-machines, chaoids, probeheads: towards a speculative production of subjectivity (Deleuze and Guattari) -- Conclusion: composite diagram and relations of adjacency. (shrink)

The way from the path integral to Feynman diagrams is sketched. The emphasis is put on the decrease of complexity in this process, from infinite-dimensional integrals down to the apparent simplicity of child’s play. On the other hand, also the subsequent increase in complexity when using Feynman diagrams to make realistic physical predictions is described, thus illustrating the dialectic between the simplicity and clarity of Feynman diagrams, and the complexity in their practical applications.

LetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$ to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

The paper considers the Copernicanism of Giordano Bruno (1548–1600) as a central moment of his philosophy of nature, concentrating on his two principal cosmological works, La cena de le ceneri (The Ash Wednesday Supper), written and published in London in 1584, and the Latin De immenso, published in Frankfurt in 1591. The principal characteristic of Bruno’s reading of Copernicus which is underlined is his physical realism, which was particularly complex due to his extension of the still finite Copernican cosmology to (...) infinite dimensions. The paper shows how Bruno’s use of diagrams was essential in defining the terms of his new infinite, post-Copernican cosmology, which constitutes an essential if much debated link between Copernicus himself and the great astronomers of the early seventeenth century such as Kepler and Galileo. (shrink)

Donald Coxeter died in 2003, at a ripe old age of 96. Though I had regularly seen him at mathematics talks in Toronto for over twenty years, I never felt rushed to seek him out. It seemed he would go on forever. His death left me regretting my missed opportunity and Siobhan Robert's excellent book makes me regret it even more. Like any good biography of an intellectual, King of Infinite Space contains personal details and mathematical achievements in some (...) detail. Thus, we learn of the traumatic effects on Coxeter of his parents' divorce, his search for a spouse, his vegetarianism, and his progressive politics. We also learn a fair bit about the kaleidoscopes he made while in Cambridge during his student days in order to study the symmetry properties of polyhedra. These involved mirrors that Coxeter had specially constructed for this purpose. Along the way we are treated to interesting tidbits, such as G.H. Hardy's detestation of mirrors . There are brief accounts of the brilliant notation Coxeter invented, known as Coxeter diagrams, and of course, the now famous Coxeter groups. Short appendices fill in a bit more detail. It is all very well done and thoroughly engrossing.Wittgenstein befriended Coxeter, who was part of the very small group to whom …. (shrink)

Since both berkeley and hume are committed to the view that a line is composed of finitely many fundamental parts, They must find responses to the standard geometrical proofs of infinite divisibility. They both repeat traditional arguments intended to show that infinite divisibility leads to absurdities, E.G., That all lines would be infinite in length, That all lines would have the same length, Etc. In each case, Their arguments rest upon a misunderstanding of the concept of a (...) limit, And thus are not successful. Berkeley, However, Adds a further ingenious argument to the effect that the standard geometrical proofs of infinite divisibility misread the unlimited representational capacity of geometrical diagrams as a substantive feature of the objects that these diagrams represent. The article concludes that berkeley is right on this matter, And that the traditional proofs of infinite divisibility do not show what they are intended to show. (shrink)

A new construction of a certain conceptual space is presented. Elements of this conceptual space correspond to concept elements of reality, which potentially comprise an infinite number of qualities. This construction of a conceptual space solves a problem stated by Dietz and his co-authors in 2013 in the context of Voronoi diagrams. The fractal construction of the conceptual space is that this problem simply does not pose itself. The concept of convexity is discussed in this new conceptual space. (...) Moreover, the meaning of convexity is discussed in full generality, for example when space is deprived of it, its substitutes for concept domains are considered. (shrink)

Fourth-order autonomous nonlinear differential equations can exhibit chaotic properties. In this study, we propose a family of fourth-order chaotic systems with infinite equilibrium points whose equilibria form closed curves of different shapes. First, the phase diagrams and Lyapunov exponents of the system family are simulated. The results show that the system family has complex phase diagrams and dynamic behaviors. Simulation analysis of the Poincarè mapping and bifurcation diagrams shows that the system has chaotic characteristics. The circuit (...) simulation model is constructed and simulated in Multisim. The circuit simulation results coincide with the numerical simulation results, which verifies the circuit feasibility of the system. Then, based on Lyapunov stability theory and the adaptive control method, the synchronous control of the system with infinite equilibria is designed. Numerical simulation results verify that the system synchronization with the adaptive control method is well. Finally, the synchronous drive system is used for image encryption, the response system is used for decryption, and color image encryption is realized by combining deoxyribonucleic acid coding and operating rules. Therefore, this study not only enriched the research on infinite equilibria chaotic systems but also further expanded secure communication technology by combining chaotic synchronization control and DNA coding in image encryption. (shrink)

Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only (...) a finite amount of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

This article analyzes Galileo's mathematization of motion, focusing in particular on his use of geometrical diagrams. It argues that Galileo regarded his diagrams of acceleration not just as a complement to his mathematical demonstrations, but as a powerful heuristic tool. Galileo probably abandoned the wrong assumption of the proportionality between the degree of velocity and the space traversed in accelerated motion when he realized that it was impossible, on the basis of that hypothesis, to build a diagram of (...) the law of fall. The article also shows how Galileo's discussion of the paradoxes of infinity in the First Day of the Two New Sciences is meant to provide a visual solution to problems linked to the theory of acceleration presented in Day Three of the work. Finally, it explores the reasons why Cavalieri and Gassendi, although endorsing Galileo's law of free fall, replaced Galileo's diagrams of acceleration with alternative ones. (shrink)

Based on an analysis of the category of “infinite judgments” in Kant, we will introduce the logical hexagon of predicate negation. This hexagon allows us to visualize in a single diagram the general structure of both Kant’s solution of the antinomies of pure reason and his argument in favor of Transcendental Idealism.

Quantum field theories are notoriously difficult to understand, physically as well as philosophically. The aim of this paper is to contribute to a better conceptual understanding of gauge quantum field theories, such as quantum chromodynamics, by discussing a famous physical limit, the ’t Hooft limit, in which the theory concerned often simplifies. The idea of the limit is that the number N of colours goes to infinity. The simplifications that can happen in this limit, and that we will consider, are: (...) the theory’s Feynman diagrams can be drawn on a plane without lines intersecting ; and the theory, or a sector of it, becomes integrable, and indeed corresponds to a well-studied system, viz. a spin chain. Planarity is important because it shows how a quantum field theory can exhibit extended, in particular string-like, structures; in some cases, this gives a connection with string theory, and so with its representation of gravity. Previous philosophical literature about how one theory might be emergent from, and-or reduced to, another one has tended to emphasize cases, such as occur in statistical mechanics, where the system before the limit has finitely many degrees of freedom. But here, our quantum field theories, including those on the way to the ’t Hooft limit, will have infinitely many degrees of freedom. Nevertheless, we will show how a recent schema by Butterfield and taxonomy by Norton apply to the quantum field theories we consider; and we will classify three physical properties of our theories in these terms. These properties are planarity and integrability, as in and above; and the behaviour of the beta-function reflecting, for example, asymptotic freedom. Our discussion of these properties, especially the beta-function, will also relate to recent philosophical debate about the propriety of assessing quantum field theories, whose rigorous existence is not yet proven. (shrink)

