'Mass terms', words like water, rice and traffic, have proved very difficult to accommodate in any theory of meaning since, unlike count nouns such as house or dog, they cannot be viewed as part of a logical set and differ in their grammatical properties. In this study, motivated by the need to design a computer program for understanding natural language utterances incorporating mass terms, Harry Bunt provides a thorough analysis of the problem and offers an original and (...) detailed solution. An extension of classical set theory, Ensemble Theory, is defined, and this provides the conceptual basis of a framework for the analysis of natural language meaning which Dr Bunt calls Two-level model-theoretic semantics. The validity of the framework is convincingly demonstrated by the formal analysis of a fragment of English including sentences with quantified and modified mass terms. Separate chapters of the book are devoted to an axiomatic definition of Ensemble Theory and a detailed discussion of its status as a mathematical formalism. (shrink)

The paper is concerned with the way in which “ontology” and “realism” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various (...) sections. (shrink)

This book explains, in lay terms, the surprisingly simple system of mathematical logic used in digital computer circuitry. Anecdotal in its style and often funny, it follows the development of this logic system from its origins in Victorian England to its rediscovery in this century as the foundation of all modern computing machinery. ONES AND ZEROS will be enjoyed by anyone who has a general interest in science and technology.

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation (...) of why, albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall. (shrink)

This paper is devoted to some generalizations of explanatory potential of connectionist approaches to theoretical problems of the philosophy of mind. Are considered both strong, and weaknesses of neural network models. Connectionism has close methodological ties with modern neurosciences and neurophilosophy. And this fact strengthens its positions, in terms of empirical naturalistic approaches. However, at the same time this direction inherits weaknesses of computational approach, and in this case all system of anticomputational critical arguments becomes applicable to the connectionst (...) models of mind. The last developments in the field of deep learning gave rich empirical material for cognitive sciences. Multilayered networks, mathematical models of associative dynamics of learning, self-organizing neuronets and all that allow to explain the principles of human conceptual organizing and after this to emulate these processes in computer systems. At all engineering achievements of this technology there is a traditional criticism from representatives of cognitive psychology who cannot accept a thesis about learning ability of a neuronet on the basis of redistribution of scales. Process of learning of natural intelligence, according to cognitive models, happens due to attraction of knowledge broadcast in a symbolical form (mental representations, concepts) at the expense of the systems of output knowledge expressed in the propositional contents. Some philosophical aspects of «neural metaphor» in modern cognitive sciences create the problem field which demands comprehensive understanding, the first step towards which is taken in this work. (shrink)

Recently discovered correspondence from Oskar Becker to Hermann Weyl sheds new light on Weyl's engagement with Husserlian transcendental phenomenology in 1918-1927. Here the last two of these letters, dated July and August, 1926, dealing with issues in the philosophy of mathematics are presented, together with background and a detailed commentary. The letters provide an instructive context for re-assessing the connection between intuitionism and phenomenology in Weyl's foundational thought, and for understanding Weyl's term ‘symbolic construction’ as marking his own considered (...) position in the foundational controversy of the 1920s. In addition, they reveal Weyl's hitherto unknown objections to Becker's detailed attempt (Mathematische Existenz, 1927) to ground the transfinite phenomenologically. (shrink)

Abstractabstract:This paper revisits debates on a tension in Cassirer's philosophy of culture. On the one hand, Cassirer describes a plurality of symbolic forms and claims that each needs to be assessed by its own internal standards of validity. On the other hand, he ranks the symbolic forms in terms of a developmental hierarchy and states that one form, mathematical natural science, constitutes the highest achievement of culture. In my paper, I do not seek to resolve this tension. Rather, (...) I aim to arrive at a better understanding of how it arises, and of the different options that it presents for understanding the development of culture. I discuss three recent attempts at resolving the tension, put forward by Sebastian Luft, Samatha Matherne, and Simon Truwant, respectively. Based on a reconstruction of Cassirer's system of symbolic forms that centralizes the concept of function, I show that the most promising of these attempts, formulated by Truwant, is not successful. I then turn to Cassirer's philosophy of the cultural sciences, the implications of which for the present problem have not yet been sufficiently explored. I argue that in this context, Cassirer develops the contours of an alternative to the function-based view of cultural development. I conclude that this alternative does not resolve the tension either, but that it allows for a reconceptualization of the teleology of culture as open. (shrink)

Ignoring primitive terms leads to an infinite regress. The alternative is to account for an intuitive understanding into the meaning of such terms. The current investigation proceeds on the basis of an idea of the structure of the various modes of being within which concrete entities function. Examples of primtive terms are given from disciplines such as mathematics, physics and logic and they are related to the general idea of a modal aspect. It is argued that (...) primitive terms are not isolated but reveal their meaning only through their interconnections with other primitive terms that are embedded in other modal aspects. However, although primitive terms are found within the various aspects, the meaning of an aspect only comes to expression through its coherence with other aspects, evinced in modal analogies that are qualified by the core meaning of an aspect. There appears to be two options, either reduce what is irreducible or merely provide synonymous terms for given primitives. The former happens when other unique terms are used to define a specific one and the latter when the attempted ‘definitions’ revert to terms with which the original terms could be meaningfully replaced. It is been pointed out that the coherence between primitives invites every academic discipline to account for the meaning attached to the analogies of primitive terms it is employing, without exploring this additional theme any further. (shrink)

A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.

The term “historical epistemology” can be read in two different ways: (1) as referring to a program of ‘historicizing’ epistemology, in the sense of a critique of traditional epistemology’s tendency to gloss over historical context, or (2) as a manifesto of ‘epistemologizing’ history, i.e. as a critique of radical historicist and relativist approaches. In this paper I will defend a position in this second sense. I show that one can account for the historical development and diversity of science without disavowing (...) the relevance of a (normatively understood) epistemology and without denying the existence of human cognitive universals across historical and cultural differences. In support of my thesis, I draw on cognitive scientific research on causal and symbolic cognition, arguing that causal understanding constitutes a basic part of science, which, in the course of its development, becomes more and more superimposed by a culturally and historically variable symbolic superstructure. (shrink)

In the field of mathematics education, one of the main questions remaining under debate is whether students’ development of mathematical reasoning and problem-solving is aided more by solving tasks with given instructions or by solving them without instructions. It has been argued, that providing little or no instruction for a mathematical task generates a mathematical struggle, which can facilitate learning. This view in contrast, tasks in which routine procedures can be applied can lead to mechanical repetition with (...) little or no conceptual understanding. This study contrasts Creative Mathematical Reasoning (CMR), in which students must construct the mathematical method, with Algorithmic Reasoning (AR), in which predetermined methods and procedures on how to solve the task are given. Moreover, measures of fluid intelligence and working memory capacity are included in the analyses alongside the students’ math tracks. The results show that practicing with CMR tasks was superior to practicing with AR tasks in terms of students’ performance onpracticed test tasksandtransfer test tasks. Cognitive proficiency was shown to have an effect on students’ learning for both CMR and AR learning conditions. However, math tracks (advanced versus a more basic level) showed no significant effect. It is argued that going beyond step-by-step textbook solutions is essential and that students need to be presented with mathematical activities involving a struggle. In the CMR approach, students must focus on the relevant information in order to solve the task, and the characteristics of CMR tasks can guide students to the structural features that are critical for aiding comprehension. (shrink)

Every branch of mathematics has its subject matter, and one of the distinguishing features of logic is that so many of its fundamental objects of study are rooted in language. The subject deals with terms, expressions, formulas, theorems, and proofs. When we speak about these notions informally, we are talking about things that can be written down and communicated with symbols. One of the goals of mathematical logic is to introduce formal definitions that capture our intuitions about (...) such objects and enable us to reason about them precisely. At the most basic level, syntactic objects can be viewed as strings of symbols. For concreteness, we can identify symbols with particular set-theoretic objects, but for most purposes, it doesn't matter what they are; all that is needed is that they are distinct from one another. (shrink)

Departing from and closing with reflections on issues regarding teaching practices of philosophy of mathematics, I propose a comparison between the main features of the Leibnizian notion of symbolic knowledge and some passages from the Tractatus on arithmetic. I argue that this reading allows (i) to shed a new light on the specificities of the Tractarian definition of number, compared to those of Frege and Russell; (ii) to highlight the understanding of the nature of mathematical knowledge as symbolic (...) or formal knowledge that Wittgenstein mobilizes in his book; (iii) to offer reasons for the claim that Wittgenstein can be considered the philosopher of mathematical practice avant la lettre. The paper ends with an overview, a return to the initial reflection on the connections between research and teaching, and a defense of the reading key used here in terms of its potential for the research in philosophy of mathematics. (shrink)

Architecture of Mathematics describes the logical structure of Mathematics from its foundations to its real-world applications. It describes the many interweaving relationships between different areas of mathematics and its practical applications, and as such provides unique reading for professional mathematicians and nonmathematicians alike. This book can be a very important resource both for the teaching of mathematics and as a means to outline the research links between different subjects within and beyond the subject. Features All notions and properties are introduced (...) logically and sequentially, to help the reader gradually build understanding. Focusses on illustrative examples that explain the meaning of mathematical objects and their properties. Suitable as a supplementary resource for teaching undergraduate mathematics, and as an aid to interdisciplinary research. Forming the reader's understanding of Mathematics as a unified science, the book helps to increase his general mathematical culture. (shrink)

