The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail (...) in the face of plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate’s disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out. (shrink)

The objective of this article is to provide a discussion that counters the infinite particle disappearance conclusion argued by Shackel, 417–433, 2018). In order to do this, clear criteria to disprove the results of the applications of his continuity principles are provided, in addition to the consideration of the fundamental Classical Mechanical principle of mass conservation as an independent and clear basis for this disproof.

In this paper I present a novel supertask in a Newtonian universe that destroys and creates infinite masses and energies, showing thereby that we can have infinite indeterminism. Previous supertasks have managed only to destroy or create finite masses and energies, thereby giving cases of only finite indeterminism. In the Nothing from Infinity paradox we will see an infinitude of finite masses and an infinitude of energy disappear entirely, and do so despite the conservation of energy (...) in all collisions. I then show how this leads to the Infinity from Nothing paradox, in which we have the spontaneous eruption of infinite mass and energy out of nothing. I conclude by showing how our supertask models at least something of an old conundrum, the question of what happens when the immovable object meets the irresistible force. (shrink)

In several of his papers, Charles S. Peirce illustrates processes of interpreting and understanding signs by examples from second language vocabulary teaching and learning. The insights conveyed by means of these little pedagogical scenarios are not meant as contributions to the psychology of second language learning, but they aim at elucidating fundamental semiotic implications of knowledge acquisition in general. Peirce's semiotic premise that a well-understood sign is one that represents an object and creates an interpretant is essential to the (...) understanding of how new words and signs in general can be taught and learned. The article argues that Peirce's theory of the object of the sign, especially of the necessity of collateral experience of the object of a sign, can help to understand the riddle posed by of the Meno paradox of the impossibility of learning what we do not yet know. It examines the semiotic implications of the didactic methods of teaching and learning through translation, ostension, mental and real images, as well as metacognition, and it shows how icons, indices, and symbols are essential to learning new words. (shrink)

In honour of Detlef Dürr, we report on a mathematical rigorous computation of the electric vacuum polarisation current and extract the well-known expression for the second order perturbation. Intermediate steps in the presented calculation demonstrate, to the knowledge of the authors for the first time, mathematical rigorous versions of the combined dimensional and Pauli-Villars regularisation schemes. These are employed as computational tools to infer convenient integral representations during the computation. The said second order expression is determined up to a remaining (...) degree of freedom of a real number–without ill-defined terms from start to end. (shrink)

Economics presents the paradox of the entropy of the law of diminishing returns and infinity of the substitution effect. Resolution assumes the substitution effect is greater than diminishing returns. Technology presupposing entropy, introduced is a new paradox of entropic technology generating infinite growth. Resolution assumes serial substitution of technologies, generating an infinite continuum. Physics and economics contest mechanic entropy and organic growth conceptions. A mechanic conception resolves set disjunctives exclusively, every set disjoined from a contiguous set, (...) constituting entropy. An organic conception resolves set disjunctives inclusively, every set conjoined with a contiguous set, constituting growth. Causally each instance of something decreases the preexistent set of all possible instances by one, dying. Functionally each instance of something increases the postexistent set of all possible instances by one, living. Economists have the advantage here. Endlessly divisible, substance is something constituted of nothing, ungoverned by the law of conservation of energy. Physicists implicitly concede this assuming substance emerges from a Big Bang. Alas lunch is not free. Product of an economic growth equation being endless and non-repetitive, calculation is irrational, rendering growth economics irrelevant. Calculation requires marginal analysis, sequences initiating anywhere within a set governed by conservation of energy and entropy converging in parallel upon a common limit, “the unseen hand.” Energy conserved, however, economic growth is bounded. (shrink)