We establish the hierarchy among twelve equivalence relations on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language L and a cardinal κ, we consider the interplay between the restrictions of these similarities to the class ModL of all L-structures of size κ. It turns out that, concerning the number of different similarities and (...) the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages. (shrink)

Producing a properly philosophical theory of capitalism as an open axiomatic system requires adding intensive multiplicities to the mathematical account of set theory, which allows only extensive multiplicities. Doing so enables us to understand pricing as a process of transforming intensive quantities into metric quantities, and thereby develop a diagram of the dynamics of axiomatisation and of the market as the two-sided and asymmetrical recording surface of the capitalist socius whose slope represents the infinite debt owed to finance capital. (...) The capitalist market axiomatises four kinds of flows – flows of liquid capital and abstract labour, of course, but also flows of raw materials and energy and of human populations. At the same time that axiomatisation conjugates these flows in denumerable sets to extract surplus-value, it connects flows in minor, non-denumerable sets that escape axiomatisation and may become revolutionary. Such minor connections would constitute the fabric of a post-capitalist axiomatics. (shrink)

The foundations of mathematics are divided into proof theory and set theory. Proof theory tries to justify the world of infinite mind from the standpoint of finite mind. Set theory tries to know more and more of the world of the infinite mind. The development of two subjects are discussed including a new proof of the accessibility of ordinal diagrams. Finally the world of large cardinals appears when we go slightly beyond girard's categorical approach to proof theory.

In the new physics and in the new field of cosmometry, 1 it is the fundamental pattern that results in the motion from which all is created. Everything starts with the point of infinite potential. The tetrahedron at the point gives birth to the cuboctahedron ; its motion and structure result in the creation of the torus structure. The torus structure is self-referencing on a moment by moment basis since all must pass through the center. But isn't self-referencing the (...) basis for consciousness? It is said that all of creation has awareness, but at different levels. We know plants are aware of threats and of death to other living creatures by the work of Cleve Backster. We know we influence random number generators from the PEAR studies at Princeton. We know baby chicks will influence the movement of robots programed to do a random walk from the work of Rene Peoc'h. Quantum physics has embraced geometry with the work in the equations which explain the Feynman diagrams where particles come in and out of existence; those equations form a structure called the Amplituhedron - which is a quarter of a star tetrahedron. We also know that the tetrahedron can form the star tetrahedron and that each point of the star tetrahedron can form its own star tetrahedron, which can go on infinitely. That is, there is infinity in the finite. As in the fractal and holographic universe, each point or tetrahedron is connected to every other point. If each point has awareness due to formation of the torus, then Hermes teaching of "As above, as below" has meaning in the torus structure. If the torus is the fundamental unit of self-reference, is that the fundamental unit from which consciousness arises? The torus or double torus appears to be fundamental to all of creation - from galaxies to planets to atoms to photons. (shrink)