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. This grand narrative began with the exhibition of quantitative laws that these heroes, Galileo and Newton for example, had disclosed: the law of falling bodies, according to which the speed of a falling body is proportional to the square of the time that has elapsed since the beginning of its fall; the (...) law of gravitation, according to which two bodies are attracted to one another in proportion to the sum of their masses and in inverse proportion to the square of the distance separating them -- according to his own preferences, each narrator added one or two quantitative laws of this kind. The essential feature was not so much the examples that were chosen, but, rather, the more or less explicit theses that accompanied them. First, mathematization would be taken as the criterion for distinguishing between a qualitative Aristotelian philosophy and the new quantitative physics. Secondly, mathematization was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena. Finally, it goes without saying that, from a social point of view, the evolution of the sciences was apprehended through what has been aptly called the "relay runner model," according to which science progresses as a result of individual discoveries. Grand narratives such as this are perhaps simply fictions doomed to ruin as soon as they are clearly expressed. In any case, the very assumption on which this grand narrative relies can be brought into question: even in the canonical domain of mechanics, the relevant epistemological units crucial to understanding the dynamics of the Scientific Revolution are perhaps not a few laws of motion, but a complex set of problems embodied in mundane objects. Moreover, each of the theses just mentioned was actually challenged during the long period of historiographical reappraisal, out of which we have probably not yet stepped. Against the sharp distinction between a qualitative Aristotelian philosophy and the new quantitative physics, numerous studies insist that Rome wasn't built in a day, so to speak. Since Antiquity, there have always been mixed sciences; the emergence of pre-classical mechanics depends on both medieval treatises and the practical challenges met by Renaissance engineers. It is indeed true that, for Aristotle, mathematics merely captures the superficial properties of things, but the Aristotelianisms were many during the Renaissance and the Early Modern period, with some of them being compatible with the introduction of mathematics in natural philosophy. In addition, the gap between the alleged program of mathematizing nature and its effective realization was underlined as most natural phenomena actually escaped mathematization; at best they were enrolled in what Thomas Kuhn began to rehabilitate under the appellation of the "Baconian sciences," i.e., empirical investigations aiming at establishing isolated facts, without relating them to any overarching theory. Hence, mathematization of nature cannot pretend to capture a historical fact: at most, it expresses an indeterminate task for generations to come. On top of these first two considerations, and against the thesis of the neutrality of the mathematical language, it was urged that mathematics is not "only a language" and that, exactly as other symbolic means or cognitive tools, it has its own constraints. For example, it has been thoroughly explained that the Euclidean theory of proportions both guides and frustrates the Galilean analysis of motion; its shortages were particularly clear with respect to the expression of continuity, which is crucial in the case of motion. Consequently, when calculus was invented and applied to the analysis of motion, it was not a transposition that left things as they stood. Even more clearly than in the case of a translation from one natural language to another, the shift from one symbolic language to another entails that certain possibilities are opened while others are closed. The cognitive constraints imposed by established mathematical theories, as seen in the theory of proportions or calculus, were not the only ones to be studied in relation to mathematization. Certain schemes dependent on the grammar of natural languages, e.g., the scheme of contrariety, or certain symbolic means of representation, e.g. geometrical diagrams and numerical tables, were also subject to such scrutiny. Lastly, it was insisted that, even if we concede the existence of scientific geniuses, mathematics is largely produced by intellectual communities and embedded within social practices. More attention was consequently paid to the forms of communication in given mathematical networks, or to the teaching of the discipline in, for example, Jesuit colleges and universities. The set of mathematical practices specific to specialized craftsmen, highly-qualified experts and engineers began to be studied in its own right. All these reflections may have helped us change our perspectives on the question of mathematization. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real history, and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. Consequently, before presenting the essays of this fascicle, I would like to say a few words in favor of a renewed study of the forms of mathematization in the history of the early sciences. (shrink)

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative (...) accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Worth of Persons by James FranklinLouis GroarkeFRANKLIN, James. The Worth of Persons, New York: Encounter Books, 2022. 272 pp. Cloth, $30.99In The Worth of Persons, James Franklin, the well-known Aristotelian mathematician, sets out to provide an account of the very first principles of ethics and morality. Franklin argues that morality begins with an acknowledgment of the intrinsic worth of human persons, understood as beings possessing “dignity” or (...) “absolute inner worth.” As it turns out, we recognize naturally, without epistemic complication, that human beings have innate moral value.Although Franklin does not use the term “personalism” and does not discuss that specific philosophical tradition, he traces his own account to a familiar source: Kant’s personhood formulation of the categorical imperative. He has, however, little use for Kant’s rigid absolutism and prefers something much closer to a natural law formulation of ethical obligation, in large part because it allows for casuistry, a negotiation of the messy circumstances of everyday life that confront us with moral dilemmas and competing duties requiring some kind of moral calculation.But Franklin’s book is not so much about normative ethics. It is an exercise in metaethics, an attempt to make sense of morality as a root cause of value in the world. Franklin wants to secure an epistemological basis from which we can “deduce” moral principles. He is particularly impressed with contemporary discussion of “supervenience” or “metaphysical grounding” as a bridge over the alleged is–ought gap that has preoccupied analytic philosophers following in the footsteps of David Hume [End Page 349], G. E. Moore, and J. L. Mackie. Franklin shows, very effectively, that similar “gaps” can be found in logic, mathematics, linguistics, computer science, philosophy of mind, metaphysics, and so forth. As examples of supervenience, he mentions the way necessary mathematical concepts supervene on the physical, the way the meanings of words are added to spoken or written symbols, the way logical inference arises from premises and conclusion in an argument, the way social organization emerges from a collection of individuals, and the way the identity of a statue of Hercules arises from its disposition of physical parts. The point is that an empiricist fundamentalism that reduces reality to perceptible facts without any higher (or “emergent”) identities provides an overly narrow account of the metaphysical content of human experience.If ethical value supervenes on some nonethical base, it does not follow that moral judgments and obligations are a human fabrication without any grounding in the world. If we can intellectually access the moral worth of human beings the way, for example, we intellectually access necessary mathematical concepts, we can deduce moral obligations from the worth of persons in a way that preserves the epistemic status of morality against noncognitive challenges. Once we recognize that individual humans have worth (because of the potentialities inherent in their natures), we can, for example, infer moral axioms such as “human beings must be respected” and, further, infer that murder is morally wrong because it involves the wanton destruction of human beings. Franklin does not focus on the normative content of specific actions, duties, obligations, or virtues. His basic point is that moral knowledge is possible once we rationally access (through understanding) the first principle of morality: that human beings are (to use Kantian terminology) ends in themselves, beings that rational agents will value as loci of dignity and moral value.This is a thorough treatment. Franklin is moderate, cautious, careful, diligent; he adds together technical points in present-day academic discussion to produce a well-rounded worldview. He writes in an accessible manner, with good examples and flashes of wit throughout, referring to a wide range of issues and authors: Pseudo-Dionysius, Aristotle, Aquinas, Kant, Husserl, Moore, Wolterstorff, Mackie, Oderberg, MacIntyre, Midgley, Finnis, Korsgaard, Dworkin, Baier, Williams, Singer, Murdoch, and so on.The book includes eight chapters that discuss contrary positions, ethical concepts, supervenience, moral personhood, aesthetic worth, the value of animals and ecosystems, moral epistemology, and metaphysical systems compatible with a realist view of morality. There is a useful index, a modest bibliography, and a full section of notes, which contains many further references.The treatment seems a... (shrink)

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation. In this essay, (...) we describe an approach to the philosophy of mathematics in which it is an important task to understand the roles of our ontological posits and assess the extent to which they enable us to achieve our mathematical goals. We use the history of Dirichlet’s theorem to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment. We also use these considerations to illuminate the formal treatment of functions and objects in Frege’s logical foundation, and we argue that his philosophical and logical decisions were influenced by many of the same factors. (shrink)

In mathematics education, students' difficulties with negative numbers are well known. To explain these difficulties, researchers traditionally refer to obstacles raised by the concept of NEGATIVE NUMBERS itself throughout its historical evolution. In order to improve our understanding, I propose to take into consideration another point of view, based on Vygotsky's principles, which define a strong relationship between signs such as language or symbols and cognitive development. I show how it is of great interest to consider students' difficulties (...) with negatives in the light of the use of symbols, in particular the minus sign. I describe the results of an empirical study about students' strategies in solving arithmetical equations with negatives. My analysis shows that the problems raised by the presence of negatives in the equations are attributable to students' very restricted and often inadequate use of the minus sign, which prevents them from finding a negative solution or making sense of it. (shrink)

Bertrand Russell’s _Principles of Mathematics_ (1903) gives rise to several interpretational challenges, especially concerning the theory of denoting concepts. Only relatively recently, for instance, has it been properly realised that Russell accepted denoting concepts that do not denote anything. Such empty denoting concepts are sometimes thought to enable Russell, whether he was aware of it or not, to avoid commitment to some of the problematic non-existent entities he seems to accept, such as the Homeric gods and chimeras. In this paper, (...) I argue first that the theory of denoting concepts in _Principles of __Mathematics_ has been generally misunderstood. According to the interpretation I defend, if a denoting concept shifts what a proposition is about, then the aggregate of the denoted terms will also be a constituent of the proposition. I then show that Russell therefore could not have avoided commitment to the Homeric gods and chimeras by appealing to empty denoting concepts. Finally, I develop what I think is the best understanding of the ontology of _Principles of __Mathematics_ by interpreting some difficult passages. (shrink)