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting (...) phrase’. Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The humble origins of Russell's paradox by J. Alberto Coffa ON SEVERAL OCCASIONS Russell pointed out that the discovery of his celebrated paradox concerning the class of all classes not belonging to themselves was intimately related to Cantor's proof that there is no greatest cardinal. lOne of the earliest remarks to that effect occurs in The Principles ofMathematics where, referring to the universal class, the class of (...) all classes and the class of all propositions, he notes that when we apply the reasoning of his [Cantor's] proofto the cases in question we find ourselves met by definite contradictions, of which the one discussed in Chapter x is an example. (P. 362) And in a footnote he adds: "It was in this way that I discovered this contradiction". Throughout his writings Russell left a number ofhints concerning the sort of connection he had drawn between Cantor's proof and his own discovery. In fact, his suggestions are so specific that there would seem to be little room left for speculation concerning how the discovery took place.2 The picture that emerges almost immediately from Russell's observations is the following. For reasons which are 1 See, e.g., Bertrand Russell, The Principles of Mathematics, 2nd ed. (London: Allen & Unwin, 1937), §§100, 344-9; G. Frege, Wissenschaftlicher Briefwechsel (Hamburg: Felix Meiner Verlag, 1976), pp. 215-16; B. Russell, Essays in Analysis, ed. D. Lackey (New York: George Braziller, 1973), p. 139; B. Russell, Introducrion to Mathemarical Philosophy (New York: Simon and Schuster, 1971), p. 136; B. Russell, My Philosophical Development (New York: Simon and Schuster, 1959), pp. 75-6; The Autobiography of Bertrand Russell, I (New York: Bantam Books, 1968), 195. 2 See, e.g., Ch. Thiel's "Einleitung des Herausgebers" in Frege, Wissenschaftlicher Briefwechsel, pp. 203 and 216 (footnote), and 1. Grattan-Guinness, "How Bertrand Russell Discovered his Paradox", Historia Marhematica, 5 (1978),127-37. 31 32 Russell, nos. 33-4 (Spring-Summer 1979) never made clear, Russell decided to analyze Cantor's argument applying it to "large" classes such as the universal class V, and the class of all classes. When, for example, we consider V, its power set and the correlationf(x) = {x} if x is not a class, f(x) =. x otherwise, then Cantor's diagonal class D turns out to be the class of all classes not belonging to themselves. Moreover, since the element ofV which is taken toD byfisD itself(i.e., sincef(D) = D), Cantor's reasoning invites us to raise the question whether D belongs tof(D) (i.e., toD) or not; and it establishes that it does precisely if it doesn't.3 The purpose of this note is to present recently uncovered information that complements and corrects our present understanding of Russell's discovery of his paradox. As it turns out, far from originating from his desire to apply the ideas in Cantor's theorem, a version of Russell's paradox first occurred in an argument that Russell had devised, late in 1900, in order to establish the invalidity of that theorem. An appeal to "large" classes such as V or Class was, as we shall see, essential to Russell's attempted refutation. As he displayed the details of his counterexample Russell's paradox emerged, at first unrecognized, as the by-product of a project that the paradox itself would eventually undermine. In a paper written in 1901 and largely devoted to a popular exposition and defence of Cantor's theory of the infinite, Russell expressed what appeared to be a rpinor reservation to Cantor's treatment : There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because, if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number, and if this proof were valid, the contradictions of infinity would reappear in a sublimated form. But in this one point, the master has been guilty of a very subtle fallacy, which I hope to explain in some future work.4 3 See Principles, §349. Perhaps I should remind... (shrink)

Alexander R. Pruss examines a large family of paradoxes to do with infinity - ranging from deterministic supertasks to infinite lotteries and decision theory. Having identified their common structure, Pruss considers at length how these paradoxes can be resolved by embracing causal finitism.

The paper argues that contemporary consciousness studies can profit from Charles S. Peirce’s philosophy of consciousness. It confronts mainstream tendencies in contemporary consciousness studies, including those which consider consciousness as an unsolvable mystery, with Peirce’s phenomenological approach to consciousness. Peirce’s answers to the following contemporary issues are presented: phenomenological consciousness and the qualia, consciousness as self-controlled agency of humans, self-control and self-reflection, consciousness and language, self-consciousness and introspection, consciousness and the other, consciousness of nonhuman animals, and the question of (...) a quasi-consciousness of the physical universe. A detailed account of Peirce’s three modes of consciousness is presented: primisense, qualisense or feeling-consciousness, altersense, and medisense, the consciousness of cognition, thought, and reasoning. In contrast to consciousness studies that establish a rather sharp dividing line between conscious and unconscious states of mind, Peirce adopts the principle of synechism, the theory of continuity. For him, consciousness is a matter of degree. An important difference between Peirce’s concept of qualia and current theories of qualia in human consciousness is discussed. The paper shows how consciousness, according to Peirce, emerges from unconscious qualia and vanishes into equally unconscious habits. It concludes with a study of the roles of qualia, habit, and self-control in Peirce’s theory of signs, in particular in qualisigns and symbols, and the question of signs as quasi-conscious agents in semiosis. (shrink)

The nature of representation has been a central but controversial issue of cognitive philosophy. After 2,500 years of reflection (cf. Rolf 2006), opinions are still divided. On the one hand, there are those who are convinced that we have reached a crisis of representation in the arts, the media, and cultural theory; on the other hand, representation has remained right at the top of the agenda of cognitive science and Artificial Intelligence research (cf. Nöth & Ljungberg, eds. 2003; Nöth 1997). (...) The words in which Daniel Dennett, in 1978, postulates the necessity for a new theory of representation as a foundation of cognitive science are symptomatic of a certain unease about the state of the art: "What is needed is.. (shrink)

Ecosemiotics is the study of sign processes (semioses) in relation to the natural environment in which they occur. The paper examines the cultural, biological, and evolutionary dimensions of ecosemioses on the basis of C. S. Peirce's theory of continuity between matter and mind and investigates the ecosemiotic dimensions of natural signs. Ecosemiotics and the semiotics of nature are distinguished from pansemiotism, and the coevolution of sign processes with their natural enviromnent is discussed as a determining factor of ecosemiosis.

