Figures of Thought looks at how mathematical works can be read as texts and examines their textual strategies. David Reed offers the first sustained and critical attempt to find a consistent argument or narrative thread in mathematical texts. Reed selects mathematicians from a range of historical periods and compares their approaches to organizing and arguing texts, using an extended commentary on Euclid's Elements as a central structuring framework. He develops fascinating interpretations of mathematicians' work throughout history, from Descartes (...) to Hilbert, Kronecker, Dedekind, Weil and Grothendieck. Reed traces the implications of this approach to the understanding of the history and development of mathematics. (shrink)
The article is based upon the following starting position. In this post-modern time, it seems that no scholar in Europe supports what is called “Enlightenment Project” with its naïve objectivism and Correspondence Theory of Truth1, - though not being really hostile, just strongly skeptical about it. No old-fasioned “classical” academical texts; only His Majesty Discourse as chain of interpretations and reinterpretations. What was called objectivity “proved to be” intersubjectivity; what was called Object (in Latin and German and Russian tradition) now (...) is related to as a phenomenon; what was called Subject either is looked upon as Cartesian “cogito” or disappeares at all; what was called Truth turned to be either method of demonstration (by positivists) or the author’s sincerity (by existentialists); what was called Essence is now merely a joke. Alternatively, we speak about Sense, Meaning, and Value. (shrink)
There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the (...) point. (shrink)
Although in the past three decades interest in mathematical explanation revived, recent literature on the subject seems to neglect the strict connection between explanation and discovery. In this paper I sketch an alternative approach that takes such connection into account. My approach is a revised version of one originally considered by Descartes. The main difference is that my approach is in terms of the analytic method, which is a method of discovery prior to axiomatized mathematics, whereas Descartes’s approach (...) is in terms of the analytic-synthetic method, which is a heuristic pattern in already axiomatized mathematics. (shrink)
Despite a vast philosophical literature on the epistemology of mathematics and much speculation about how, in principle, knowledge of this domain is possible, little attention has been paid to the psychological findings and theories concerning the acquisition, comprehension and use of mathematical knowledge. This contrasts sharply with recent philosophical work on language where comparable issues and problems arise. One topic that is the center of debate in the study of mathematical cognition is the question of innateness. This (...) paper critically examines the controversy. (shrink)
The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main (...) problem facing Platonists—the problem of explaining how our experiences make contact with mathematical reality. (shrink)
A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that (...) Natorp's metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)
In this paper I examine a strategy which aims to bypass the technicalities of the indispensability debate and to offer a direct route to nominalism. The starting-point for this alternative nominalist strategy is the claim that--according to the platonist picture--the existence of mathematical objects makes no difference to the concrete, physical world. My principal goal is to show that the 'Makes No Difference' (MND) Argument does not succeed in undermining platonism. The basic reason why not is that the makes-no-difference (...) claim which the argument is based on is problematic. Arguments both for and against this claim can be found in the literature; I examine three such arguments, uncovering flaws in each one. In the second half of the paper, I take a more direct approach and present an analysis of the counterfactual which underpins the makes-no-difference claim. What this analysis reveals is that indispensability considerations are in fact crucial to the proper evaluation of the MND Argument, contrary to the claims of its supporters. (shrink)
A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics (...) can do to make an explanation mathematical; realists think something can be done, but they are not clear about what that something is. I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's  notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism (in this journal). I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist. (shrink)