First published in the most ambitious international philosophy project for a generation; the _Routledge Encyclopedia of Philosophy_. _Logic from A to Z_ is a unique glossary of terms used in formal logic and the philosophy of mathematics. Over 500 entries include key terms found in the study of: * Logic: Argument, Turing Machine, Variable * Set and model theory: Isomorphism, Function * Computability theory: Algorithm, Turing Machine * Plus a table of logical symbols. Extensively cross-referenced to help (...) comprehension and add detail, _Logic from A to Z_ provides an indispensable reference source for students of all branches of logic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 38.4 (2004) 71-80 [Access article in PDF] The Arts and the Creation of Mind: Eisner's Contributions to the Arts in Education Arthur Efland Professor Emeritus, Department of Art Education The Ohio State University In the last four years at least three books in arts education have dealt with the subject of cognition in relation to the arts. I refer to Charles Dorn's Mind in (...) Art, my book Art and Cognition, and Elliot Eisner's The Arts and the Creation of Mind, with the publication of the latter providing the occasion for this article.1 Each of these texts makes its case for the arts in education on somewhat similar grounds. All three share the view that the creation and understanding of works of art, though endowed with feeling and emotion, are nevertheless cognitive endeavors. But why have so many books appeared on the topic of cognition at this time? To answer this question it becomes necessary to describe the changes in our understanding of cognition and the problems these have created for the arts in education.Once cognition was the term used to designate propositional thinking with verbal and numerical symbols, whereas the current sense of the term embraces all forms of thought including mental images obtained through perception. In short it includes all forms of sentience by which the human organism comes to know itself and its environment. This change was to have a major impact upon curriculum reform initiatives in all studies but it was to have a revolutionary impact upon education in the arts for reasons explained later in this essay. However, an adequate story of cognition in the arts cannot be told without a discussion of Eisner's contributions to this discourse. The Waning of Behaviorism For the first half of the twentieth century the psychology that dominated educational practice was behaviorism. Its proponents included psychologists like Edward L. Thorndike, James B. Watson, and B.F. Skinner.2 Each strove to create a science of psychology that would match the rigor of physics or chemistry, where causes in relation to effects would be carefully observed and measured, making the explanation, prediction, and control of phenomena possible, especially those behaviors that fall under the control of the classroom teacher. The formation of connections called stimulus-response (S-R) bonds had become the molar unit to explain learning, much [End Page 71] like the way that differences in the structure of the atom was used to explain the behavior of the chemical elements of the Periodic Table. This bond between environmental stimuli and the organism's response was seen as the elementary unit of behavior while the IQ had become its measure. Invisible notions like mind and imagination were banished to metaphysical philosophy, outside the legitimate bounds of science.Though behaviorists never denied the existence of higher order thinking that is, abstract, sense-free thinking as found in pure mathematics, they were unable to explain how these human capacities emerge through the accumulation of such rudimentary stimulus-response events. Nevertheless, they recognized that humans do have a capacity for thinking that relies heavily upon symbols like words, numbers and the like. Moreover, certain subjects that require reason and logic, like math and theoretical physics, would be inconceivable without such symbols and the rules by which these are manipulated. By contrast it was assumed that the arts, which relied on sensory images, either did not employ this type of propositional thought, or did so to a far lesser degree. Thus the arts could be dispatched to the noncognitive or affective domains with little or no reduction in the learner's cognitive capacities. Indeed, these noncognitive realms, which gave play to feelings and emotions, were often seen as inimical to the acquisition of rational powers of thought. Intellectual power as it was then understood, required that feelings and emotions be brought under the control of reason.As long as behaviorism served as the dominant paradigm, the school curriculum could be divided into two or three broad divisions... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 38.4 (2004) 71-80 [Access article in PDF] The Arts and the Creation of Mind: Eisner's Contributions to the Arts in Education Arthur Efland Professor Emeritus, Department of Art Education The Ohio State University In the last four years at least three books in arts education have dealt with the subject of cognition in relation to the arts. I refer to Charles Dorn's Mind in (...) Art, my book Art and Cognition, and Elliot Eisner's The Arts and the Creation of Mind, with the publication of the latter providing the occasion for this article.1 Each of these texts makes its case for the arts in education on somewhat similar grounds. All three share the view that the creation and understanding of works of art, though endowed with feeling and emotion, are nevertheless cognitive endeavors. But why have so many books appeared on the topic of cognition at this time? To answer this question it becomes necessary to describe the changes in our understanding of cognition and the problems these have created for the arts in education.Once cognition was the term used to designate propositional thinking with verbal and numerical symbols, whereas the current sense of the term embraces all forms of thought including mental images obtained through perception. In short it includes all forms of sentience by which the human organism comes to know itself and its environment. This change was to have a major impact upon curriculum reform initiatives in all studies but it was to have a revolutionary impact upon education in the arts for reasons explained later in this essay. However, an adequate story of cognition in the arts cannot be told without a discussion of Eisner's contributions to this discourse. The Waning of Behaviorism For the first half of the twentieth century the psychology that dominated educational practice was behaviorism. Its proponents included psychologists like Edward L. Thorndike, James B. Watson, and B.F. Skinner.2 Each strove to create a science of psychology that would match the rigor of physics or chemistry, where causes in relation to effects would be carefully observed and measured, making the explanation, prediction, and control of phenomena possible, especially those behaviors that fall under the control of the classroom teacher. The formation of connections called stimulus-response (S-R) bonds had become the molar unit to explain learning, much [End Page 71] like the way that differences in the structure of the atom was used to explain the behavior of the chemical elements of the Periodic Table. This bond between environmental stimuli and the organism's response was seen as the elementary unit of behavior while the IQ had become its measure. Invisible notions like mind and imagination were banished to metaphysical philosophy, outside the legitimate bounds of science.Though behaviorists never denied the existence of higher order thinking that is, abstract, sense-free thinking as found in pure mathematics, they were unable to explain how these human capacities emerge through the accumulation of such rudimentary stimulus-response events. Nevertheless, they recognized that humans do have a capacity for thinking that relies heavily upon symbols like words, numbers and the like. Moreover, certain subjects that require reason and logic, like math and theoretical physics, would be inconceivable without such symbols and the rules by which these are manipulated. By contrast it was assumed that the arts, which relied on sensory images, either did not employ this type of propositional thought, or did so to a far lesser degree. Thus the arts could be dispatched to the noncognitive or affective domains with little or no reduction in the learner's cognitive capacities. Indeed, these noncognitive realms, which gave play to feelings and emotions, were often seen as inimical to the acquisition of rational powers of thought. Intellectual power as it was then understood, required that feelings and emotions be brought under the control of reason.As long as behaviorism served as the dominant paradigm, the school curriculum could be divided into two or three broad divisions... (shrink)

The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to their diverse backgrounds, there are several connections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce’s development of abductive reasoning [39], Toulmin’s argumentation layout [52], Lakatos’s theory of reasoning in mathematics [23], Pollock’s notions of counterexample [44], and (...) argumentation schemes constructed by Walton et al. [54], and explore some connections between, as well as within, the theories. For instance, we investigate Peirce’s abduction to deal with surprising situations in mathematics, represent Pollock’s examples in terms of Toulmin’s layout, discuss connections between Toulmin’s layout and Walton’s argumentation schemes, and suggest new argumentation schemes to cover the sort of reasoning that Lakatos describes, in which arguments may be accepted as faulty, but revised, rather than being accepted or rejected. We also consider how such theories may apply to reasoning in mathematics: in particular, we aim to build on ideas such as Dove’s [13], which help to show ways in which the work of Lakatos fits into the informal reasoning community. (shrink)

The philosophy of mathematics plays a vital role in the mature philosophy of Charles S. Peirce. Peirce received rigorous mathematical training from his father and his philosophy carries on in decidedly mathematical and symbolic veins. For Peirce, math was a philosophical tool and many of his most productive ideas rest firmly on the foundation of mathematical principles. This volume collects Peirce’s most important writings on the subject, many appearing in print for the first time. Peirce’s determination to (...) understand matter, the cosmos, and "the grand design" of the universe remain relevant for contemporary students of science, technology, and symbolic logic. (shrink)

To illustrate the view that a speaker can have a partial understanding of a concept, Burge uses the example of Leibniz’s and Newton’s understanding of the concept of derivative. In a recent article, Sheldon Smith criticizes this example and maintains that Newton’s and Leibniz’s use of their derivative symbols does not univocally determine their references. The present article aims at challenging Smith’s analysis. It first shows that Smith misconstrues Burge’s position. It second suggests that the philosophical lessons (...) one should draw from the practice of the historians of philosophy are more ambivalent than what Smith thinks. (shrink)

References to God. Some Remarks by Wittgenstein on Religion in the Years 1949 – 51. After a brief overview of Wittgenstein's stock of remarks on the subject of religion from 1949 – 1951, this article will focus on two particular points: supposedly nonsensical conceptions of God, for instance in the context of proofs of God, definitions of the term ”God” by hinting at something. Connections between and both systematically and exegetically within the framework of Wittgenstein's remarks are made.Ich danke Anja (...) Weiberg für sehr hilfreiche Kommentare. (shrink)