Protosemiotics is the study of the rudiments of semiosis, primarily in nature. The extension of the semiotic field from culture to nature is both necessary and possible in the framework of Peirce's semiotic theory. Against this extension, the critique of pansemiotism has been raised. However, Peirce's semiotics is not pansemiotic since it is based on the criterion of thirdness, which is not ubiquitous in nature. The paper examines the criteria of protosemiosis in the domain of physical and mechanical processes.

"'There is only one thing that is certain, namely that we can have nothing certain; and therefore it is not certain that we can have nothing certain,' Samuel Butler once said, expressing in that mindbloggler all the elements required to form a classical paradox. Throughout the ages wise men and jesters alike have been intrigued by such mental twists and riddles which defy common sense and yet appear to be true." -- Dust jacket.

Journal Name: Semiotica Issue: Ahead of print.

Open peer commentary on the target article “From Objects to Processes: A Proposal to Rewrite Radical Constructivism” by Siegfried J. Schmidt. Upshot: The paper examines some of S. J. Schmidt’s key concepts from a semiotic perspective. It argues that not all of them are as incompatible with key notions of semiotics as the author claims and that, even though others remain indeed irreconcilable, some of the latter may contribute to extending radical constructivism beyond its own new horizons.

The paper is a study of how graphic narratives (graphic novels and the comics) represent time in external visual space as well as in inner (mental) representations. Peirce’s semiotics is the main tool of research. After a survey of various approaches to the study of time in narratives in general and in graphic narratives in particular, an outline of the various aspects of the embodiment of time in space in general is given before the forms of the embodiment of time (...) in the space of graphic narratives is examined in detail. Signs of time are signs that represent time as their object and create mental representations of time as their interpretant. On this semiotic premise, the paper proposes a reinterpretation of Genette’s theory of time in narratives. A comparison with narratives in prose reveals that narrative time in graphic narratives evinces marked discontinuities in their spatial representation. The paper distinguishes between time as continuity, time as age, and points in time. It concludes that time as continuity is typically embodied in the form of spatial diagrams whereas time as age and points in time are mainly of the nature of indexical signs. The insights into the forms of embodiment of time in space derived from these premises are applied to examples from recent graphic novels and some traditional comics. (shrink)

Kant claimed that it is impossible for us to have a consistent notion of the infinite. I shall concentrate on three versions of the paradoxes of the infinite: Kant’s first antinomy, the paradoxes of Cantorian set theory, and applications of Cantorian arguments to the metaphysics of the world. I shall dare two side-glance looks at Ancient Chinese Philosophy, where analogies to the Western paradoxes can be found. I shall first discuss key passages from the Chinese sophists, and then consider (...) the formulation of the Law of Non-Contradiction in the Moist Canons. I conclude that the paradoxes of the infinite remain a major challenge for reason. (shrink)

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of (...) infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism. (shrink)

Paradoxes are arguments that lead from apparently true premises, via apparently uncontroversial reasoning, to a false or even contradictory conclusion. Paradoxes threaten our basic understanding of central concepts such as space, time, motion, infinity, truth, knowledge, and belief. In this volume Roy T Cook provides a sophisticated, yet accessible and entertaining, introduction to the study of paradoxes, one that includes a detailed examination of a wide variety of paradoxes. The book is organized around four important types of (...) class='Hi'>paradox: the semantic paradoxes involving truth, the set-theoretic paradoxes involving arbitrary collections of objects, the Soritical paradoxes involving vague concepts, and the epistemic paradoxes involving knowledge and belief. In each of these cases, Cook frames the discussion in terms of four different approaches one might take towards solving such paradoxes. Each chapter concludes with a number of exercises that illustrate the philosophical arguments and logical concepts involved in the paradoxes. _Paradoxes_ is the ideal introduction to the topic and will be a valuable resource for scholars and students in a wide variety of disciplines who wish to understand the important role that paradoxes have played, and continue to play, in contemporary philosophy. (shrink)

Behind the third ventricle of the human brain a miniscule pedunculate bud, close to the optic thalamus, that is, to the two beds of optic nerves, a gland soft in substance yet containing gritty particles. Function: unknown. Because of its pine-cone shape it is called the conarium or pineal body, even though the recent photographs of it by Nilsson and Lindberg show it to be morphologically reminiscent of nothing so much as the plucked tail of a gamebird, which Simon (...) Dedalus refers to as ‘the pope's nose’. Today it is presumed to be an endocrine gland of some sort, even though there is no doubt that morphogenetically in all vertebrates it is a vestigial unpaired eye. As fossil evidence indicates—and we still find it almost fully developed in some extant amphibians—ancestral vertebrates possessed in addition to the paired bilateral eyes a solitary dorsal eye opening at the top of the skull to the sky. This singular evagination of the brain—something betwixt a visual organ and a gland—seems to hold a special fascination for philosophers. Here we shall consider two of them: René Descartes (1596–1650), the father, as we say, of modern philosophy; and Georges Bataille (1897–1962), the father, as many say, of post-modern philosophy. Three hundred years separate them. Devotion to the pineal body conjoins them. (shrink)