continent. 2.1 (2012): 6–21. The French philosopher and novelist Tristan Garcia was born in Toulouse in 1981. This makes him rather young to have written such an imaginative work of systematic philosophy as Forme et objet , 1 the latest entry in the MétaphysiqueS series at Presses universitaires de France. But this reference to Garcia’s youthfulness is not a form of condescension: by publishing a complete system of philosophy in the grand style, he has already done what none of us (...) in the older generation of speculative realists has done so far. His book is sophisticated, erudite, rigorous, imaginatively rich, and abundant in worldly wisdom– despite the author’s conclusion that wisdom does not exist. The quality and scope of Forme et objet took few observers by surprise, since Garcia has been treated as an emerging philosopher to watch across half a decade of Parisian oral tradition. But Garcia was not just the subject of rumor, being already well known to the French public as a writer of fiction. His debut novel, La meilleure part des hommes , 2 was awarded the 2008 Prix de Flore 3 and has already appeared in English as Hate: A Romance . 4 His follow-up novel, Mémoires de la jungle , 5 made clever use of a chimpanzee narrator. Nor was Garcia only published as a novelist before last November: his philosophical study L’Image 6 had already appeared when the author was just twenty-six, a year before he was crowned by the muses at the historic Café de Flore. And then in 2011, just months before the appearance of Forme et objet , Garcia published a widely distributed work entitled Nous, animaux et humains , 7 with its focus on Jeremy Bentham’s ideas about animals. Given this prolific and versatile track record, an optimistic scenario might envisage the young Garcia as one of those combined literary/philosophical talents who appear intermittently in France across the centuries: Jean-Paul Sartre is merely the most famous recent case. While more time is needed to see how Garcia will channel his impressive mental energies, Forme et objet displays such breadth of insight that its author has a good chance to emerge as one of the leading philosophers of his generation. If we accept Aristotle’s dictum that the peak mental age is fifty-one, then to read Garcia’s massive book is to gain some idea of what European philosophy might look like in the futuristic-sounding 2030’s. The present article is confined to Forme et objet . At 486 pages, the work is obviously daunting in size. Indeed, it is even longer than it sounds, given that many of its early sections are printed in a smaller typeface to designate them as supplemental commentary to the main flow of the argument. But while the length of the book reportedly led to delays in French publication, and will probably slow the inevitable appearance of an English translation, the length of the book should not deter interested readers– much of it results from Garcia’s teacherly writing style. Whereas Quentin Meillassoux’s prose displays an arctic economy of means, Garcia’s style is reminiscent of the repeated lessons of oral classroom proceedings. Rarely is the reader given fewer than three or four chances to master an idea before the author moves on to the next. In practice, the style feels welcoming rather than long-winded. Otherwise, the structure of Forme et objet is surprisingly simple. There is a useful Introduction of less than twenty pages. Then comes Book I: Formally , running to approximately 135 pages. Here Garcia outlines the most basic features of a thing “no matter what it is,” or n’importe quoi , an everyday phrase that Garcia shapes into a technical term. This part of the book feels at times like a more amiable version of Hegel’s Science of Logic , a parallel emphasized further by the threefold articulation of its theme: 1. Thing; 2. Thing and World; 3. Being and Understanding. This is followed by the much longer Book II: Objectively , totaling more than 300 pages. It contains sixteen essay-like meditations on specific kinds of objects—including time, animals, humans, history, gender, and death. Here each chapter rolls smoothly into the next, making this second part of the book feel more like a different work of Hegel: The Phenomenology of Spirit . But these are merely analogies. Garcia is no Hegelian, even if the book contains a few dialectical flourishes that seem to reflect his early enthusiasm for the Frankfurt School. Forme et objet ends with a six-page Coda, followed by the usual page of acknowledgments. In what follows, I will briefly summarize each of these four parts of the book before ending with some more general remarks. Before doing so, it will be useful to situate Garcia biographically (as much as I am able) and philosophically. Though Toulouse is his native city, his formative years were spent largely in Algeria, where his family has deep roots. During our sole private conversation, Garcia mentioned that his parents are professors of literature. 8 As a student of philosopher Garcia flourished so early that many of his current ideas date to his teenaged years: “There are sentences in Forme et objet that I wrote when I was seventeen,” he said in response to a question on that cold night on the Canal St.-Martin. I recalled that remark when reading his brilliant account, late in the book, of the central role of adolescence in contemporary culture. While many prodigies blow through their formal academic training without serious obstruction, Garcia’s student memories are rich in tales of isolation and struggle, though equally rich in gratitude for a half-dozen or so exceptional teachers who provided the intellectual space he needed: Meillassoux and Alain Badiou are simply two of the most prominent figures on that list. Though there are many points of agreement between Garcia’s philosophical position and my own, he not only reached his position years before reading my work, 9 he arrived along a rather different path: not through phenomenology, but via the Frankfurt School, which may be one of the reasons for his profound fascination with aesthetics. Garcia’s cultural background is as broad as one could wish: he is no less informed about punk rock and European football leagues than about the spiritualist roots of Bergson’s philosophy. Curious about everything and contemptuous towards nothing, Garcia can be expected to write insightfully on dozens of topics in the years to come. Given that his philosophy is so personally tantalizing in its agreements and disagreements with my own, and given the great internal richness of Forme et objet itself, the present review is no better than a first effort at coming to terms with the challenges posed by this minstrel from the rising generation. This is especially intriguing for older Generation X’ers like me, since confrontation with the younger generation is one of the many themes treated insightfully in Garcia’s book. 1. Introduction Garcia begins in defense of a so-called “flat ontology,” in which all things are equally things. While Roy Bhaskar 10 used this term pejoratively to refer to anti-realist philosophies that flatten everything onto an epistemic plane of human access, Manuel DeLanda 11 (an admirer of Bhaskar) reversed it into the positive principle that all realities are equally realities. Similar notions can be found in the “absistence” of Alexius Meinong, 12 the “irreduction” of Bruno Latour, 13 and my own critique 14 of the undermining/overmining pair. Also noteworthy is Levi Bryant’s use of the term “flat ontology” throughout The Democracy of Objects 15 and his earlier essay “The Ontic Principle.” 16 But for Garcia, flatness is only one face of the cosmos, and one that he ultimately declares to be rather impoverished. Even so, he always remains an advocate of a flat ontology. Insofar as everything is equally something, no matter what it is ( n’importe qui ), everything is equally a thing, equally solitary in its relation with world. This is why his book abounds in those long lists of random, ontologically equivalent entities that Ian Bogost has playfully termed “Latour Litanies.” 17 The first litany in Garcia’s book runs as follows: “We live in this world of things, where a cutting of acacia, a gene, a computer-generated image, a transplantable hand, a musical sample, a trademarked name, or a sexual service are comparable things.” (7) Yet Garcia is frankly dualistic; his flat ontology only lasts until page 159 and the end of Book I (entitled “Formally”), which deals entirely with things that are equally things. Thereafter Garcia turns his attention from things to objects, which are not flat in the least, but engage in hierarchical relations with one another. In agreement with both DeLanda and the speculative realists, Garcia proclaims that his book “proposes to put to the test a thought about things rather than a thought about our thought about things .” (8) Just as ducklings are “imprinted” (9) after hatching and treat the first creature they see as their mother, philosophers are imprinted by the idea with which they begin. Hence, philosophies that begin with human access will never truly find their way back to things. This makes Garcia rather suspicious of twentieth century philosophy, since “the twentieth century—to which in some way this work proposes to bid adieu—has been a period of theorizing modes of access to things rather than things...” (9) Among other possible benefits of the philosophy of things that Garcia proposes, it is fully able to account for thought as a special variant of things, while the reverse is not possible.(10) In Book I of Forme et objet , Garcia’s “things” are so flat, so de-determined, that he is forced to renounce some of the most basic features ascribed to things by most realists. As he tells us in his foreboding third footnote: “We will maintain that the solitude [of things] is less than unity, less than identity, and that it does not imply acceptance (any more than refusal) of the principle of non-contradiction.” (11) In a contemporary world cluttered with too many things, Garcia’s flat and formal plane provides us with some breathing room: “The formal plan of thought enables or re-enables us to cut short all accumulation—whether of knowing, experience, or action—by a simplicity, an impoverished surface...” (13) As Garcia says elsewhere in responding to a Deleuzian critic of the book, his starting point in flat ontology is designed to obstruct the claims of both analytic philosophy and Hegelianism: “Hence, this work seeks to protect each thing—real, imaginary, inconsistent, contradictory—both against Ockham’s Razor and against the Aufhebung or dialectical process.” 