The paper deals with Kuhn’s and Lakatos’s ideas related to the so-called “historical turn” and its application to the philosophy of mathematics. In the first part the meaning of the term “postpositivism” is specified. If we lack such a specification we can hardly discuss the philosophy of science that comes “after postpositivism”. With this end in view, the metaphor of “generations” in the philosophy of science is used. It is proposed that we restrict the use of the term “post-positivism” to (...) two and only two philosophical “generations”: the one to which Kuhn, Lakatos and Feyerabend belong, and the previous “generation” to which Wittgenstein, Polanyi, Popper and Quine (as well as the major part of logical positivists) belong. From this point of view, Bloor, Latour, Pickering, Daston and Galison belong to the “third generation” which represents the philosophy of science “after post-positivism”. The characteristic feature of post-positivism is the combination of decisive impact of logical positivism and its severe criticism. This combination inevitably makes post-positivism a transitional form in the philosophy of science. In the second part the contribution of the “big four” of post-positivist philosophers (Popper, Kuhn, Lakatos, and Feyerabend) to the radical change in the philosophy of mathematics in the second half of the 20th century is analyzed. Primarily, they shifted philosophical interest from the logical analysis of formal systems to the historical dynamics of informal mathematics. They also reconsidered the sharp opposition between mathematics and the physical sciences. However, the transitional character of their philosophy manifests itself both in their treatment of mathematics and their way of understanding history. On the one hand, their “heritage” is ambiguous, on the other hand, it opens new perspectives. Neither Kuhn, nor Lakatos, have eliminated completely the methodological barrier positing the fundamental heterogeneity of mathematics and natural science. Neither Lakatos, nor Kuhn, adhered to the viewpoint of relentless historicism. Nevertheless, it is their work that has made these options open for today’s historians and philosophers of science, even for philosophers of mathematics. (shrink)

In this paper, I discuss the social philosopher Pierre Bourdieu’s concept of habitus, and use it to locate and examine dispositions in a larger constellation of related concepts, exploring their dynamic relationship within the social context, and their construction, manifestation, and function in relation to classroom mathematics practices. I describe the main characteristics of habitus that account for its invisible effects: its embodiment, its deep and pre-reflective internalization as schemata, orientation, and taste that are learned and yet unthought, and are (...) subsumed by our practices, which we understand as something that “goes without saying.” I also propose that, similarly to Bourdieu’s concept of linguistic habitus, a math habitus is made up of a complex intertwining of collective and individual histories that turn into “nature,” which structure all individual and collective action and inform mathematical classroom practice. I suggest that individual math dispositions may be liable to reconstruction through the reconstruction of the collective math habitus, which follows from opening spaces for dialogue, problematization and reconstruction of the unthought categories of the doxa. This requires that students acquire new concrete and symbolic means with which to challenge their current sense of mathematics as a discipline, and mathematical practice tout court. Finally, I argue that community of inquiry, employed as a pedagogical model, provides an avenue for both: for opening those spaces for reflective dialogical inquiry into concepts and questions whose meanings and references have so far been taken for granted, and for acquiring critical thinking and dialogical skills and dispositions that are a necessary means for participating in such reflective inquiry that offers significant promise for reconstructing individual and collective habitus in school settings. (shrink)

Kant's special metaphysics is intended to provide the a priori foundation for Newtonian science, which is to be achieved by exhibiting the a priori content of Newtonian concepts and laws. Kant envisions a two-step mathematical construction of the dynamical concept of matter involving a geometrical construction of matter’s bulk and a symbolic construction of matter’s density. Since Newton himself defines quantity of matter in terms of bulk and density, there is no reason why we shouldn’t interpret Kant’s Dynamics (...) as a defence of a Newtonian concept of matter. When Kant’s reasoning is understood in relation to his criteria for mathematical construction, it is possible to maintain that matter theory is central to the Metaphysical Foundations, but that this does not undermine Kant’s stated aim of giving an a priori foundation for Newtonian science. (shrink)

French scientist Henri Poincaré is one of the leading thinkers in the field of philosophy of science with his determinations on science and scientific activity. Poincare's understanding of science is expressed as conventionalist because he asserts that all sciences, including mathematics, consist of conventions and definitions. In this article, Poincare's views on how scientists should evaluate the data obtained from observation and experiment in their studies and the hypotheses that describe the relationships between these data are discussed. In the (...) article, it is also thought that especially Poincare's claims about the structure of science and the nature of scientific activity are important in terms of understanding some scientific views that emerged in the second half of the twentieth century. (shrink)