Semiotic Theory of Learning asks what learning is and what brings it about, challenging the hegemony of psychological and sociological constructions of learning in order to develop a burgeoning literature in semiotics as an educational foundation. Drawing on theoretical research and its application in empirical studies, the book attempts to avoid the problematization of the distinction between theory and practice in semiotics. It covers topics such as signs, significance and semiosis; the ontology of learning; the limits of learning; ecosemiotics; ecology (...) and sexuality. -/- The book is written by five of the key figures in the semiotics field, each committed to the belief that living is a process of interaction through acts of signification with a signifying environment. While the authors are agreed on the value of semiotic frameworks, the book aims not to present an entirely coherent line in every respect, but rather to reflect ongoing scholarship and debates in the area. In light of this, the book offers a range of possible interpretations of major semiotic theorists, unsettling assumptions while offering a fresh, and still developing, series of perspectives on learning from academics grounded in semiotics. -/- Semiotic Theory of Learning is a timely and valuable text that will be of great interest to academics, researchers and postgraduates working in the fields of educational studies, semiotics, psychology, philosophy, applied linguistics and media studies. (shrink)

Behind the third ventricle of the human brain a miniscule pedunculate bud, close to the optic thalamus, that is, to the two beds of optic nerves, a gland soft in substance yet containing gritty particles. Function: unknown. Because of its pine-cone shape it is called the conarium or pineal body, even though the recent photographs of it by Nilsson and Lindberg show it to be morphologically reminiscent of nothing so much as the plucked tail of a gamebird, which Simon (...) Dedalus refers to as ‘the pope's nose’. Today it is presumed to be an endocrine gland of some sort, even though there is no doubt that morphogenetically in all vertebrates it is a vestigial unpaired eye. As fossil evidence indicates—and we still find it almost fully developed in some extant amphibians—ancestral vertebrates possessed in addition to the paired bilateral eyes a solitary dorsal eye opening at the top of the skull to the sky. This singular evagination of the brain—something betwixt a visual organ and a gland—seems to hold a special fascination for philosophers. Here we shall consider two of them: René Descartes (1596–1650), the father, as we say, of modern philosophy; and Georges Bataille (1897–1962), the father, as many say, of post-modern philosophy. Three hundred years separate them. Devotion to the pineal body conjoins them. (shrink)

This paper divides the sophisms and paradoxes put forth by Chinese thinkers of the pre-Qin period of China into six groups: paradoxes of motion and infinity, paradoxes of class membership, semantic paradoxes, epistemic paradoxes, paradoxes of relativization, other logical contradictions. It focuses on the comparison between the Chinese items and the counterparts of ancient Greek and even of contemporary Western philosophy, and concludes that there turn out to be many similar elements of philosophy and logic at the beginnings of (...) Chinese and Greek civilizations. (shrink)

Infinity and the Brain proposes a logical and scientific way to resolve the paradox of mind and matter -- by explaining how the perception of a finite image is dependent upon the contrasting infinitude of God. The theory holds that awareness is equal to a tension between existence and nonexistence, such that the self is illuminated to itself (becomes conscious) to the exact measure that it anticipates the infinitude of its own nonexistence. This "anticipation" is actually a "tendency (...) toward" a loss of structure in accordance with the second law of thermodynamics, a process that is restrained by effort and the consequent wholesness of an image. An understanding of how effort is linked to an image and how both are dependent upon neural systems inseparable from "infinity anticipation" sets the stage for a new worldview -- in which mind is seen as dependent upon the immanence of an infinite God. And because "mind" can be defined by the presence of an image which intrinsically restrains disorder, we see by extrapolation that the entire universe must be fundamentally personal, not objective as currently viewed by modern science. Book jacket. (shrink)

Infinity is an intriguing topic, with connections to religion, philosophy, metaphysics, logic, and physics as well as mathematics. Its history goes back to ancient times, with especially important contributions from Euclid, Aristotle, Eudoxus, and Archimedes. The infinitely large is intimately related to the infinitely small. Cosmologists consider sweeping questions about whether space and time are infinite. Philosophers and mathematicians ranging from Zeno to Russell have posed numerous paradoxes about infinity and infinitesimals. Many vital areas of mathematics (...) rest upon some version of infinity. The most obvious, and the first context in which major new techniques depended on formulating infinite processes, is calculus. But there are many others, for example Fourier analysis and fractals.In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectual exercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. (shrink)