18 Yet contrary to the equalizing spirit of many flat ontologies, “we will add to our formal ontology of the equal, an objective ontology of the unequal.” (13) But initially, Garcia joins all flat ontologists in holding that everything is irreducible: “this irreducibility, which we will term the ‘chance’ of each thing... also marks the refusal of a positive thought that reduces things exclusively to natural things, or social things, or historical things, etc.” (15) This irreducible “chance” of a thing emerges as an important technical term in the book, always paired with its inverted brother, the “price to pay” ( prix à payer ). On pages 17-19, we find the only diagrams in the book. What they illustrate is that Garcia wishes to avoid two equally dangerous extremes. The first is the philosophy of substance, featuring the thing-in-itself as a mighty river fed by attributes as if by subordinate tributary streams. This model can be found in many of the classic thinkers of West and East alike. In it, “there is obviously a hierarchization between that which is dragged towards something other than itself, and this other which serves it as an ontological support while supporting its proper being.” (16) For Garcia, the second extreme worth avoiding is the philosophy of events: “One thus conceives trajectories of being, identified as events, facts, powers, intensities, or intentionality. These vectors of being come first, bearing and supporting being, displacing it, but without ever finding a stopping point, a buffer, an objective consistency.” (17) The first model gives us a thing too wrapped up in itself, too compact . This word “compact” (the French and the English are the same) is another technical term for Garcia. But if the “compact” model of things leads us to something more than things, the philosophy of events gives us less than things, by dissolving them into a play of vectors. Garcia’s alternative lies midway between these two extremes: Being enters the thing, being comes out of it. And a thing is nothing other than the difference between the enters and the being that comes out. Thus, the circuit of being is never halted. In the thing, there is never the thing-in-itself. And the thing is never in-itself, but outside of itself. Nonetheless, being is not eventally “pollinated” by vectors: it possesses an objecting halting-point... (19) This single idea is the key to Garcia’s book: the thing is neither a self-contained durable lump nor some sort of evental flux. Instead, the thing is the difference between its various components and its relations with its environment. Or stated differently: “the price to pay for this disposition is a circulation of being that systematically distinguishes two senses of things: that which is in the thing , and that in which the thing is , or that which encompasses it and that which it encompasses,” (19) translating comprendre here as “encompass.” 19 In a beautiful description of a piece of black slate, Garcia sums up the various minerals, qualities, and shapes that compose [ comprend ] it, and calls them “that which is in the thing,” (20) noting that this tells us nothing about “that in which [the slate] is”—namely, all the various situations in which the black slate can be found. Instead, the slate is the difference between these two: the most characteristic principle of Garcia’s philosophy. 2. Formally Book One of Forme et objet , “Formally,” is concerned with the formal equality of all things in a flat world. “Two questions mark the boundaries of reflection: of what is everything composed [ composé ], and: what do all things compose?” (27) Looking downward, we wish to know what everything is made of; looking upward, we want to know the ultimate result of the combination of all things. Here we must turn our attention to the thing n’importe quoi— no matter what it is. (30) Anything with finite qualities is obviously too specific to be relevant to global ontological questions. To an equal degree, something possessing all qualities (think of Whitehead’s God) 20 would not be n’importe quoi either, since it would still be too definite, even if incredibly vast. The same holds for contradictions, since these all differ from each other. The square circle, the non-white black white, and the non-city city are all too distinct to count as the thing no matter what it is. The n’importe quoi must be devoid of all specific qualities, including contradictory ones. In one of the more intriguing points in his book, so reminiscent of Meinong, Garcia proclaims that “the ‘no matter what it is’ is neither a reality nor an abstract construction, nor both of these at once; the ‘no matter what it is’ is simply the plane of equality of that which is real, that which is possible, that which is inexistent, that which is past, that which is impossible, that which is true, that which is false, that which is bad.”(39-40) Since everything has two faces, it would be a grievous mistake to focus on just one of them at the expense of the other, as physicalism or materialism do when reducing the world to minuscule physical underpinnings. For scientistic materialism, “it is either atoms, particles, or fields of force... which are the things.” (47-48) Moreover, “these more-than-things are accompanied by less-than-things: for example, ideas or facts of consciousness are determined by the state of matter and are not autonomous things, but manifestations reduced to secondary effects of material processes...” (48) On this point, Garcia’s position is in complete accord with my own critique of undermining and overmining. 21 Where we disagree is that Garcia is more deeply suspicious of the notion of substance, which I view as salvageable with a few needed changes, while Garcia sees this operation as hopeless: “A substance, in the history of philosophy, is the more-than-thing par excellence.” (51) Another agreement between our positions is visible when Garcia claims (correctly, in my opinion) “that it is vain to distinguish between things which are material and those which are not.” (52) Yet we also find an even more important disagreement, since for Garcia withdrawal cannot be the quality of a thing. Instead, the absence of a thing is simultaneous with it, embodied in all that is not it– the absence of the sculpture of a woman is to be found in the mold that appears at the same time as it, and thus withdrawal must be viewed as an “event” rather than as something pertaining to an object. For Garcia, nothing withdraws beyond access. Since we must distinguish between “that which is something” and “that which something is,” and since the former is identified with “no matter what it is is” and the latter with “ not no matter what it is,” we can say that “everything is thus a milieu, a fragile link between ‘no matter what it is’ and ‘ not no matter what it is.’” (62) And here we find Garcia’s critique of the thing-in-itself: “A thing is never defined en bloc . We can affirm that a thing is this or that, but that does not suffice. It is still necessary to state precisely that which is this thing .” (62) Stated differently, “something is not in itself : for that which is in the thing is not the thing, and that in which the thing is is not the thing.” (62) And here Garcia and I, facing the same evidence, draw opposite conclusions. For me, the fact that nothing can be identified with either its components or its concrete location means that the thing must be something in-itself distinct from both of these. Yet for Garcia, to be in-itself would mean to be identified with just one of these two extreme terms, and hence the thing can only be the difference between them. Garcia is equally suspicious of the classical tendency to view “unity” as a property of the thing, since in his eyes unity is too relational a property to belong to things. (65) While specific things are situated determinately with respect to other things, we are still speaking here about the thing no matter what it is, and this can be viewed only in terms of solitude, which all things share: a human being, a hand, or a chair or all equally things insofar as they are on their own , not insofar as they are one . (64) A thing is alone, and relates only to the one thing that is not another thing: world. In a striking parallel to my own argument for a partial revival of occasionalism, Garcia tells us that “the things communicate only by their solitude: it is because everything is equally on its own in the world that things can be together, enmeshed in one another.” (67) Alone in their solitude, things all relate to world, which serves as a mediator allowing them to become mixed up in one another. As we have seen, one reason that nothing can be in itself is because everything is in something else. For Garcia, “to be in something and to be something are equivalent.” (69) Stated more broadly, “being is thus the difference between the two aspects of each thing: that which is it, and that which it is.” (70) And even more vividly: “a thing is almost like a sack: there is that which one puts in the sack and that which remains outside the sack.” (70) But not quite like a sack, “since a thing is not a thin skin or film. Instead, a thing is comparable to a sack that is immaterial and without thickness: it is nothing other than the difference between that which is this thing and that which thing is, between content and container.” (71) Nothing can be in-itself because everything is two selves at once. For example, we cannot say that our self is defined by our consciousness: “Everything has a self because nothing is in itself. The self is not the quality of that which is related to itself (which is conscious, for example) or which thinks itself related to itself. Nonetheless, for an entity called ‘conscious’ to be related to itself, it is necessary that this very relation should be another thing than the self to which it is related.” (71) Consciousness cannot be the self, precisely because it is other than that of which it is conscious. Nothing is able to grasp itself. The self is “the function by which being and composition [ compréhension ] are mutually excluded...” (72) The self is “the point of shadow of everything that projects some light...” (72) The in-itself faces two opposite dangers: “For something to be in-itself is to be a self. Something which is a self flies out through one of its two sides... Stated differently, being in-itself is simply the possibility of a double failure.” (73) The in-itself can be termed compact : “There remains to us a means of thinking that which does not fully enter into the world, though without exiting from it. This means is what we call the compact.” (76) In a sense, the compact is the opposite of the world. For in the case of the world, everything enters it and it enters nothing; as for the compact, it enters the world (since it is something, after all) while nothing enters it. (77) The compact marks the presence of the impossible in the world. (78) It is not impossible, but possible only on the condition that it fails. (78) The time has come to speak of where a thing is located. “The sole condition of a thing is that of being in another thing than itself, and thus in another thing than something.” (78) A condition is “that which determines something, that which forms something, that in which something is.” (78) As for humans, “the condition of someone is his situation; my social condition is that which socially determines me, my place and my function...” (79) More generally, “to be conditioned is to find oneself reduced to that in which one is.” (79) Everything is conditioned, but nothing is reducible to this condition. To determine the condition of something is to determine in what it is. A thing is located in that which contradicts it, just as a statue exists in its mold, which is precisely that which it is not. Since the thing is finite and definite, its condition or form must be infinite and indefinite. That in which all things are is the world, which Garcia also terms “the whole.” (81) “To try to be in-itself is to attempt to remain outside the world. And indeed, to try to be in-itself is only a path of entry into the world.” (83) For Garcia, “the world is not the pre-existent container of the things it contains, a priori , nor the construction by the mind of a fictional ensemble of all things, a posteriori .” (85) Instead, the world is simultaneous with all things; the two always go together. The world cannot be a determinate world, such as the physical universe or mathematical space, since these are already too specific and limited. “Every determinate world, which is in fact a universe , is a ‘big thing’ [ grosse chose ]: it is a set, however vast, of composite things which itself embodies a thing.” (85) Every determinate world is really just a “big thing.” Stated differently, “it is nothing other than a balanced milieu between the things that compose it and the thing that it composes.” (85) We generally picture the world as a physical univer. (shrink)