Philosophers of mathematics have become increasingly interested in the explanatory role of mathematics in empirical science, in the context of new versions of the Quinean ‘Indispensability Argument’ which employ inference to the best explanation for the existence of abstract mathematical objects. However, little attention has been paid to analysing the nature of the explanatory relation involved in these mathematical explanations in science (MES). In this paper, I attack the only articulated account of MES in the literature (an (...) account sketched by Mark Steiner), according to which a genuine MES incorporates an explanatory proof of the mathematical result being used. The central case study involves an explanation for why bees build the cells of their honeycombs in the shape of hexagons. I make a distinction between two kinds of MES, mathematics-driven explanation in science and science-driven mathematical explanation, and argue that it is the second category which is both scientifically and philosophical more central. I conclude that the explanatory relation involved in MES is genuinely scientific and hence that the phenomenon of MES poses a challenge to general accounts of scientific explanation. (shrink)
We provide a critical assessment of the ambiguity aversion literature, which we characterize in terms of the view that Ellsberg choices are rational responses to ambiguity, to be explained by relaxing Savage's Sure-Thing principle and adding an ambiguity-aversion postulate. First, admitting Ellsberg choices as rational leads to behaviour, such as sensitivity to irrelevant sunk cost, or aversion to information, which most economists would consider absurd or irrational. Second, we argue that the mathematical objects referred to as in the (...) ambiguity aversion literature have little to do with how an economist or game theorist understands and uses the concept. This is because of the lack of a useful notion of updating. Third, the anomaly of the Ellsberg choices can be explained simply and without tampering with the foundations of choice theory. These choices can arise when decision makers form heuristics that serve them well in real-life situations where odds are manipulable, and misapply them to experimental settings. (shrink)
Of the various notions of reduction in the logical literature, relative interpretability in the sense of Tarskiet al.  appears to be the central one. In the present note, this syntactic notion is characterized semantically, through the existence of a suitable reduction functor on models. The latter mathematical condition itself suggests a natural generalization, whose syntactic equivalent turns out to be a notion of interpretability quite close to that of Ershov , Szczerba  and Gaifman .
Many arguments found in the physics literature involve concepts that are not well-defined by the usual standards of mathematics. I argue that physicists are entitled to employ such concepts without rigorously defining them so long as they restrict the sorts of mathematical arguments in which these concepts are involved. Restrictions of this sort allow the physicist to ignore calculations involving these concepts that might lead to contradictory results. I argue that such restrictions need not be ad hoc, but (...) can sometimes be justified by considering some of the metaphysical issues surrounding the question of the applicability of mathematics to physical reality. 1 Introduction 2 Rejecting inferential permissiveness 3 The agreement problem 4 Independent objections to the liberal view. (shrink)
Several case studies and theoretical reports indicate that the structuralist concept of a constraint has a central role in the reconstruction of physical theories. It is surprising that there is, in the literature, only little theoretical discussion on the relevance of constraints for the reconstruction of social scientific theories. Almost all structuralist reconstructions of social theorizing are vacuously constrained. Consequently, constraints are methodologically irrelevant.In this paper I try to show that there really exist constraint-type assumptions in mathematical modelling (...) in the social sciences. Methodologically constraints have exactly the same role in the context of social mathematical modelling as they have in physical theories. In typical cases of mathematical modelling in the social sciences, the related constraints work as empirical hypotheses and should be tested by statistical means. (shrink)
This paper examines the complexity and fluidity of maternal identity through an examination of narratives about "real motherhood" found in children's literature. Focusing on the multiplicity of mothers in adoption, I question standard views of maternity in which gestational, genetic and social mothering all coincide in a single person. The shortcomings of traditional notions of motherhood are overcome by developing a fluid and inclusive conception of maternal reality as authored by a child's own perceptions.
Of the various notions of reduction in the logical literature, relative interpretability in the sense of Tarski et al.  appears to be the central one. In the present note, this syntactic notion is characterized semantically, through the existence of a suitable reduction functor on models. The latter mathematical condition itself suggests a natural generalization, whose syntactic equivalent turns out to be a notion of interpretability quite close to that of Ershov , Szczerba  and Gaifman .