This paper motivates and defends alethic nihilism, the theory that nothing is true. I first argue that alethic paradoxes like the Liar and Curry motivate nihilism; I then defend the view from objections. The critical discussion has two primary outcomes. First, a proof of concept. Alethic nihilism strikes many as silly or obviously false, even incoherent. I argue that it is in fact well-motivated and internally coherent. Second, I argue that deflationists about truth ought to be nihilists. Deflationists (...) maintain that the utility of the truth predicate is exhausted by its expressive role, and I argue that the truth predicate can still play this expressive role even if nothing is true. As such, deflationists do not stand to lose anything by accepting nihilism. Since they also stand to gain an elegant solution to the alethic paradoxes, on balance deflationists ought to be nihilists. (shrink)

In this book, best-selling author and mathematician Haim Shapira presents an introduction to mathematical theories which deal with the most beautiful concept ever invented by humankind: infinity. Written in clear, simple language and aimed at a lay audience, this book also offers some strategies that will allow readers to try their ability at solving truly fascinating mathematical problems. Infinity is a deeply counter-intuitive concept that has inspired many great thinkers. In this book we will meet many sages, both (...) familiar and unfamiliar: Zeno and Pythagoras, Georg Cantor and Bertrand Russell, Sofia Kovalevskaya and Emmy Noether, al-Khwarizmi and Euclid, Sophie Germain and Srinivasa Ramanujan. The world of infinity is inhabited by many paradoxes, and so is this book: Zeno paradoxes, Hilbert's "Infinity Hotel", Achilles and the gods paradox, the paradox of heaven and hell, the Ross-Littlewood paradox involving tennis balls, the Galileo paradox and many more. Aimed at the curious but non-technical reader, this book refrains from using any fearsome mathematical symbols. It uses only the most basic operations of mathematics: adding, subtracting, multiplication, division, powers and roots - that is all. But that doesn't mean that a bit of deep thinking won't be necessary and rewarding. Writing with humour and lightness of touch, Haim Shapira banishes the chalky pallor of the schoolroom and offers instead a truly thrilling intellectual journey. Fasten your seatbelt - we are going to Infinity, and beyond! (shrink)

I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that (...) is bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

In a letter drafted at age forty-eight, Jean-Jacques Rousseau confessed that he passed his days "vainly looking for solid attachments."1 Two years later, he again lamented that he had wasted much time pursuing attachments that "did not exist."2 At age fifty-eight, he confessed that he had "always felt some void."3 And, at the very end of his life, he still bemoaned that he had been cast "into the whirlwind of the world" only to discover that he "was not made to (...) live in it."4 In response to these persistent disappointments, Rousseau undertook a careful examination of his formative experiences. In so doing, he founded a science of consciousness.I argue that Rousseau's Confessions and epistolary novel, Julie, are a... (shrink)

It amazes children, as they try to count themselves out of numbers, only to discover one day that the hundreds, thousands, and zillions go on forever—to something like infinity. And anyone who has advanced beyond the bounds of basic mathematics has soon marveled at that drunken number eight lying on its side in the pages of their work. Infinity fascinates; it takes the mind beyond its everyday concerns—indeed, beyond everything—to something always more. Infinity makes even the infinite (...) universe seem small; yet it can also be infinitesimal. Infinity thrives on paradox, and it turns the simplest arithmetic on its head, with 1 seeming feasibly to equal 0, after all. Infinity defies common sense. The contemplation of it has relieved at least two great mathematicians of their sanity. Thoroughly readable and entirely accessible, science writer Brian Clegg's lively history explores infinity in its many intriguing facets, from its ancient origins to its place today at the heart of mathematics and science. He examines infinity's paradoxes and profiles the people who first grappled with and then defined and refined them, offering information, mystery, and poetry to conceive the inconceivable and define the indefinable. (shrink)

Our ability to select relevant information from the environment is limited by the resolution of attention – i.e., the minimum size of the region that can be selected. Neural mechanisms that underlie this limit and its development are not yet understood. Functional MRI was performed during an object tracking task in 7- and 11-year-old children, and in young adults. Object tracking activated canonical fronto-parietal attention systems and motion-sensitive area MT in children as young as 7 years. Object tracking performance (...) improved with age, together with stronger recruitment of parietal attention areas and a shift from low-level to higher-level visual areas. Increasing the required resolution of spatial attention – which was implemented by varying the distance between target and distractors in the object tracking task – led to activation increases in fronto-insular cortex, medial frontal cortex including anterior cingulate cortex (ACC) and supplementary motor area, superior colliculi, and thalamus. This core circuitry for attentional precision was recruited by all age groups, but ACC showed an age-related activation reduction. Our results suggest that age-related improvements in selective visual attention and in the resolution of attention are characterized by an increased use of more functionally specialized brain regions during the course of development. (shrink)