Writing, the exigency of writing: no longer the writing that has always (through a necessity in no way avoidable) been in the service of the speech or thought that is called idealist (that is to say, moralizing), but rather the writing that through its own slowly liberated force (the aleatory force of absence) seems to devote itself solely to itself as something that remains without identity, and little by little brings forth possibilities that are entirely other: an anonymous, distracted, deferred, (...) and dispersed way of being in relation, by which everything is brought into question – and first of all the idea of God, of the Self, of the Subject, then of Truth and the One, then finally the idea of the Book and the Work so that this writing (understood in its enigmatic rigor), far from having the Book as its goal rather signals its end: a writing that could be said to be outside discourse, outside language. – Maurice Blanchot ( The Infinite Conversation , xi) The Compendium * Beginning with the idea and practice of writing, and moving to the subject or self, then to truth and the One, and arriving at the Book and the Work, Blanchot’s words reflect the trajectory of the following work of writing. True to the embeddedness of the subject in the work of writing, the term “Compendium” has been a metonym for me over the past few years as I have thought about the concept of totality alongside its expression in figures such as the One, the Whole, or the All. In the following I aim to share some of the content of this metonym, and to enrich it by making some distinctions. 1 Here the term “Compendium” will refer to a concept of totality that is ontological (pertaining to being, and the copula) and also textual (pertaining to the symbolic, and the signifier-signified relationship). The Compendium is a figure for thinking the world as a Book, in the broadest sense—an approach to reality that is by no means new, but one that is due for renewal. 2 In partial answer to the question of the Compendium we will say that the word ‘Compendium’ stands in for expressions which seek to totalize, or concepts which seek to approach the concept of ‘everything,’ particularly when these expressions are bound up in the question of the Book (both the book as a physical object and the Book as a metaphysical figure: a way of thinking about the work in and of writing). The question of the Compendium is strongly associated with the physical and metaphysical form of the Book, like the mysterious and symbolic books which are opened in the Biblical book of Revelation, the book of life and the book of death, which together constitute an important couplet. 3 Parataxis & Hypotaxis There are two helpful distinctions that will bring us closer to an answer to the question “What is a Compendium?” The first distinction is between two figures in and for writing: parataxis and hypotaxis . Typically paired in contrast to one another as literary techniques, with parataxis indicating a side-by-side placement of textual elements and hypotaxis referring to subordinate arrangements of textual elements, the two figures can be understood as having a philosophical significance in addition to their practical function as devices for writers. For us these two terms will remain between philosophical theory and writing practice, and serve as a perspective for our writing and creation of texts. Here I take texts to refer to anything from the concrete written words in a book, to the ontological text of the world that we experience. Parataxis , as a figural way of thinking about the ontological structure of texts, refers to texts which are tightly woven and interdependent – texts within which each sentence bears the weight of the entire work. Parataxis often involves repetition, great density, and fragility. One of the best examples of this sort of text is Theodor Adorno’s posthumous magnum opus , Aesthetic Theory . In addition to the text itself, the translation history of the book may also help us to better understand parataxis and its relation to hypotaxis . The final published version of Aesthetic Theory , in German, was a text that Adorno intended to revise and rewrite, but this intention was never realized because of his untimely death. Where the German text of Aesthetic Theory certainly exemplifies the concept of parataxis , the first English edition took this densely woven text and carved Adorno’s lengthy paragraphs and lengthy sentences into manageable ‘bite-sized’ pieces of English text. The first translator took further liberty and inserted headings and new paragraph breaks where there were none in the original. Aesthetic Theory was eventually retranslated by Robert Hullot-Kentor and is now available in a form that is much more faithful to the original work. 4 The pertinent idea that this translation history points to is the distinction between parataxis and hypotaxis . Where parataxis describes texts which are repetitive, densely woven, and often fragile, hypotaxis describes texts which are hierarchical and in which the primary relation is linear – the latter of which is very similar to the first English translation of Aesthetic Theory . On the other hand, the original German text that Adorno wrote was very dense, often repetitive, and contained long sections of text unbroken by paragraphs or headings. This example of parataxis was then turned towards hypotaxis through the initial translation which took a text that was, in many ways, nonhierarchical and nonlinear, and artificially subjected it to a hierarchy that was not its own (and here I take the word of Fredric Jameson who comments on the two translations in a note at the beginning of his book Late Marxism ). 5 The point here is not about translation, but rather the distinction between parataxis and hypotaxis precisely as they are figures for discourse, written or otherwise. The concept of parataxis looks far more postmodern and rhizomic than the concept of hypotaxis which remains very modern and arborescent. This is a more figural way of talking about the distinction. However, if we wanted to take a more precise and analytical approach we could say that on the level of form parataxis involves a relationship between sentences and paragraphs in which each part bears an equally crushing responsibility to present the whole content of the text. As well, on the level of content, parataxis requires that each concept in a work take on the full conceptual weight of the total work. Continuing the analysis, we could say that parataxis describes texts in which there is no (or very little) linear or causal move from antecedent to consequent, whether in content or form. Instead, parataxis describes texts which weave together concepts via conjugations or associations. The concept of hypotaxis , on the other hand, requires that texts submit themselves to hierarchy, one example of which is logical argumentation. The contingency of the conclusion upon premises in hypotaxis is certainly distinct from the repetition, re-presentation, and conjugation of concepts in parataxis , and I think that this is evident in the difference between the writing styles prevalent in contemporary Analytic philosophy and Continental philosophy. It is not the case that the distinction between parataxis and hypotaxis absolutely corresponds with the writing styles of Continental and Analytic philosophy (respectively). There are thinkers who take exception to this pattern, such as Badiou’s more formal approach in the discourse of Continental philosophy to give one example. However, when Analytic philosophy takes its writing lessons from the sciences, and seeks to do philosophy via the clean linear move from antecedent to consequent, then I think that Analytic philosophy writes with hypotaxis in mind. On the other hand, when Continental (particularly German and French) philosophy takes its writing lessons from narrative or poetry then Continental philosophy writes with parataxis in mind. It is not fair to say that all those writing Continental philosophy are looking to narrative and poetry for stylistic direction, just as it is not the case that all those writing Analytic philosophy have mathematics and science as their writing format, however there are some striking ways in which the differing epistemologies of Continental and Analytic philosophy encourage writing with parataxis and hypotaxis in mind (respectively). For some, writing with parataxis or hypotaxis in mind is a conscious decision and a product of stylistic self-consciousness, and for others it is an unconscious discursive and epistemic requirement that must be met in order to be involved in a particular discourse. Before moving on to our second distinction, and then an examination of the question of the Book, it is important to point out that both totality and figurations like the One, the Whole, and the All, come out of very human desires to totalize or to actualize our being-towards-totality. The relationship between identity and totality then can be construed, with Heidegger in mind, as striving toward wholeness, completion, or fulfillment, each of which is a quality of compendia. 6 Compilation & Selection The first distinction between parataxis and hypotaxis pertains to both form and content, and to both the mechanics of writing and the ideas to which writing refers. The second distinction pertains to both as well. In the process of writing and in the process of perception as well, there is a tension between compilation and selection . To begin with, in the work of theory-writing there is compilation, and this is because writing theory requires a broad perspective and certain measure of totalization. On the other hand, writing theory requires that the writer select an idea or a combination of ideas to individuate out of a radical and infinite multiplicity of identities and combinations. The concern here is not primarily for the writing of a theory about a specific idea, like a secondary work or a reference text. But rather the concern is with the writing of grand theories: Marx and Marxism, Derrida and Deconstruction, Husserl and Phenomenology, Sartre and Existentialism, Saussure and Structuralism —each of which strives along a trajectory towards being all-encompassing, whether it intends to or not. Today, we could even say that Speculative Realism has embarked on this journey, and perhaps now we could say that the discourse of Speculative Realism has reached the inevitable point after which a grand theory leaves the hands of its writer or writers and becomes a possession of the collective consciousness of the academy, or (perhaps now) the mass consciousness of the blogosphere—and this is a true pharmakon , a poison and a cure, a blessing and a curse. 7 When we write theory, whether the scope is that of a grand theory or a particular theory, we write in the tension between compilation (making good on the desire to be all encompassing) and selection (being required to decide and discern and to judge what is included in the work and what is deleted or appended or abridged or given over to ellipses). Generally speaking, the work of theory-writing often comes out of a desire to totalize, that is, to develop a theory of everything that is able to apprehend new experiences and ideas while still remaining whole. I grant that this is not a universal desire, and that not everyone who sits down to write a work (or book) of theory does so because of their will-to-totality, but it remains that this drive to totalize does condition a great many writers of theory (especially those who seek to develop grand theories like those mentioned previously). At the beginning of his book of interviews, Between Existentialism and Marxism , Sartre is quoted as saying, after completing the first volume of his Critique of Dialectical Reason : “I no longer feel the need to make long digressions in my books, as if I were forever chasing after my own philosophy. It will now be deposited in little coffins, and I will feel completely emptied and at peace—as I felt after Being and Nothingness . A feeling of emptiness: a writer is fortunate if he can attain such a state. For when one has nothing to say, one can say everything .” 8 This is where our two initial distinctions come to bear on the being-towards-totality as it is expressed in the concrete practice of writing: the first being between parataxis and hypotaxis , and the second being between compilation and selection. Where this second distinction is concerned there is a certain paradox at play given that selection is inescapable, and compilation is unachievable. Selection is inescapable (with the figure of the Compendium in mind) because, even when the imperative to compile is followed to excruciating lengths, selection still has the last word. This is why the Compendium is a figure rather than some material thing that can actually be accomplished or actualized. No matter how far one goes along with the will-to-archive and the will-to-compile, it is only ever a question of minimizing selection and never eliminating it. This leads to the second point, which is that total compilation is unachievable. The closest thing to total compilation that we have before us is the ontological and textual fabric of the world. Even in the case of the world, compilation cannot be enacted in a total fashion because of the need to include both the sphere of the actual and the sphere of the possible or potential. This as another way in which compilation is always-already selection, and further evidence that total compilation is practically impossible. Discursive Figuration and Total Writing Rather than figuration referring to a figure of speech or an image, here the term points to a way in which to think about the work of writing, both the work put into writing, and the work that is the result of writing: the finished yet incomplete final piece. As figures or figurations, the Book and the Compendium are ways of thinking about the work of writing and the human desire for the wholeness, completion, and fulfillment that are made manifest in the completed form of the book (whether a published or printed or saved document put to rest by the author). These two figures bear more strongly upon works that seek to be all-encompassing, and works that strive towards expressions of totality such as the One, the Whole, or the All. Moving on to the topic of the subtitle, and on a more prescriptive note, I would say that there is more hope for theory-writing to be found in parataxis than hypotaxis . Part of the reason for this claim is the poststructuralist critique of hierarchies, and yet another part is the compelling line of thinking called “weak thought” (in theology by John D. Caputo, and in philosophy by Gianni Vattimo, among others). 9 The consequences of these two convictions are such that if one is to strongly assert the truth of a grand theory without leaving the realm of hierarchy-critique and weak thought, then one must write paradoxically with parataxis in mind, and in so doing write weakly and non-hierarchically. This does not mean avoiding a sort of topography or topology when writing theory, rather it means avoiding both subjugation and oppression in thought by pursuing a nonviolent sort of ontology. In order to do this, theory writing does not present itself as an exercise in logical analysis where one mechanically moves from a set of premises (via contingency) to the inevitable conclusion (via necessity). Instead, theory approaches the idea of a Compendium. Given that the figural Compendium places parataxis and compilation above hypotaxis and selection, then the question becomes: is placing one part of a binary term before another not just another way of selecting, or another expression of hypotaxis ? Counter to the urge to resign oneself to the reign of selection, with some qualification one can nonetheless write without deciding whether selection and hypotaxis should be subservient to compilation and parataxis (which is to say that writing-without-decision is somewhere between and beyond possibility and impossibility). This dilemma can be disarmed by stating that theory writing is not about primacy, and not about having-decided-beforehand, both stylistically and also where content is concerned. This is the paradox of writing: always striving along a trajectory ( telos ) towards totality via parataxis and compilation, but always being drawn back to finitude and making selections, and placing one idea before another with hypotaxis in mind. In light of this paradox, the question of the Compendium as a figure for discourse and a figural Book has resonance with Jacques Derrida’s essay on Edmund Jabès’ Book of Questions , found in his Writing and Difference . The Question of the Book “Little by little the book will finish me.” 10 Derrida quotes Jabès, and proceeds to outline the reflexive relationship between the author of the book and the book itself, each of which are subjected to the other through a sort of chiasmus, both ontological and textual (not that the two are entirely separable). The Book, for Derrida like Blanchot in the introductory quotation, “infinitely reflects itself” and “develops as a painful questioning of its own possibility”, and in light of this we can draw an association with the painful tension between the practical reality of selection and the ideal trajectory of compilation. 11 This tension in writing, between the will to total compilation and the necessity of abridgment, is an ontological tension between part and whole just as it is a textual tension between signifier and signified. The ontological tension in writing that Derrida expresses in his essay on Jabès is between everything and nothing – a contradiction which Derrida finds in Jabès’ Book of Questions and also in the divine, in God. When one writes between compilation and selection one writes towards totality, and here we can follow Derrida who states that the lapse in signification, presumably in the signifier-signified discrepancy, is a “ rupture with totality itself” and furthermore that this lapse cannot be rectified through deductive reason or even philosophical discourse. 12 This rupture with totality, found in the troubled relation between signifier and signified, can also tell us a lot about the Book and about writing. Given an understanding of writing as an ontological act that strives towards (but never accomplishes) totality, we can see the written book as the manifested and given body of that striving. Perhaps in other disciplines or even other schools of philosophy there is a cultural climate within which the article is prized above the book, but I am fairly confident, especially when referring to the grand theories of Continental thought, that there is a respect for the book as a uniquely meaningful object capable of apprehending totality. Derrida writes, Between the too warm flesh of the literal event and the cold skin of the concept runs meaning. This is how it enters into the book. Everything enters into, transpires in the book. This is why the book is never finite. It always remains suffering and vigilant. 13 Here the vitality of the literal event is compromised by the deadening weight of the concept, and the figure of the Book assists in this lapse of meaning. The Book is a totalizing object, much like the figure of the Compendium, and yet this totalization lacks its final object of completed totality both because the figural Book is necessarily incomplete, and because the physical book is always-already a product of selection. Instead of achieving its end of being a place where everything takes place, it suffers from the lapse of writing, the discrepancy between event and concept. Derrida continues his exposition on Jabès by bringing to light yet another reflexive relation, a reversal of the relationship between the Book and the world. Derrida writes that, for Jabès, “the book is not in the world, but the world is in the book.” 14 In addition to what was stated before, I take figuration to mean that the concepts employed, such as the Book or the Compendium, are not subject to the supposed rigor of analysis that is so prized by Enlightenment or Capitalist realisms. The figure of the Book and the figure of the Compendium cannot be held accountable to standards imported from scientific method, given an understanding that these standards require a concept to be replicable, consistent, falsifiable, measurable, and so on… This is an important point to make, especially in the present atmosphere within which theory must justify itself under conditions that are not its own. Instead of being held to these standards, figuration serves as a way in which to think about writing that leaves thought open to idealistic speculation, imperfect analogies, and most importantly the ever-present gain and loss that occur in the relationship between thought and being. By extension the figural way of addressing writing indulges in enough generalization to accommodate the excess/lack relationship between the ideal form of the Book or the Compendium, and the manifested and given body of a text (as it is practically completed and closed). Here we speak against the hostile atmosphere which would have theory submit itself to hierarchical rigor, rather than Blanchot’s “enigmatic rigor”, by being explicit about the discursive conditions and epistemic conditions under which theory operates. To write with parataxis in mind is, to a certain degree, to write with weakness as one’s methodology (with weak thought being in opposition to hypotaxis and hierarchy as much as weak thought can be in opposition to anything). Rather than asserting the strength of hierarchical theory, and rather than employing antagonistic argumentation with the goal of refutation different discursive and epistemic conditions for theory writing must be cultivated (and these are by no means new). First the critical and theoretical spirit reveals itself as being concerned with the task of complication—especially the complication and critique of binaries, dichotomies, dualities, polarities, paradoxes, parallaxes, hybridities, and especially antinomies. Second, theory positions itself as a sort of showing or revealing, rather than being ultimately focused on coming to full agreement or disagreement. This is where figuration can help theory-writing, both by placing emphasis on teleologies and trajectories (like parataxis and compilation), and by shifting focus away from pure primacy, power, or absolute origin. Figuration then, assists theory writing by placing the concern of theory outside of the concerns of the hard or soft sciences, and into the realm of thought or the idea. To speak of ‘the’ Compendium or ‘the’ Book, here, is to generalize not regarding the perfect form of the Compendium or Book, but to engage speculatively with an abstract idea which remains singular and yet complicated by multiplicity. The question of the Book that Derrida asks through Jabès is a question of everything and nothing, totality and nonbeing, and this question is a concern for writing and a concern for the figure of the Compendium so defined by the tension between compilation and selection, and parataxis and hypotaxis . On the note of nonbeing, we can look to the final page of Derrida’s first essay on Jabès in Writing and Difference and notice the introduction of his neologism, différance (with an ‘a’). Derrida writes: Life negates itself in literature only so that it may survive better. So that it may be better. It does not negate itself any more than it affirms itself: it differs from itself, defers itself, and writes itself as différance . Books are always books of life (the archetype would be the Book of Life kept by the God of the Jews) or of afterlife (the archetype would be the Books of the Dead kept by the Egyptians). 15 This introduction of différance into the equation of the Book, alongside ontological affirmation and negation, hearkens back to the discussion of the symbolic lapse earlier in his essay. Differing and deferring, the Book is always a Book of Life and a figure for the intersection of the vital (life) and the total in writing. In a way, the writer of the Book may be someone who lives life, and in another way the writer of the book-as-object may be someone who engages in the act of writing, both practically (by inscribing words on paper, or typing script on a computer) and ontologically or symbolically (by investing their existence and inexistence into a work worthy of the figural Book). In addition to the writer as writer, the writer also serves as an editor, and the editorial role alongside the idea of the Compendium shows the editor to be one who collects and compiles, while being restrained by eventual selection and decision. Beyond this the editor of works and texts, in both the Book and the world, is engaged in a process of inscribing notation—of annotating (on) the Compendium. Eternal commentary is a feature of the textual Compendium, just as open-ended totality describes the ontological Compendium. The writer as writer does not simply write texts, but rather engages in a profoundly ontological act of inscribing their existence into the world, and the writer as editor does not merely annotate or alter texts as they are, but rather comments upon the text of the world. In/Conclusion So as we create texts, and as we write and create theory, let us be and remain attentive to the figure of the Compendium and the figure of the Book. These two concurrent metonymies are essential for total writing (grand theories, etc.), and may be forgettable for those concerned with fragmentary or hierarchical writing (which are valid in their own right). From the ontological and symbolic text of the world and phenomenological experience, to concrete texts such as books or mixed media, the figure of the Compendium and the figure of the Book are important for the actualization of the human will to be all-encompassing. The figure of the Compendium teaches writers of theory that the repetition, density, and fragility of parataxis offers a strong sort of weakness which does not become yet another variety of oppressive and hegemonic thought. Rather than write with hypotaxis in mind, subjugating one thought to another via cause and effect or antecedent and consequent, I would hope that theory-writing could guiltlessly indulge in the dialectical and contradictory conjugation of ideas, and take this as a legitimate methodology. Rather than being ashamed of showing resonances or giving way to abridgment or ellipses, it is my own authorial (but not authoritative) conviction that one need not give up on totalizing and constructing grand theories because of the fact that complete totalization is impossible, and one need not give up on totalizing and constructing grand theories because of the worry of violence, for the Book and the Compendium lack their completed object and are ever incomplete trajectories. NOTES * UPDATED 7/19/13: we've updated the HTML version to reflect the (correct) PDF version. Our apologies to the author—eds. 1. I would like extend thanks to Dr. Peter Schwenger of the University of Western Ontario for his comments on this paper and his hospitality as it was presented as a Theory Session at the Centre for Theory and Criticism on October 26th 2012. I would also like to thank Andrew Weiss for conversation and critique. 2. I should clarify my use of the term ‘figure’ at the outset. Given the use of the term by Jean-Francois Lyotard in Discourse, Figure (and also Gilles Deleuze in Francis Bacon ), I should state clearly that my use of the term will not correspond to the term ‘figure’ understood as the representation of an object. Instead, my use of the term ‘figure’ and its variations (‘figural’, ‘figurative’, ‘figuration’) will refer to the illustrative or metaphorical use of the Compendium or the Book as models or ways of thinking about writing and discourse. 3. Cf. Revelations 20:12-15. 4. Theodor Adorno, Aesthetic Theory , Trans. Robert Hullot-Kentor (London & New York: Continuum Press, 1997). 5. Fredric Jameson, Late Marxism: Adorno, or the Persistence of the Dialectic (London & New York: Verso, 1990), ix-x. 6. I have written about this concept of being-towards-totality elsewhere in two pieces of writing: the first is called Notes on the Compendium (an unfinished draft) which addresses some very wide theoretical concerns, being structured as a sort of itinerary or archive, and the second is Dialectics Unbound (Punctum Books, 2013) which outlines a concept of totality without the violence of totalization. While these works are more concerned with ontological totality, here I would like to focus on the idea of the Compendium as a textual totality. 7. Cf. Louis Morelle, “Speculative Realism: After Finitude and Beyond, A vade mecum” in Speculations: Journal of Speculative Realism . Issue 3 (Brooklyn, New York: Punctum Books, 2012), 241-272. 8. Jean-Paul Sartre, Between Existentialism and Marxism . Trans. John Matthews (London & New York: Verso, 1974), 9. 9. Cf. John D. Caputo, The Weakness of God (Indiana: Indiana University Press, 2006). 10. Jacques Derrida, “Edmond Jabès and the Question of the Book” in Writing and Difference . Trans. Alan Bass (Chicago: University of Chicago Press, 1978), 65. 11. Ibid, 65. 12. Ibid, 71. 13. Ibid, 75. 14. Ibid, 76. 15. Ibid, 78. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques (...) for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