We give a precise and modern mathematical characterization of the Newtonian spacetime structure (ℕ). Our formulation clarifies the concepts of absolute space, Newton's relative spaces, and absolute time. The concept of reference frames (which are “timelike” vector fields on ℕ) plays a fundamental role in our approach, and the classification of all possible reference frames on ℕ is investigated in detail. We succeed in identifying a Lorentzian structure on ℕ and we study the classical electrodynamics of Maxwell and Lorentz (...) relative to this structure, obtaining the important result that there exists only one intrinsic generalization of the Lorentz force law which is compatible with Maxwell equations. This is at variance with other proposed intrinsic generalizations of the Lorentz force law appearing in the literature. We present also a formulation of Newtonian gravitational theory as a curve spacetime theory and discuss its meaning. (shrink)
There is no shortage of testimony to literature's puzzling, unsettling, intoxicating, affecting, delighting powers. Nor has there been a shortage of attempts to define literature as a concept, a body of texts or a cultural practice. However, no definition has been able to pin down the peculiarity of literature or to chart our experience of the literary. In this volume, Derek Attridge ask us to confront with him the resistance to definition in order to explore afresh the (...) singularity of literature. In seeking new purchase on the elusive "literary", the author finds himself reflecting upon the history of Western art as a practice and as an institution. At its heart he finds a closely linked trinity of crucial issues: innovation or invention , the uniqueness or singularity of the artwork and, underlying these, the concept of otherness or alterity. Calling for a type of reading that does justice to these aspects of the literary work, he explores literature as event or performance and brilliantly retheorizes its place in the realm of the ethical. The author acknowledges the impossibility of definition and rather offers us an account of his particular "living-through" of the literary in the terms above and invites us to share with him the insights it might offer. The insights in this case are invaluable, as we are offered not only an original framework within which to consider texts, but a clear case for the ethical value of the literary institution to a culture. Never losing sight of the pleasures and potency of our experience of literature, The Singularity of Literature is itself a delight to read. Returning to arguments begun in his influential volume Peculiar Language , Derek Attridge here energizes discussion of the literary by forever shifting the terms of debate and offering new perspectives on questions that haunt every reader. (shrink)
This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation (...) facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)
In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in (...) class='Hi'>mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)
Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols (...) are not only used to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)
The relationship of words to the things they represent and to the mind that forms them has long been the subject of linguistic enquiry. Joseph Graham's challenging book takes this debate into the field of literary theory, making a searching enquiry into the nature of literary representation. It reviews the arguments of Plato's Cratylus on how words signify things, and of Chomsky's theory of the innate "natural" status of language (contrasted with Saussure's notion of its essential arbitrariness). In the process, (...) Graham explores the issues of meaning and intentionality in representation, and questions of how the mind represents the world. Graham's use of linguistic theories and models leads him to a new response to Wimsatt's notion of the verbal icon, Stanley Fish's concept of literature as self-consuming artifact, and de Man's idea of its function as an allegory of reading. In showing them in fact to be complementary, he transcends the current controversies among literary theorists, arguing that the solution lies not in epistemology or philosophy, but in psychology and the study of how literature teaches and why humans learn best by example. (shrink)
The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...) with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition. (shrink)
German classicist's monumental study of the origins of European thought in Greek literature and philosophy. Brilliant, widely influential. Includes "Homer's View of Man," "The Olympian Gods," "The Rise of the Individual in the Early Greek Lyric," "Pindar's Hymn to Zeus," "Myth and Reality in Greek Tragedy," and "Aristophanes and Aesthetic Criticism.".
"A valuable collection both for original source material as well as historical formulations of current problems."-- The Review of Metaphysics "Much more than a mere collection of papers . . . a valuable addition to the literature."-- Mathematics of Computation An anthology of fundamental papers on undecidability and unsolvability by major figures in the field, this classic reference opens with Godel's landmark 1931 paper demonstrating that systems of logic cannot admit proofs of all true assertions of arithmetic. Subsequent papers (...) by Godel, Church, Turing, and Post single out the class of recursive functions as computable by finite algorithms. Additional papers by Church, Turing, and Post cover unsolvable problems from the theory of abstract computing machines, mathematical logic, and algebra, and material by Kleene and Post includes initiation of the classification theory of unsolvable problems. Suitable for graduate and undergraduate courses. 1965 ed. (shrink)
The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.