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse phenomena can be elucidated as both complete and incomplete, after which choice is the (...) border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or “observer”, or “user” to infinity implied by Henkin’s proposition as the only consistent ones as to infinity. (shrink)

Given any proposition, is it possible to have rationally acceptable attitudes towards it? Absent reasons to the contrary, one would probably think that this should be possible. In this paper I provide a reason to the contrary. There is a proposition such that, if one has any opinions about it at all, one will have a rationally unacceptable set of propositional attitudes—or if one doesn’t, one will end up being cognitively imperfect in some other manner. The proposition I am concerned (...) with is a self-referential propositional attitude ascription involving the propositional attitude of rejection. Given a basic assumption about what constitutes irrationality, and a few assumptions about the nature of cognitively ideal agents, a paradox results. This paradox is superficially like the Liar, but it is importantly different in that no alethic notions are involved at all. As such, it stands independent of the Liar and is not a ‘revenge’ version of it. After setting out the paradox I discuss possible responses. After considering several I argue that one is best off simply accepting that the paradox shows us something surprising and interesting about rationality: that some cognitive shortfall is unavoidable even for ideal agents. I argue that nothing disastrous follows from accepting this conclusion. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 61.2 (2000) 343-346 [Access article in PDF] The Paradox Topos Lisa Gorton As William Egginton points out, 1 when Dante and Beatrice step outside the cosmos, they step into another set of concentric spheres. 2 These surround our (supposedly) geocentric cosmos, and yet they center upon God. The image affronts our logic of space. If these concentric spheres encompass us, how can (...) they center upon something somewhere else? The notion seems to turn space inside-out. But Dante's spatial paradox is not in truth a paradox of space. Instead, it is a spiritual paradox, set in terms of space. When Dante and Beatrice step outside the cosmos, they step outside space into an old philosophical topos, a spatial image of God. For that reason, Dante's image is not really akin to the spatial paradoxes of modern astrophysics, although Egginton's comparison of the two does provoke some fascinating comments upon medieval concepts of space. Egginton assesses Dante's paradox in light of medieval debates about space, place, and God. However, he overlooks the most important source for Dante's paradox--the symbolic geometry of Neoplatonic philosophy. 3Neoplatonic philosophers use spatial forms to imagine spiritual relationships. They develop a symbolic geometry, a spatial language of thought, which borrows nothing from physical space but its shapes: the center, the circle, and concentric spheres. Neoplatonists use this symbolic geometry because it offers a spatial way to imagine the relationship between a cause and its effect, without importing time differences into that relationship. In this way they seek to avoid the problems of time and causality, which the creation-myth in Timaeus introduced into Plato's philosophy: how and when could a timeless being make time? 4 [End Page 343]Plotinus, the first Neoplatonist, uses symbolic geometry to design an "Intellectual Cosmos," which he treats as quite distinct from our physical cosmos. 5 He imagines the source of life as a center: The Highest cannot be divided and allotted... thus a centre is an independent unity; everything within the circle has its own term in the centre, and to the centre the radii each bring their own. (Enneads, V. 1. 11). The "Highest" exists as the center of this "Intellectual Cosmos." Plotinus treats a center's geometric qualities as metaphysical attributes. Like an undimensioned central point, the "Highest" does not exist in space or time, and yet space and time radiate from it as a circumference radiates from its center.The bare symbolic space of Plotinus's "Intellectual Cosmos" has emotional connotations. Plotinus holds that "the center of a circle is distinctively a point of rest" (Enneads II. 2. 1). Lesser beings must turn sequentially through all the angles their center holds altogether at once: There is, we may put it, something that is centre; about it, a circle of light shed from it; round centre and first circle alike, another circle, light from light; outside that again, not another circle of light but one which, lacking light of its own, must borrow. (Enneads, IV. 3. 17) In Plotinus's "Intellectual Cosmos" life emanates from the center in concentric spheres. 6Dante and Beatrice step into such an "Intellectual Cosmos" when they step outside the physical world. They step into a spatial image that stands outside time and space, at the spiritual center of the poem, when they step into spheres that center upon a light too bright to look upon. Dante wonders where that center could be, and Beatrice responds to his silent question: "The nature of the universe, which holds the center still and moves all else around it, begins here as if from its turning point" (XXVII, 106-9). Egginton comments, "Dante's description of this absolute center as the origin of all movement and force in the universe" is "an 'eerie' likeness of what he [the modern astrophysicist, Robert Osserman] terms the retro-verse." More relevantly, it also refers to the "Intellectual Cosmos" of Neoplatonic philosophy, where the "Highest" is a center of rest, the starting point of light, movement, and life. [End Page 344]Plotinus... (shrink)