continent. 1.2 (2011): 70-75. cartography of ghosts . . . And as a way to talk . . . of temporality the topography of imagination, this body whose dirty entry into the articulation of history as rapturous becoming & unbecoming, greeted with violence, i take permission to extend this grace —Akilah Oliver from “An Arriving Guard of Angels Thusly Coming To Greet” Our disappearance is already here. —Jacques Derrida, 117 I wrestled with death as a threshold, an aporia, a bandit, (...) a part of life. —Akilah Oliver Moraine in geological lingo is that which is left behind. Moraine- a euphemism for the de-stabilizing referent of the writer-ly body as a “troubled and troubling landscape marked by cultural and historical signifiers, the body as flesh memory [...] the body as transitory” (Oliver, Author Statement). Moraine— a geological metaphor of the poet as a holder of memory, as an accumulation of rocks and debris carried along the edge, terminal, dropped at the foot of language (in language). “Flesh Memory” according to Akilah Oliver is "that which my body recalls [...] everything has to do with the task of remembrance and its narrative reinvention [...] I was always translating an idea of the world as it presented itself at any given time. To write was a choice about how to be seen, how to enter the world as translator, actor, participant, in the dialogues that apparently made the real 'real'" (Levitsky). Flesh. Memory. The stuff some poems are made from. The stuff that gets abandoned, gleaned, and picked up by more flesh and memory. "My body, my life has always felt like a kaleidoscopic rip in the dominant fabric [...] has always been a dialogue with the impossible and the apparent” (Levitsky). The impossible-body or poet's body anticipates and performs (through language) an irretrievable death. IN APORIA I realized everything I must have been doing must have been Death. It was Christmas or Labor Day—a holiday—and every time you turned on the radio they said something like ‘four million’ or ‘going to die’.” — Andy Warhol I’m trying on egos, [a justification for the planet’s continuance]. Oh hello transgressor, you’ve come to collect utilitarian debts, humbling narrative space. Give me condition and wheatgrass, I his body disintegrating. I his body is ossification. Death my habit radius, yeah yeah. I his body can’t refuse this summons. I can’t get out this fucking room. Tell me something different about torture dear Trickster. Tell me about the lightness my mother told me to pick the one i love the best how it signals everything I ever wish to believe true just holy on my ship. I jump all over this house. this is it [what I thought is thought only, nothing more deceptive than]: I his body keeps thinking someone will come along, touch me. As like human or lima bean. I’m cradling you to my breast, you are looking out. A little wooden lion you & Peter carve on Bluff Street is quieting across your cheekbone. Not at all like the kind of terror found in sleep, on trembling grounds. It is yesterday now. I have not had a chance to dance in this century. Tonight I shall kill someone, a condition to remember Sunday morning. To think of lives as repetitions [rather than singular serial incarnations]. To understand your death is as exacerbating as trying to figure out why as schoolchildren in mid-nineteen-sixties Southern California we performed reflexive motions: cutting out lace snowflakes, reading Dick and Jane search for their missing mittens, imagining snow. Disintegrat ing . The -ing gerund catapults from the non-finite verb into past, present, future. The -ing as a tail pinned to death, a dog spinning to bite and never fully reaching itself, always shy of the end, circumreferential; a double copulative: deathing. Possessively AO calls it “habit radius” (a virtual fetish attribute) or an inescapable death presence that “confronts us with the paradox of an unattainable object [...] through it’s being unattainable” (Agamben, 27). A flirtation or dialogue with an unknowable thing and aporia utilized as investigative instrument to engage (death) while (in Southern California) we “perform reflexive motions,” cut lace snowflakes, imagine snow, and pay rent like “yeah, yeah” what else is new. And this too, fiction. The book I wish to right. The restored fallen, heroic. Did you expect a different grace from the world? Or upon exit? I’m working on “tough.” They think I am already. All ready. Who is the dead person? Is "I'm sorry" real to a dead person? Browning grass. My hands on this table. A contentious century. A place to pay rent. Redemptive moments. Am I now the dead person? Dead person, dead person, will you partake in my persimmon feast? The body inside the body astounds, confesses sins of the funhouse. I too have admired the people of this planet. Their frilly, orderly intellects. The use they’ve made of cardamom, radiation as well. How they’ve pasteurized milk, loaned surnames to stars, captured tribes, diseases, streets, and ideas too. The living-body as archive: is it possible to experience the living-body as archive without a (kind of) death? Sifting the rubble, rummaging through hoarded debris, skin sheds, memory-napping, and re-awoken (in flesh and) on terrain. “An investigative poetics seeks to unravel staid communities of thought and grasp at what might always be just beyond reach; a poetics of inquiry that lies between language as meaning, and language as rapturous entry into the world of posited ideas and idealism”( Levitsky). Something snaps. Lights blow out prior to embarking upon an investigative poetics. It begins with a question (often a sexy aporia) that leads to openings. "Every politics of memory [...] implies an intervention of the state. It's a state that legislates and acts with regard to the nonfinite mass of materials to be stored, materials which must be collected, preserved” (Derrida & Stiegler, 62). It seems poetic investigation already contains the potentiality of an (invisible) archive if the writer is “always writing” especially when not. Here’s my stupid digital romantic inclination: the living-body (of a poet) is a self-sustaining archive of non-finite memories. But not even I really believe that. AO innovated and sculpted an investigative poetic praxis. In a conversation with poet Rachel Levitsky, poetic-voice is viewed not as a precious identifier, but as a means to think through/about form, concluding that form is linked to framing. While poetic-voice may have tendency to precede form, it also erupts as a result of framing techniques. “They are frames that hold the shape of thinking (which is also to say of imagining) [...].”7 This reminds me of my rabbit who symmetrically chewed the corners of his hutch, which makes me wonder if it’s an expression of the shape of some animal anxiety tick I won’t ever have access to. Beyond the form/frame, death is an unoriginal yet unique limit; death is a damn deathless thing. It functions