We review mathematical models of HIV dynamics, disease progression, and therapy. We start by introducing a basic model of virus infection and demonstrate how it was used to study HIV dynamics and to measure crucial parameters that lead to a new understanding of the disease process. We discuss the diversity threshold model as an example of the general principle that virus evolution can drive disease progression and the destruction of the immune system. Finally, we show how mathematical (...) models can be used to understand correlates of long‐term immunological control of HIV, and to design therapy regimes that convert a progressing patient into a state of long‐term non‐progression. BioEssays 24:1178–1187, 2002. © 2002 Wiley‐Periodicals, Inc. (shrink)

In 1966 Richard Levins argued that applications of mathematics to population biology faced various constraints which forced mathematical modelers to trade-off at least one of realism, precision, or generality in their approach. Much traditional mathematical modeling in biology has prioritized generality and precision in the place of realism through strategies of idealization and simplification. This has at times created tensions with experimental biologists. The past 20 years however has seen an explosion in mathematical modeling of biological systems (...) with the rise of modern computational systems biology and many new collaborations between modelers and experimenters. In this paper I argue that many of these collaborations revolve around detail-driven modeling practices which in Levins’ terms trade-off generality for realism and precision. These practices apply mathematics by working from detailed accounts of biological systems, rather than from initially idealized or simplified representations. This is possible by virtue of modern computation. The form these practices take today suggest however Levins’ constraints on mathematical application no longer apply, transforming our understanding of what is possible with mathematics in biology. Further the engagement with realism and the ability to push realistic models in new directions aligns well with the epistemological and methodological views of experimenters, which helps explain their increased enthusiasm for biological modeling. (shrink)

Peirce's determination to understand matter, the cosmos, and "the grand design" of the universe remain relevant for contemporary students of science, technology, and symbolic logic.