If "being" is usually defined in terms of something roughly equivalent to the Aristotelian categories, are we forced to say that whatever escapes classification by these categories is to that extent beyond or above being? For instance, if "perfection" is used outside of the categories which apply to particular beings and is attributed to a first principle of all things, is "perfection" in this application so different from "perfection" as it applies within the categories that the two applications share (...) nothing in common? (shrink)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

This paper reconsiders certain of Kierkegaard's criticisms of Hegel's theoretical philosophy in the light of recent interpretations of the latter. The paper seeks to show how these criticisms, far from being merely parochial or rhetorical, turn on central issues concerning the nature of thought and what it is to think. I begin by introducing Hegel's conception of "pure thought" as this is distinguished by his commitment to certain general requirements on a properly philosophical form of inquiry. I then outline (...) Hegel's strategy for resolving a crucial problem he takes himself to face. For his account of the nature of thought depends upon the idea of a form of inquiry in which nothing whatsoever is presupposed; but this idea appears basically paradoxical inasmuch as the mere act of beginning to inquire in a certain way embodies an assumption about how it is appropriate to begin. Turning to Kierkegaard, I consider a key objection to the effect that Hegel's strategy for resolving this paradox depends on the incoherent idea of a purely reflexive act of thinking. Finally, I draw out some central features of the alternative account of "situated" thought and inquiry which Kierkegaard presents as distinctively Socratic. (shrink)

In reference to the paradoxical classification of art in Kafka’s Josefine story, religious aspects of the artist’s performances and their effect on the audience are scrutinized. The leading question is to what extent it evokes an allusion to the primal scene of divine revelation, following Stéphane Mosès’ commentary in his readings of the Bible. Josefine’s singing with a “nothing of a voice”, reduced to “the slightest of nullities”, the zero-point of signification, nevertheless affects an experience of presence and community (...) that touches the listeners’ entire being. From the perspective of the artist, her song recitals are manifestations of supreme art of which she claims a soteriological significance in respect of her audience who belongs in Kafka’s portrayal obviously to the Jewish people. Around such a tension between maximal restraint of an utterance (the voice onset of the aleph of the word anokhi/i) and all-encompassing meaning (the name of God and all Commandments whose acceptance is constitutive for the Jewish people) circle the theological exegeses of the Sinai-scene (esp. Ex. 20,1) in the tradition of Rashi and Maimonides, in modernity e.g. of Gershom Scholem in dispute with Walter Benjamin on Kafka. It will be shown that the depicted paradoxes of Josefine’s art performances unfold the mentioned tension as an original literary reflection on revelation and language under the conditions of modernity. (shrink)

Zeno of Elea's motion and infinity paradoxes, excluding the Stadium, are stated (1), commented on (2), and their historical proposed solutions then discussed (3). Their correct solution, based on recent conclusions in physics associated with time and classical and quantum mechanics, and in particular, of there being a necessary trade off of all precisely determined physical values at a time (including relative position), for their continuity through time, is then explained (4). This article follows on from another, more (...) physics orientated and widely encompassing paper entitled "Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity" (Lynds, 2003), with its intention being to detail the correct solution to Zeno's paradoxes more fully by presently focusing on them alone. If any difficulties are encountered in understanding any aspects of the physics underpinning the following contents, it is suggested that readers refer to the original paper for a more in depth coverage. (shrink)

The epistemic paradox of 'belief instability' has recently received notable attention from many philosophers. Understanding this paradox is very important because belief is a central notion of psychologically motivated semantic theories in philosophy, linguistics, and cognitive science, and this paradox poses serious problems for these theories. In this dissertation I criticize previous proposals and offer a new proposal, which I call a 'revision theory of belief'. ;My revision theory of belief is in many respects an application (...) of Gupta's and Belnap's revision theory of truth, which is proposed to deal with the Liar paradox. They argue that truth is a circular notion because Tarski biconditionals provide circular, partial definitions of truth. And this circularity of truth is the source of the Liar paradox. I offer a similar proposal for handling the paradox of belief instability. In particular, I argue that our notion of belief involved in the paradox of belief instability is circular, and that is why this paradox arises. ;If the notion of belief involved in the paradox is circular, we have to employ a hypothetical evaluation of the belief predicate, and if we do so through a revision process, we can avoid the contradiction which seems to be generated by some paradoxical belief sentences. But it is not easy to show that our notion of belief is circular. Especially, there is nothing like Tarski biconditionals in the case of belief. Nonetheless, I show in this dissertation that there is a certain schema, which I call the B-schema, whose instances can be taken as circular, partial definitions of a notion of belief, and it is that notion of belief which is involved in the paradox of belief instability. In this dissertation I also show that the revision approach can be extended for handling the Knower paradox. I argue that the Knower paradox arises because our notion of knowledge involved in the paradox is circular. (shrink)