as a source of poetic investigation; that thing always “just beyond reach.” And how is death not a fetish (in this case an obsessive reverence for something non-material)? “Insofar as it [death] is a presence, a fetish [...] it is in fact something concrete and tangible; but insofar as it is the presence of an absence, it is, at the same time, immaterial and intangible, because it alludes continuously beyond itself to something that can never really be possessed [...] The fetish is [...] a sign of an absence, it is not an unrepeatable unique object; on the contrary, it is something infinitely capable of substitution, without any successive incarnations ever succeeding in exhausting the nullity of which it is the symbol” (Agamben, 33). AO utilized absence (the absent body [catapulted by the death of a beloved]) as an apparatus to investigate. In the process of conversing with absence or that which is absent, the absent body is affectionately objectified, incessantly summonsed back to a place of recognition, of objects, a desire for the absent body to remain intact while exiting the structural limits of grammar and syntax by moving into chant forms “to say what cannot be said” (Levitsky). from AN ARRIVING GUARD OF ANGELS THUSLY COMING TO GREET dear oluchi- the light is blinking rapidly on the black boxy machine. your room seems bigger than before and i am still planning to read some of those robert jordan books of yours. yesterday at the used bookstore where i was browsing the mysteries to “stall reality” (they are really not mysteries at all, they just employ death as the plot mistress but are unable to grasp its mystery at all)—well the point is, things were calm down here for a while and the world was little. i want to be big like you. or i want you not vast, not dead, not gone, but human small and here. i am so selfish. that is what i really want. to see you again. to oil your scalp. to hear you walk in the door, say ma i’m home . give me a chance to say welcome home son. or when leaving, don’t forget your hat . what do you wear out there? i wish you could have taken your new shoes with you. i’m so proud of you. i’m sorry for the way you died. i miss you all the time. even before, i missed you. out there, one time, some different men said: “shake for me girl, i wanna be your backdoor man.” who dat you love. 5/18/03 A letter-poem in sixteen lines “dear oluchi-” is safe-housed in epistolary form. Poetic voice is rendered as internal thought meanderings, a not-so-much confession, private/(pillow?) talk in the desire to be heard/witnessed by the referent and reader with an intent to absolve. The diminutive “i” bears a relation to poet Fanny Howe’s “little g God” in that “One of the (many) things I like about little g God is that you can have a vodka tonic while you talk to little g God, sing along to Bowie’s “I’m Afraid Of Americans,” and hum Coltrane’s “A Love Supreme,” though maybe not all at the same time” (Oliver, 2009). Towards the middle of the poem AO is at a used bookstore and remarks on the funny employment of death as a ‘plot mistress’ that ‘they’ (the dubious employed mystery authors) are ‘unable to grasp’, thereby giving death a mouthpiece, a modeling job, something to do to pass the time. from THE VISIBLE UNSEEN When I first saw graffiti, I recognized in it an ugly aesthetic, a dialectics of violence, a distortion of limbs, a hieroglyph. It was only later when I read the names of the dead that I then saw the path of ghosts charted there; its narrative of loss for the visible unseen whose place in history has been fictionalized and rendered unseen under the totalizing glare of history. Inscriptions, traces, specters. Graffiti begs a public face just as ghosts require non-ghosts (humans) to sense them. The “visible unseen” is a game of hide-and-seek between public viewer and graffiti-inscriber, an ephemeral-violent aesthetic on an ephemeral-policed canvas. Graffiti-inscribers already submit to being forgotten, expect to be washed away; perhaps it’s a holy urban mandala created by gangster-type monks without Buddhism. [...] in its refusal to disappear it forces a discourse in the public imagination we are forced to see what we would rather not, to make sense of an encoded language that we cannot read on the level of meaning. it irritates, forces its agency on us, speaks outside and beyond semiotic reach. An epic font-size pervading the public’s imagination, illegible, I could just close my eyes, remain passive, drive past, abandon it beyond reach, push it further away beyond death walls. In Barcelona I watched a clean up crew wash walls with an awesome water hose but I was more intrigued by their bodies; not a distortion of limbs, not hieroglyph but also not entirely legible; the laboring body permanently erasing specters of the city, and of course they knew it was also an invitation for the ghosts to return. Graffiti is death’s little sister, is also an aporia. [...] Graffiti (fr GK -graph(os), something drawn or written, to diagram or chart) attempts to stage the impossible: to erase the essence of its own subjectivity. Graffiti is a cartography of ghosts, a mapping of elegiac rapture (the transporting of a person from one place to another, as in heaven) and rupture (the state of being broken open.) Dwelling is a fiction stasis. [...] The notion of the past as being something done with, a look-back event, inhibits the possibility of reading graffiti as rapture, as rupture. If graffiti posits history as always in the process of becoming undone. [...] Because what is the body, if not also a complex temple, an unstable site through which to negotiate subjects, materiality, economies, gods, and modes of representations? The site where we are all already belated. Graphein meaning “to write.” “Derrida says every archive makes a law, and the law of genre is its own rupture” (Bloch, 39). However, graffiti is an (non/anti)-archive of erasure due to (the politics of) washing out its subjectivity, which only adds onto (or is symptomatic of) its character. The inhibition of “reading graffiti as rapture, as rupture” is partly due to it being a “look-back” event in that it’s process involves scratching through layers to reveal previous specters underneath. Graffiti (as an ancient genre) has always been a thing of ‘becoming undone’, and therefore ‘belated and always in arrival’ (Levitsky). It’s a Dionysian activity done at night with it’s back turned toward us. "The specter [...] is of the visible, but of the invisible visible, it is the visibility of a body which is not present in flesh and blood [...] appearing for vision, to the brightness of day [...] something becomes almost visible which is visible only insofar as it is not visible in flesh and blood. It is a night visibility. As soon as there is a technology of the image, visibility brings night. It incarnates in a night body, it radiates in a night light" (Derrida & Stiegler, 115). (shrink)