A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zeno's paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests (...) that in discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time. (shrink)

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 - 9 yields 12. Here, we investigate whether preschool children’s approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an (...) inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding. (shrink)

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...) that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

This article considers the meaning of the ancient Chinese magic square Luoshu. It is known that this square is the most ancient of this type of squares. The importance of the magic square in the philosophical tradition and in the whole culture of China is large. The ancient understanding of number differs from the modern one by its dual character, combining the features of philosophical symbolism and mathematical constructions. Unfortunately, modern interpretations of the Luoshu as well as other (...) numerical constructions preserved by the Chinese tradition are too often limited to a superficial statement of their symbolic, “numerological” nature, ignoring the effects of the mathematical structure represented by them. At the same time, some scholars strongly emphasize the intuitive aspects of numerology, which do not presuppose accuracy. On the contrary, Leibniz saw in the Chinese binary system of thinking the beginning of an exact universal language. The author, appealing to the concept of A.A. Krushinskiy, analyzes the arithmetic laws of the structure of the square. The article shows that the magic square Luoshu is not a numbers game. Without any doubt, its main meaning lies in the application of arithmetic. Chinese magic square is an illustrative implementation of one of the main features of Chinese language and traditional thinking – its rational and mathematical basis. The Luoshu square is a vivid example of visual constructivism in the Chinese style of thinking. By means of the magic square’s scheme, the procedure of generalization that is a basis of concepts formation is carried out. This procedure uses a mathematical magic square algorithm based on the numerical codes of the five elements. The quinary spontaneous symbolism is rooted in the Chinese traditional culture of thinking and life order. (shrink)

Intuitionistic logic formalises the foundational ideas of L.E.J. Brouwer’s mathematical programme of intuitionism. It is one of the earliest non-classical logics, and the difference between classical and intuitionistic logic may be interpreted to lie in the law of the excluded middle, which asserts that either a proposition is true or its negation is true. This principle is deemed unacceptable from the constructive point of view, in whose understanding the law means that there is an effective procedure to determine (...) the truth of all propositions. This understanding of the distinction between the two logics supports the view that negation plays a vital role in the formulation of intuitionistic logic.Nonetheless, the formalisation of negation in intuitionistic logic has not been universally accepted, and many alternative accounts of negation have been proposed. Some seek to weaken or strengthen the negation, and others actively supporting negative inferences that are impossible with it.This thesis follows this tradition and investigates various aspects of negation in intuitionistic logic. Firstly, we look at a problem proposed by H. Ishihara, which asks how effectively one can conserve the deducibility of classical theorems into intuitionistic logic, by assuming atomic classes of non-constructive principles. The classes given in this section improve a previous class given by K. Ishii in two respects: (a) instead of a single class for the law of the excluded middle, two classes are given in terms of weaker principles, allowing a finer analysis and (b) the conservation now extends to a subsystem of intuitionistic logic called Glivenko’s logic. This section also discusses the extension of Ishihara’s problem to minimal logic.Secondly, we study the relationship between two frameworks for weak constructive negation, the approach of D. Vakarelov on one hand and the framework of subminimal negation by A. Colacito, D. de Jongh, and A. L. Vargas on the other hand. We capture a version of Vakarelov’s logic with the semantics of the latter framework, and clarify the relationship between the two semantics. This also provides proof-theoretic insights, which results in the formulation of a cut-free sequent calculus for the aforementioned system.Thirdly, we investigate the ways to unify the formalisations of some logics with contra-intuitionistic inferences. The enquiry concerns paraconsistent logics by R. Sylvan and A. B. Gordienko, as well as the logic of co-negation by G. Priest and of empirical negation by M. De and H. Omori. We take Sylvan’s system as basic, and formulate the frame conditions of the defining axioms of the other systems. The conditions are then used to obtain cut-free labelled sequent calculi for the systems.Finally, we consider L. Humberstone’s actuality operator for intuitionistic logic, which can be seen as the dualisation of a contra-intuitionistic negation. A compete axiomatisation of intuitionistic logic with actuality operator is given, and comparisons are made for some related operators.Abstract prepared by Satoru Niki.E-mail:[email protected]. (shrink)

In this paper we study proof procedures for some variants of first-order modal logics, where domains may be either cumulative or freely varying and terms may be either rigid or non-rigid, local or non-local. We define both ground and free variable tableau methods, parametric with respect to the variants of the considered logics. The treatment of each variant is equally simple and is based on the annotation of functional symbols by natural numbers, conveying some semantical information on the (...) worlds where they are meant to be interpreted.This paper is an extended version of a previous work where full proofs were not included. Proofs are in some points rather tricky and may help in understanding the reasons for some details in basic definitions. (shrink)

This book is about how to understand quantum mechanics by means of a modal interpretation. Modal interpretations provide a general framework within which quantum mechanics can be considered as a theory that describes reality in terms of physical systems possessing definite properties. Quantum mechanics is standardly understood to be a theory about probabilities with which measurements have outcomes. Modal interpretations are relatively new attempts to present quantum mechanics as a theory which, like other physical theories, describes an observer-independent reality. (...) In this book, Pieter Vermaas summarises the results of this work. The book will be of great value to undergraduates, graduate students and researchers in philosophy of science, and physics departments with an interest in learning about modal interpretations of quantum mechanics. (shrink)

Distinction of ‘awām- khawāṣ (the ordinary and the elite) is a general distinction in philosophical literature that shows the difference of people in their level of understanding the truth. It is possible to take this distinction back to Plato in Ancient Greek philosophy. Plato's hesitation in expressing his philosophical thoughts in written form, and Aristotle's use of obscure expressions and symbols in his works against the possibility of reaching those who are not competent, is a result of the (...) distinction between ‘awām-khawāṣ. This distinction was covered in Islamic thought by associating it with the three addressing styles mentioned in the Quran and the fact that the prophet was proclaiming the message according to these addressing styles. The purpose of this study is to reveal how the truth should be conveyed to people of all levels through the evaluations of the Islamic philosophers and thinkers on the distinction between ‘awām and khawāṣ. In the study it will be attempted to find answers to the questions of what are the reasons leading the Islamic philosophers and thinkers to make the distinction between ‘awām and khawāṣ, what arguments they used to explained the necessity of this distinction and why is the distinction between ‘awām and khawāṣ so important. Firstly, the lexical and terminological meanings of the concepts of ‘awām and khawāṣ are introduced. The ‘awām, which is used to express the public, ordinary people, and the lower layer of the society, is a term that qualifies the ignorant society, who has adopted a system of imitation-based life and belief cannot understand the truth of life. Khawāṣ, who are defined as the elite, the prominent, who represents the intellectual group and strives to grasp the truth, base their life and belief on knowledge and attain a higher level of awareness. After the literal and terminological meanings, it has been proceeded to the historical background of this distinction. Then the views of Islamic philosophers and thinkers who directly point to this distinction in this regard have been examined by referring to their main works. These are al-Fārābī, Ikhvān-Ṣafāʾ, Ibn Sinā, Rāghıb al-Iṣfahānī, Ghazālī, Ibn Tufayl, Ibn Rūshd, Ibn'ūn-Nefīs. For instance, al-Fārābī makes a clear distinction between ‘awām and khawāṣ in his works on political philosophy. He addresses this distinction by stating that the virtuous city is divided into two groups, the difference is in the methods used by these group to grasp the truth, and that they can achieve happiness in different ways. Stating that people are made up of different groups, Ikhvān-Ṣafāʾ goes to the distinction of ‘awām- khawāṣ and classifies the sciences accordingly. In addition, Ikhvān-Ṣafāʾ say that the prophets pay attention to the mentioned distinction while communicating the message of Allah to people. Ikhvān-Ṣafāʾ have related this distinction to the distinction between exoteric and esoteric. The understanding of prophethood and eschatology of Ibn Sinā shows ‘awām-khawāṣ distinction very clearly, too. According to him the truth should be conveyed gradually. Ghazālī claims that the ‘awām should not demand the sciences that are not suitable for their cognitive level and that they should leave them to competent people. In Islamic thought the dominant idea is that both wisdom and religion are presented in exo-teric and esoteric knowledge and that the learning of the esoteric way is only necessary for the khawāṣ. The aim of the ‘awām-khawāṣ distinction is to prevent misunderstanding that would arise when the knowledge would be conveyed to wrong people and thus to prevent the loss of truth by those who are not competent. The majority of Islamic thinkers call ‘awām as people of imagination, and khawāṣ as people of thought. They think that the methods of thinking and imagining should be taken into account when explaining the religious and philosophical truths. They have also emphasized the perfection of the prophetic style of addressing society. The symbols and metaphors that the prophets used to express religious truths or the metaphors and symbolic stories that are preferred by philosophers to tell some truths can be seen as a product of the distinction between ‘awām and khawāṣ. As it can be seen in the example of Ḥay bin Ya-qẓān, explaining the truth clearly to ‘awām may cause them to deviate from the right way. For this reason, it is necessary to tell the truth to people according to their level of understanding. It is possible to say that the ideas of Islamic thinkers about this distinction still remain important to-day, due to the fact that people have different dispositions and their level of understanding is not the same despite a change in time and place. Therefore, the disadvantage as pointed out by the Islamic thinkers is that the attempt to convey the truth to people may not be able to reach its exact purpose by ignoring this distinction. (shrink)