A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness (...) and completeness -- Why is first order logic "First Order"? (shrink)
This book examines the complex and varied ways in which fictions relate to the real world, and offers a precise account of how imaginative works of literature can use fictional content to explore matters of universal human interest. While rejecting the traditional view that literature is important for the truths that it imparts, the authors also reject attempts to cut literature off altogether from real human concerns. Their detailed account of fictionality, mimesis, and cognitive value, founded on (...) the methods of analytical philosophy, restores to literature its distinctive status among cultural practices. The authors also explore metaphysical and skeptical views, prevalent in modern thought, according to which the world itself is a kind of fiction, and truth no more than a social construct. They identify different conceptions of fiction in science, logic, epistemology, and make-believe, and thereby challenge the idea that discourse per se is fictional and that different modes of discourse are at root indistinguishable. They offer rigorous analyses of the roles of narrative, imagination, metaphor, and "making" in human thought processes. Both in their methods and in their conclusions, Lamarque and Olsen aim to restore rigor and clarity to debates about the values of literature, and to provide new, philosophically sound foundations for a genuine change of direction in literary theorizing. (shrink)
How should we characterise the view that we can learn about the mind from literature? Should we say that such learning consists in acquiring knowledge of truths? That option is more attractive than it is sometimes made to seem by those who oppose propositional knowledge to practical knowledge or “knowing how”. But some writers on this topic—Lamarque and Olsen—argue that, while literature may express interesting propositions, it is not their truth that matters, but their “content”. Matters to what? (...) To literary criticism, they reply: there is no place in criticism for “debate about the truth or falsity of general statements about human life or the human condition.” I argue, to the contrary, that ideas of truth and truthfulness are woven into the fabric of a kind of criticism that is widespread now and comes with a long and distinguished history. (shrink)
This is the first comprehensive introduction to Deleuze's work on literature. It provides thorough treatments of Deleuze's early book on Proust and his seminal volume on Kafka and minor literature. Deleuze on Literature situates those studies and many other scattered writings within a general project that extends throughout Deleuze's career-that of conceiving of literature as a form of health and the writer as a cultural physician.
This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a (...) human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical objection” to his view that machines can think. Logico-mathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do. (shrink)
Noted logician and philosopher addresses various forms of mathematical logic, discussing both theoretical underpinnings and practical applications. After historical survey, lucid treatment of set theory, model theory, recursion theory and constructivism and proof theory. Place of problems in development of theories of logic, logic’s relationship to computer science, more. Suitable for readers at many levels of mathematical sophistication. 3 appendixes. Bibliography. 1981 edition.
Literary ontology is essentially a phenomenological issue rather than one of epistemology, sociology, or psychology. It is a theory of the phenomenological essence intuited from a sense of beauty, based on the phenomenological ontology of beauty, which puts into brackets the sociohistorical premises and material conditions of aesthetic phenomena. Beauty is the objectified emotion. This is the phenomenological definition of the essence of beauty, which manifests itself on three levels, namely emotion qua selfconsciousness, sense of beauty qua emotion, and sentiment (...) qua sense of beauty. Art on the other hand is the objectification of emotion whose most general and closest manner to humanity is literature and poetry. Poetry is the origin of language and the linguistic essence is a metaphor. Language as the house of being is both thinking and poetry. Literature expresses the essence of art in the most direct way and, in traditional Chinese aesthetic terminology, literature is the language of emotion conveyed by the writer based on his own emotion towards the language of scene. (shrink)
In this paper the reader is asked to engage in some simple problem-solving in classical pure number theory and to then describe, on the basis of a series of questions, what it is like to solve the problems. In the recent philosophy of mind this “what is it like” question is one way of signaling a turn to phenomenological description. The description of what it is like to solve the problems in this paper, it is argued, leads to several morals (...) about the epistemology and ontology of classical pure mathematical practice. Instead of simply making philosophical judgments about the subject matter in advance, the exercise asks the reader to briefly engage in a mathematical practice and to then reflect on the practice. (shrink)