In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z -axis. Numerical analysis of the system reveals many strong dynamics. The (...) new system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system. (shrink)

This collection focuses on the ontology of space and time. It is centred on the idea that the issues typically encountered in this area must be tackled from a multifarious perspective, paying attention to both a priori and a posteriori considerations. Several experts in this area contribute to this volume: G. Landini discusses how Russell’s conception of time features in his general philosophical perspective;D. Dieks proposes a middle course between substantivalist and relationist accounts of space-time;P. Graziani argues that it is (...) necessary to provide an account of the “synthetic procedures” implicit in the recourse to diagrams in Euclid’s Elements, while E. Mares comes to the conclusion that in Euclid’s Elements we should treat the parallel postulate as empirical and the postulate that space is continuous as a priori. M. Arsenijevi?/M. Adži? present an important formal result concerning two theories of the infinite two-dimensional continua, which sheds new light on the current dispute between gunkologists and pointilists; F. Orilia discusses two problems for presentism, one regarding the duration of the present and the other related to Zeno’s paradoxes. A. Iacona delves deep into logical matters by focusing on the so-called T×W modal frames in order to deal with the deteterminism-indeterminism controversy. D. Mancuso outlines a non-standard temporal model compatible with time travel, andV. Fano/G. Macchia discuss time travels in the light of an important foundational principle of modern cosmology, Weyl’s Principle. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:119. HUME'S CONDITIONS FOR' CAUSATION: FURTHER TO GRAY AND IMLAY As part of his second proof of the existence of God, Descartes in Meditations III argues a causal premise derived from the nature of time. He argues it follows from the nature of time "that, in order to be conserved in each moment in which it endures, a substance has need of the same power and action as would (...) be necessary to produce and create it anew, supposing it did not yet exist; so that the light of nature shows us clearly that the distinction between creation and conservation is solely a distinction of reason." The nature of time to which he refers allows only a finite divisibility into durationless moments, each of which is entirely independent of every other. In the ultimate analysis, nothing true of one moment is deducible from what is true of any other, even what happens to be eternally true. The doctrine of the creation of the eternal verities is of a piece with this conception of time, and following with it from the conception of the divinity as sheer omnipotence, points to Hume's view that experience is the only guide to the world. It would seem to be Descartes' view, that nothing at a given time can stand in the necessary relation it must have to be a cause of something at another time; or, only something outside time can be a cause, for whose operation things in time can serve as but occasions. To what extent this dialectic in fact led to later occasionalist views is an historical question we need not consider. My only suggestion here concerns Hume's réponse to the occasionalists. With all but psychological necessity eliminated from his analysis of causation, Hume effectively argues the familiar case that occasional causes are the only causes by arguing the finite divisibility of time. In the literature only Laporte has held that for Descartes time is infinitely divisible and in this sense continuous. The separability of the parts of time, he 120. thought, suffice for their independence and contingency. But if time is continuous, its parts are inseparable; as Gueroult correctly observes, by discontinuity is meant con2 tingence, independence and separability. In addition, if a moment had duration, light would not be propagated instant aneously, thus according to Descartes upsetting his entire 3 physics. Gueroult maintains, rather, that Descartes' view of time is determined by his view of motion. Whatever the historical and logical priorities, there clearly is a connection between the above and what Bergson called the cinematographic conception of motion as a succession of static objects. It is a conception that precludes events as such. Motion occurs only in time, never at the moment, and occurs as a result of the re-creation of matter with regularly differing spatial relations. Due to metaphysical consideration expressed in theological terms, especially with reference to the immutability and simplicity of divine ways, this recreation of matter is constrained in certain ways, some of which we pick out as conditions for what was later called occasional causation. Put very simply, only what is constantly conjoined can stand in this relation; and that there should be instances of constant conjunction follows from the metaphysical considerations. With Hume's elimination of these metaphysical considerations, the status of the conditions, which he takes over as conditions for causation simpliciter, becomes problematic. But a recent challenge to Hume's program is still more problematic. The contention is that there is a case in which Hume's conditions necessary fo constant conjunction cannot all be satisfied. Even worse, i is a case in which his analysis ought to be conspicuously successful, viz. collision, which Hume like Descartes con4 ceives in cinematographic terms. The aim of this paper is to show that Hume's analysis stands against the challenge. Consider the following diagram which illustrates the positions (A3, A_...) of two balls (x,y) at successive moments (t_, t.,...)· 121. A3 A2 A1 A B -B1 B2 B3 fc0 T T T ©© © G®©T©T h fc3 fc4 fc5 H Robert Gray's thesis is that in at least one case of causation, collision which results in... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:119. HUME'S CONDITIONS FOR' CAUSATION: FURTHER TO GRAY AND IMLAY As part of his second proof of the existence of God, Descartes in Meditations III argues a causal premise derived from the nature of time. He argues it follows from the nature of time "that, in order to be conserved in each moment in which it endures, a substance has need of the same power and action as would (...) be necessary to produce and create it anew, supposing it did not yet exist; so that the light of nature shows us clearly that the distinction between creation and conservation is solely a distinction of reason." The nature of time to which he refers allows only a finite divisibility into durationless moments, each of which is entirely independent of every other. In the ultimate analysis, nothing true of one moment is deducible from what is true of any other, even what happens to be eternally true. The doctrine of the creation of the eternal verities is of a piece with this conception of time, and following with it from the conception of the divinity as sheer omnipotence, points to Hume's view that experience is the only guide to the world. It would seem to be Descartes' view, that nothing at a given time can stand in the necessary relation it must have to be a cause of something at another time; or, only something outside time can be a cause, for whose operation things in time can serve as but occasions. To what extent this dialectic in fact led to later occasionalist views is an historical question we need not consider. My only suggestion here concerns Hume's réponse to the occasionalists. With all but psychological necessity eliminated from his analysis of causation, Hume effectively argues the familiar case that occasional causes are the only causes by arguing the finite divisibility of time. In the literature only Laporte has held that for Descartes time is infinitely divisible and in this sense continuous. The separability of the parts of time, he 120. thought, suffice for their independence and contingency. But if time is continuous, its parts are inseparable; as Gueroult correctly observes, by discontinuity is meant con2 tingence, independence and separability. In addition, if a moment had duration, light would not be propagated instant aneously, thus according to Descartes upsetting his entire 3 physics. Gueroult maintains, rather, that Descartes' view of time is determined by his view of motion. Whatever the historical and logical priorities, there clearly is a connection between the above and what Bergson called the cinematographic conception of motion as a succession of static objects. It is a conception that precludes events as such. Motion occurs only in time, never at the moment, and occurs as a result of the re-creation of matter with regularly differing spatial relations. Due to metaphysical consideration expressed in theological terms, especially with reference to the immutability and simplicity of divine ways, this recreation of matter is constrained in certain ways, some of which we pick out as conditions for what was later called occasional causation. Put very simply, only what is constantly conjoined can stand in this relation; and that there should be instances of constant conjunction follows from the metaphysical considerations. With Hume's elimination of these metaphysical considerations, the status of the conditions, which he takes over as conditions for causation simpliciter, becomes problematic. But a recent challenge to Hume's program is still more problematic. The contention is that there is a case in which Hume's conditions necessary fo constant conjunction cannot all be satisfied. Even worse, i is a case in which his analysis ought to be conspicuously successful, viz. collision, which Hume like Descartes con4 ceives in cinematographic terms. The aim of this paper is to show that Hume's analysis stands against the challenge. Consider the following diagram which illustrates the positions (A3, A_...) of two balls (x,y) at successive moments (t_, t.,...)· 121. A3 A2 A1 A B -B1 B2 B3 fc0 T T T ©© © G®©T©T h fc3 fc4 fc5 H Robert Gray's thesis is that in at least one case of causation, collision which results in... (shrink)