This “fun, brain-twisting book... will make you think” as it explores more than 75 paradoxes in mathematics, philosophy, physics, and the social sciences (Sean Carroll, New York Times–bestselling author of Something Deeply Hidden) Paradox is a sophisticated kind of magic trick. A magician’s purpose is to create the appearance of impossibility, to pull a rabbit from an empty hat. Yet paradox doesn’t require tangibles, like rabbits or hats. Paradox works in the abstract, with words and concepts (...) and symbols, to create the illusion of contradiction. There are no contradictions in reality, but there can appear to be. In Sleight of Mind, Matt Cook and a few collaborators dive deeply into more than 75 paradoxes in mathematics, physics, philosophy, and the social sciences. As each paradox is discussed and resolved, Cook helps readers discover the meaning of knowledge and the proper formation of concepts—and how reason can dispel the illusion of contradiction. The journey begins with “a most ingenious paradox” from Gilbert and Sullivan’s Pirates of Penzance. Readers will then travel from Ancient Greece to cutting-edge laboratories, encounter infinity and its different sizes, and discover mathematical impossibilities inherent in elections. They will tackle conundrums in probability, induction, geometry, and game theory; perform “supertasks”; build apparent perpetual motion machines; meet twins living in different millennia; explore the strange quantum world—and much more. (shrink)

This article seeks to encounter the late ancient doctrine of creatio ex nihilo from a new angle, gently questioning both its distinctive Christianness and its inevitable alignment with a theology of absolute sovereignty. Franz Rosenzweig's idiosyncratic elaboration of this doctrine opens up new lines of thought: multiplying the “nothing” while also emphasizing its fecundity, he articulates not only a negative theology but also a negative anthropology and cosmology; insisting on the irreducible relationality of God, human, and world, he (...) also posits their non‐essentiality. Admittedly, early versions of creatio ex nihilo foreground the omnipotence of the creator and the goodness of the created world, and these concerns continue to be expressed by later proponents of the ex nihilo, Jewish as well as Christian. However, beginning in the fourth century there are hints of something new as well, something more Rosenzweigian—namely, a preoccupation with the nothing itself. What comes increasingly to the fore in these articulations of the ex nihilo is not mastery but mystery, not power but fragility, not separation but intimacy—and paradoxically so, given that the gap between creator and created has never before seemed to yawn so widely. This shift is manifest in the opening passages of Genesis Rabbah, I would suggest. It is also evident in the locus classicus of post‐Constantinian Christian ex nihilo doctrine, namely Athanasius's On the Incarnation of the Word of God. (shrink)

This paper responds to commentaries by Kaiserman and Magidor, and Hawthorne. The case of the infinite clowns can teach us several things.

In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveller from reaching his destination. In this paper it is argued, contra Priest and Yablo, that the paradox must be resolved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical form as Yablo’s Paradox. It is suggested that constructivism can serve as the basis of a common (...) solution to New Zeno and the paradoxes of truth, and a constructivist interpretation of Kripke’s theory of truth is given. (shrink)

The paradox of hedonism is the idea that intrinsically desiring nothing other than pleasure can prevent one from obtaining pleasure. In this article, I show how the paradox of hedonism can be used as the basis for an objection against hedonism about well-being, and one that is more defensible than has been commonly recognized. Moreover, I argue that the challenge presented by the paradox can be used to target not only hedonism about well-being, but also (...) desire satisfactionism and the hybrid theory. However, I argue that certain sophisticated versions of all three theories can escape it. (shrink)

The epistemic paradox of 'belief instability' has recently received notable attention from many philosophers. Understanding this paradox is very important because belief is a central notion of psychologically motivated semantic theories in philosophy, linguistics, and cognitive science, and this paradox poses serious problems for these theories. In this dissertation I criticize previous proposals and offer a new proposal, which I call a 'revision theory of belief'. -/- My revision theory of belief is in many respects an (...) application of Gupta's and Belnap's revision theory of truth, which is proposed to deal with the Liar paradox. They argue that truth is a circular notion because Tarski biconditionals provide circular, partial definitions of truth. And this circularity of truth is the source of the Liar paradox. I offer a similar proposal for handling the paradox of belief instability. In particular, I argue that our notion of belief involved in the paradox of belief instability is circular, and that is why this paradox arises. -/- If the notion of belief involved in the paradox is circular, we have to employ a hypothetical evaluation of the belief predicate, and if we do so through a revision process, we can avoid the contradiction which seems to be generated by some paradoxical belief sentences. But it is not easy to show that our notion of belief is circular. Especially, there is nothing like Tarski biconditionals in the case of belief. Nonetheless, I show in this dissertation that there is a certain schema, which I call the B-schema, whose instances can be taken as circular, partial definitions of a notion of belief, and it is that notion of belief which is involved in the paradox of belief instability. In this dissertation I also show that the revision approach can be extended for handling the Knower paradox. I argue that the Knower paradox arises because our notion of knowledge involved in the paradox is circular. (shrink)