Literature, like the visual arts, posess its own characteristic philosophical problems. Literary theorists have discussed widely the nature of literature, while analytic philosophers have dealt with literary problems within the framework of aesthetics or have restricted themselves to topics which are accessible only to a philosophical audience. Philosophy of Literature is unique in that it introduces the philosophy of literature from an analytic perspective which is both accessible to students of literature and students of philosophy. (...) Specifically, the book addresses: the definition of literature, the distinction between oral and written literature and the identity of literary works. Philosophy of Literature offers fresh approaches to traditional issues and raises new questions about the nature of philosophical problems which literature gives rise to. (shrink)
Recent philosophical discussion about the relation between fiction and reality pays little attention to our moral involvement with literature. Frank Palmer's purpose is to investigate how our appreciation of literary works calls upon and develops our capacity for moral understanding. He explores a wide range of philosophical questions about the relation of art to morality, and challenges theories that he regards as incompatible with a humane view of literary art. Palmer considers, in particular, the extent to which the values (...) and moral concepts involved in our understanding of human beings can be said to enter into our understanding of, and response to, fictional characters. The scope of his discussion encompasses literary aesthetics, ethics, and epistemology, and he makes extensive reference to literary examples. (shrink)
This monumental collection of new and recent essays from an international team of eminent scholars represents the best contemporary critical thinking relating to both literary and philosophical studies of literature. Helpfully groups essays into the field's main sub-categories, among them ‘Relations Between Philosophy and Literature’, ‘Emotional Engagement and the Experience of Reading’, ‘Literature and the Moral Life’, and ‘Literary Language’ Offers a combination of analytical precision and literary richness Represents an unparalleled work of reference for students and (...) specialists alike, ideal for course use. (shrink)
Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or (...) creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)
One of the older questions in the debate about Corporate Social Responsibility (CSR) is whether it is worthwhile for organizations to pay attention to societal demands. This debate was emotionally, normatively, and ideologically loaded. Up to the present, this question has been an important trigger for empirical research in CSR. However, the answer to the question has apparently not been found yet, at least that is what many researchers state. This apparent ambivalence in CSR consequences invites a literature study (...) that can clarify the debate and allow for the drawing of conclusions. The results of the literature study performed here reveal that there is indeed clear empirical evidence for a positive correlation between corporate social and financial performance. Voices that state the opposite refer to out-dated material. Since the beginnings of the CSR debate, societies have changed. We can therefore clearly state that, for the present Western society, “Good Ethics is Good Business.”. (shrink)
In 'Literature Suspends Death: Sacrifice and Storytelling in Kierkegaard, Kafka and Blanchot' Chris Danta takes Genesis 22 as the starting point for an investigation of the role of literary imagination. His aim is to read the Genesis story from a literary-theoretical perspective in order to show how it can ‘illuminate the secular situation of the literary writer.’ To do this, Danta stages a fruitful confrontation between Søren Kierkegaard as defender of religion and inwardness and Franz Kafka and Maurice Blanchot (...) as defenders of literature. In this review, three important points in this confrontation are highlighted. 1. The problem of identification. 2. The moment of substitution. 3. The spectrality of the writer. (shrink)
The present paper argues that ‘mature mathematical formalisms’ play a central role in achieving representation via scientific models. A close discussion of two contemporary accounts of how mathematical models apply—the DDI account (according to which representation depends on the successful interplay of denotation, demonstration and interpretation) and the ‘matching model’ account—reveals shortcomings of each, which, it is argued, suggests that scientific representation may be ineliminably heterogeneous in character. In order to achieve a degree of unification that is compatible (...) with successful representation, scientists often rely on the existence of a ‘mature mathematical formalism’, where the latter refers to a—mathematically formulated and physically interpreted—notational system of locally applicable rules that derive from (but need not be reducible to) fundamental theory. As mathematical formalisms undergo a process of elaboration, enrichment, and entrenchment, they come to embody theoretical, ontological, and methodological commitments and assumptions. Since these are enshrined in the formalism itself, they are no longer readily obvious to either the novice or the proficient user. At the same time as formalisms constrain what may be represented, they also function as inferential and interpretative resources. (shrink)
Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give (...) a natural explanation of Saccheri’s proofs as well as standard geometric proofs and even number-theoretic proofs. (shrink)