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From (...) this perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises. (shrink)

Diagrams are ubiquitous in mathematics. From the most elementary class to the most advanced seminar, in both introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase understanding, and prove results. They thus fulfill a variety of important roles in mathematical practice. Long overlooked by philosophers focused on foundational and ontological issues, these roles have come to receive attention in the past two decades, a trend in line with the growing philosophical interest in actual mathematical practice.

I will explore the paradoxical nature of epistemic access. By critiquing the traditional conception of mental states that are labelled as ’knowledge’, I demonstrate the susceptibility of these states to an infinite regress, thus, challenging their existence and validity. I scrutinise the assumption that an epistemic agent can have complete epistemic access to all facts about a given object while simultaneously being ignorant of certain truths that impact the very knowledge claims about the object. I further analyse the implications (...) of this paradox when attempting to fine-grain the parameters of epistemic access, which happens to reveal the vacuity of such definitions and the inevitability of the regress. (shrink)

We contend that diagrams are tools not only for communication but also for supporting the reasoning of biologists. In the mechanistic research that is characteristic of biology, diagrams delineate the phenomenon to be explained, display explanatory relations, and show the organized parts and operations of the mechanism proposed as responsible for the phenomenon. Both phenomenon diagrams and explanatory relations diagrams, employing graphs or other formats, facilitate applying visual processing to the detection of relevant patterns. Mechanism (...) class='Hi'>diagrams guide reasoning about how the parts and operations work together to produce the phenomenon and what experiments need to be done to improve on the existing account. We examine how these functions are served by diagrams in circadian rhythm research. (shrink)

In this paper, we present a survey of the development of the technique of argument diagramming covering not only the fields in which it originated - informal logic, argumentation theory, evidence law and legal reasoning – but also more recent work in applying and developing it in computer science and artificial intelligence. Beginning with a simple example of an everyday argument, we present an analysis of it visualised as an argument diagram constructed using a software tool. In the context of (...) a brief history of the development of diagramming, it is then shown how argument diagrams have been used to analyze and work with argumentation in law, philosophy and artificial intelligence. (shrink)

In his Berlin Lectures of the 1820s, the German philosopher Arthur Schopenhauer (1788–1860) used spatial logic diagrams for philosophy of language. These logic diagrams were applied to many areas of semantics and pragmatics, such as theories of concept formation, concept development, translation theory, clarification of conceptual disputes, etc. In this paper we first introduce the basic principles of Schopenhauer’s philosophy of language and his diagrammatic method. Since Schopenhauer often gives little information about how the individual diagrams are (...) to be understood, we then make the attempt to reconstruct, specify and further develop one diagram type for the field of conceptual analysis. (shrink)

In his Berlin Lectures of the 1820s, the German philosopher Arthur Schopenhauer (1788–1860) used spatial logic diagrams for philosophy of language. These logic diagrams were applied to many areas of semantics and pragmatics, such as theories of concept formation, concept development, translation theory, clarification of conceptual disputes, etc. In this paper we first introduce the basic principles of Schopenhauer’s philosophy of language and his diagrammatic method. Since Schopenhauer often gives little information about how the individual diagrams are (...) to be understood, we then make the attempt to reconstruct, specify and further develop one diagram type for the field of conceptual analysis. (shrink)

This paper analyses a hitherto unknown technique of using logic diagrams to create argument maps in eristic dialectics. The method was invented in the 1810s and -20s by Arthur Schopenhauer, who is considered the originator of modern eristic. This technique of Schopenhauer could be interesting for several branches of research in the field of argumentation: Firstly, for the field of argument mapping, since here a hitherto unknown diagrammatic technique is shown in order to visualise possible situations of arguments in (...) a dialogical controversy. Secondly, the art of controversy or eristic, since the diagrams do not analyse the truth of judgements and the validity of inferences, but the persuasiveness of arguments in a dialogue. (shrink)

In a multi-study naturalistic quasi-experiment involving 269 students in a semester-long introductory philosophy course, we investigated the effect of teaching argument diagramming on students’ scores on argument analysis tasks. An argument diagram is a visual representation of the content and structure of an argument. In each study, all of the students completed pre- and posttests containing argument analysis tasks. During the semester, the treatment group was taught AD, while the control group was not. The results were that among the different (...) pretest achievement levels, the scores of low-achieving students who were taught AD increased significantly more than the scores of low-achieving students who were not taught AD, while the scores of the intermediate- and high-achieving students did not differ significantly between the treatment and control groups. The implication of these studies is that learning AD significantly improves low-achieving students’ ability to analyze arguments. (shrink)

In his 1903 Syllabus, Charles S. Peirce makes a distinction between icons and iconic signs, or hypoicons, and briefly introduces a division of the latter into images, diagrams, and metaphors. Peirce scholars have tried to make better sense of those concepts by understanding iconic signs in the context of the ten classes of signs described in the same Syllabus. We will argue, however, that the three kinds of hypoicons can better be understood in the context of Peirce's sixty-six classes (...) of signs. We analyze examples of hypoicons taken from the field of information design, describing them in the framework of the sixty-six classes, and discuss the consequences of those descriptions to the debate about the order of determination of the 10 trichotomies that form those classes. (shrink)

This article presents diagrams developed from the insights of three middle school children with limited mobility about their experiences navigating social and spatial relations in their home, school and neighbourhoods. The paper explores the concept of assemblage as well as operationalising the Deleuzian idea of the diagram. The diagrams we produce are developed in connection with dominant idealisations of neighbourhood and home range that function in North America to choreograph children's progression from infancy through adolescence. We undertake this (...) diagramming in order to immediately and affectively illustrate the limits and possibilities of the challenges these young people face. (shrink)

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of (...) most importance for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations. I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. (shrink)

Biologists depend on visual representations, and their use of diagrams has drawn the attention of philosophers, historians, and sociologists interested in understanding how these images are involved in biological reasoning. These studies, however, proceed from identification of diagrams on the basis of their spare visual appearance, and do not draw on a foundational theory of the nature of diagrams as representations. This approach has limited the extent to which we under- stand how these diagrams are involved (...) in biological reasoning. In this paper, I characterize three different kinds of figures among those previously identified as diagrams. The features that make these figures distinctive as representational types, furthermore, illuminate the ways in which they are involved in biological reasoning. (shrink)

Suppose you found that the universe around you was infinite—that it extended infinitely far in space or in time and, as a result, contained infinitely many persons. How should this change your moral decision-making? Radically, it seems, according to some philosophers. According to various recent arguments, any moral theory that is ’minimally aggregative’ will deliver absurd judgements in practice if the universe is (even remotely likely to be) infinite. This seems like sound justification for abandoning any such theory. (...) -/- My goal in this thesis is simple: to demonstrate that we need not abandon minimally aggregative theories, even if we happen to live in an infinite universe. I develop and motivate an extension of such theories, which delivers plausible judgements in a range of realistic cases. I show that this extended theory can overcome key objections—both old and new—and that it succeeds where other proposals do not. With this proposal in hand, we can indeed retain minimally aggregative theories and continue to make moral decisions based on what will promote the good. (shrink)