The novel use of symbolism in early modern mathematics poses both philosophical and historical questions. How can we trace its development and transmission through manuscript sources? Is it intrinsically related to the emergence of symbolic algebra? How does symbolism relate to the use of diagrams? What are the consequences of symbolic reasoning on our understanding of nature? Can a symbolic language enable new forms of reasoning? Does a universal symbolic language exist which enable us to express all knowledge? This (...) book brings together a collection of papers that address all these and related questions which were initially posed at a conference held in Ghent (Belgium) in August 2009. Scholars working on philosophy of science, history of philosophy and history of mathematics provide an insight into the role and function of symbolic representations in the development of early modern mathematics. The papers cover the period from early abbaco arithmetic and algebra (14h century) up to Leibniz (early 18th century). (shrink)

The article offers an analysis of Simone Weil's philosophy of mathematics. Weil's reflection starts from a critique of Bourbaki's programme, led by her brother André: the "mechanical attention" Bourbaki considered an advantage of their treatment of mathematics was for her responsible for the incomprehensibility of modern algebra, and even a cause of alien-ation and social oppression. On the contrary, she developed her pivotal concept of 'atten-tion' with the aim of approaching mathematical problems in order to make "progress in another (...) more mysterious dimension". In the Pythagorean 'crisis of incommensurables', Weil saw the possibility of defining the relationships between things in terms that are not exclusively numerical. This implies drawing an analogy between mathematical relation-ships and God's relationship with mankind (logos), the basis of a 'supernatural' reformu-lation of the entire scientific understanding of the world. The consequence is a critique of machinism and the possibility to contrast algorithmic reason with a "supernatural reason". (shrink)

The article presents a verbal and mathematical model of medium-term business cycles (with a characteristic period of 7–11 years) known as Juglar cycles. The model takes into account a number of approaches to the analysis of such cycles; in the meantime it also takes into account some of the authors' own generalizations and additions that are important for understanding the internal logic of the cycle, its variability and its peculiarities in the present-time conditions. The authors argue that the (...) most important cause of cyclical crises stems from strong structural disproportions that develop during economic booms. These are not only disproportions between different economic sectors, but also disproportions between different societal subsystems; at present these are also disproportions within the World System as a whole. The proposed model of business cycle is based on its subdivision into four phases: – recovery phase (which could be subdivided into the start sub-phase and the acceleration sub-phase); – upswing/prosperity/expansion phase (which could be subdivided into the growth sub-phase and the boom/overheating sub-phase); – recession phase (within which one may single out the crash/bust/acute crisis subphase and the downswing sub-phase); – depression/stagnation phase (which we could subdivide into the stabilization subphase and the breakthrough sub-phase). The article provides a detailed qualitative description of macroeconomic dynamics at all the phases; it specifies driving forces of cyclical dynamics and the causes of transition from one phase to another (including psychological causes); a special attention is paid to the turning point from the peak of overheating to the acute crisis, as well as to the turning point from the downswing to recovery. The proposed mathematical model of Juglar cycle takes into account the following effects that are typical for the market economy: • positive feedbacks between various economic processes; • presence of a certain inertia, time lags in reactions of the economic subsystem to the change in conditions; • amplification by the financial subsystem of positive feedbacks and time lags in the economic subsystem; • excessive reaction to changing conditions during the acute crisis sub-phase. The authors suggest that the current crisis turns out to be rather similar to classical Juglar crises; however, there is also a significant difference, as the current crisis occurs at a truly global scale. Yet, due to this truly global scale of the current crisis, the possibilities of regulation with the national state's measures have turned out to be ineffective,whereas the suprastate regulation of financial processes hardly exists. It is shown that all these have led to the reproduction of the current crisis according to a classical Juglar scenario. (shrink)

This contribution, by a mathematician, to the Common Knowledge symposium “Fuzzy Studies” examines some mechanisms that seem essential for the “ratchet effect” that, in Michael Tomasello's use of the term, refers to the ability of human cultures to preserve their achievements even through serious crises and even where preservation entails substantial loss. By taking the word culture to refer to any group of individuals who closely cooperate over an extended period, this article evaluates mathematicians and mathematics as its main example. (...) The assessment of the enterprise of mathematics as a culture corresponds to the author's personal experience — and mathematics, he argues, is both historically old and simply structured, despite the formidable complexity of mathematical knowledge as it has been compiled over the centuries. The preserving, restructuring, and enlarging of this knowledge may be regarded as the main cultural goals of mathematics and are the objects of study here. This article concentrates first on the techniques used in pursuing the tasks of preservation, enlargement, and restructuring — techniques such as counting methods and geometric diagrams — and it emphasizes their use in communication, when they function as media. The cultural techniques of mathematics, known to and used by all members of that culture, are media that, through intensity of communication, add symbolic functions to mere procedures. (The prototypical example of a cultural technique, in this sense, is language.) Knowledge, as the additional symbolic value, is distributed and at the same time preserved in the process of permanent communication. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Symbolic Classification and The Emergence of a Metaphysics of CausalityOwen Goldinwhat is distinctive about metaphysics as a mode of thought that emerged in the fifth century before the Common Era? How did it emerge out of early ways of conceptualizing the world as a whole, and why? Many answers have been proposed. One common view is that earlier modes of thought personify natural agencies; once this is abandoned, the (...) way is open for discerning more complex regularities among natural events.1 Another view is that the process of "rationalization" introduced by the Greeks involves an implicit appeal to the individual's natural intelligence, in contrast to a dependence on an "inherited stock of wisdom."2 A third view is that the first Greek philosophers introduced inference as a way of navigating conceptual relationships, independent of the sorts of personal interactions among divinities related in mythical narrative.3 According to a fourth account, the mode of rationality that allowed for the development of ancient Greek metaphysics was a social [End Page 3] practice with the aim of persuasion, brought on by more democratic involvement in societal deliberations concerning law and policy.4To these I add a fifth way of answering the question. Crucial to the emergence of Greek metaphysics is explicit distinction of ontological/causal priority and posteriority. Ancient Greek scientific explanations developed out of older, archaic modes of conceptualization, which I maintain have deep parallels with modes of conceptualization shared by many nonliterate cultures. Such modes of thought rest on an implicit worldview which can be called a metaphysics, in a loose sense, and are still worthy of philosophical consideration.I begin with a red flag that will be raised for many as soon as we start talking about the metaphysical traditions of nonliterate cultures. What counts as a metaphysical tradition? I make some distinctions. In the loosest sense, an account or tradition is metaphysical if it reflects a coherent worldview concerning the kinds of beings that there are and how they are related to one another. According to a stricter sense of the term, a tradition is metaphysical if it involves working through this worldview in an explicit and systematic matter. In an even stricter sense, a tradition is metaphysical if, in addition to being explicit and systematic, it merits being classified as "philosophical," itself not an unproblematic term. I here stipulate that an account is philosophical if it does not rest on premises that are specific to certain cultural traditions and employs inferential arguments that are in principle subject to refutation.Generally speaking, there are two ways in which a tradition that is metaphysical in the strictest sense can listen to one that is metaphysical in a looser sense. First, a metaphysician might examine the latter tradition in order to explore how, historically, it served as the root for the former. An example is Aristotle's interest in Hesiod as expressing metaphysical intuitions that later philosophers unpacked.5 By studying [End Page 4] the ontological presuppositions of the sort of account from which our own metaphysics is derived, one learns more about our own ontological presuppositions and what we gain from them.6 This sort of dialogue carries with it the danger that the later thinker presupposes that—in the spirit of "you have to start somewhere"—the earlier thinkers were simply doing a bad job working through their own theories; Aristotle has often been accused of precisely this.7Another sort of dialogue is possible as well. An alternative metaphysical tradition can reveal the possibility of a radically different way of conceptualizing the causal and ontological structure of the world, one with its own advantages. In such a case there is shared set of problems at issue, shared points of reference within the philosophical literature, and a shared understanding of what philosophical argument is. But philosophical insight might also emerge from the kind of dialogue in which a developed metaphysics that conforms to certain standards of inferential argument confronts a way of thinking that is metaphysical in a looser sense. A substance-based ontologist can be led to reflect on whether the world has to be cut up as she does by studying Aztec... (shrink)