Introduction: the figure of this world -- 1. The question of political ontology -- 2. The poetic experience of the world -- 3. The myth of the earth -- 4. The unbearable -- 5. The creature before the law -- 6. The animal for which animality is an issue -- 7. Understanding the happy -- 8. The picture and its captives -- 9. The passing of the figure of this world.

Interpreting Figurative Meaning critically evaluates the recent empirical work from psycholinguistics and neuroscience examining the successes and difficulties associated with interpreting figurative language. There is now a huge, often contradictory literature on how people understand figures of speech. Gibbs and Colston argue that there may not be a single theory or model that adequately explains both the processes and products of figurative meaning experience. Experimental research may ultimately be unable to simply adjudicate between current models in psychology, linguistics and philosophy (...) of how figurative meaning is interpreted. Alternatively, the authors advance a broad theoretical framework, motivated by ideas from 'dynamical systems theory', that describes the multiple, interacting influences which shape people's experiences of figurative meaning in discourse. This book details past research and theory, offers a critical assessment of this work and sets the stage for a new vision of figurative experience in human life. (shrink)

I argue that figurative speech, and irony in particular, presents a deep challenge to the orthodox view about sentence content. The standard view is that sentence contents are, at their core, propositional contents: truth-conditional contents. Moreover, the only component of a sentence’s content that embeds in compound sentences, like belief reports or conditionals, is the propositional content. I argue that a careful analysis of irony shows this view cannot be maintained. Irony is a purely pragmatic form of content that embeds (...) in compound sentences. The standard view cannot account for this fact. I sketch out a speech-act theoretic framework that can. But in accepting this alternative framework we are giving up on the whole idea of an autonomous semantics. (shrink)

Macpherson (Nous 40(1):82–117, 2006) argues that the square/regular diamond figure threatens representationalism, construed as the theory which holds that the phenomenal character is explained by the nonconceptual content of experience. Her argument is the claim that representationalism is committed to the thesis that differences in the experience of ambiguous figures, the gestalt switch, should be explained by differences in the NCC of perception of these figures. However, with respect to the square/regular diamond and some other ambiguous figure representationalism (...) fails to offer a unified account of how representational content makes them ambiguous. In this paper, I aim, first, to offer a representationalist account of ambiguous figures and, second, to examine and rebut Macpherson’s arguments. My main point is that in each ambiguous figure Macpherson discusses there are differences in representational content that can explain differences in phenomenal character or content. The representational differences are due to the ways the Cartesian frame of reference in which perceptual content is always cast cuts the figure, underlying different properties of the figure with respect to the axes of the Cartesian frame of reference. (shrink)

Il primo volume di Figure del pensiero medievale, come recita il titolo, presenta “i fondamenti” e gli “inizi” della figura della teologia che verrà elaborata nella riflessione occidentale fino alla “Via moderna”, ossia fino al XIV e agli inizi del XV secolo. Né si tratta soltanto degli inizi cronologici: i vari saggi editi nel volume delineano, infatti, gli inizi metodologici, in certo modo esemplari, che avviano e determinano la forma della teologia, quale verrà in seguito elaborata. Il capitolo introduttivo (...) offre una sintesi della teologia medievale, con le sue fonti e i suoi riferimenti e processi fondamentali. Seguono poi i grandi modelli: Agostino, con la fede che genera l'"intelletto teologico"; Boezio, con le sue esemplari attuazioni o i suoi "opuscoli" della teologia risultante dall'investimento razionale della stessa fede; Gregorio, che resterà il geniale ispiratore della teologia come “intelligenza spirituale”, come lettura di tutte le risorse dei significati biblici, destinati ad avverarsi come esperienza; lo Pseudo-Dionigi, il maestro della teologia come inesausta ricerca e desiderio di Dio come di inarrivabile trascendenza, e impossibilità di "possesso", di là da ogni rappresentatività e simbolo; Isidoro di Siviglia, con il suo enciclopedismo, mirante alla visione il più possibile comprensiva della realtà della natura e della storia. (shrink)

Figurativity has attracted scholars' attention for thousands of years and yet there are still open questions concerning its nature. Figurativity and Human Ecology endorses a view of figurativity as ubiquitous in human reasoning and language, and as a key example of how a human organism and its perceived or imagined environment co-function as a system. The volume sees figurativity not only as embedded in an environment but also as a way of acting within that environment. It places figurativity within an (...) ecological context, and approaches it as a phenomenon which cuts across bodily, psychological, linguistic, social, cultural and natural environments. Figurativity and Human Ecology will appeal to those interested in the analysis of the all-encompassing creativity of the human mind and in the methodological difficulties associated with the study of cognition. (shrink)

Representationalism is the position that the phenomenal character of an experience is either identical with, or supervenes on, the content of that experience. Many representationalists hold that the relevant content of experience is nonconceptual. I propose a counterexample to this form of representationalism that arises from the phenomenon of Gestalt switching, which occurs when viewing ambiguous figures. First, I argue that one does not need to appeal to the conceptual content of experience or to judgements to account for Gestalt switching. (...) I then argue that experiences of certain ambiguous figures are problematic because they have different phenomenal characters but that no difference in the nonconceptual content of these experiences can be identified. I consider three solutions to this problem that have been proposed by both philosophers and psychologists and conclude that none can account for all the ambiguous figures that pose the problem. I conclude that the onus is on representationalists to specify the relevant difference in content or to abandon their position. (shrink)

Demographic diversity might often be present in a group without group members noticing it. What are the epistemic effects if they do? Several philosophers and social scientists have recently argued that when individuals detect demographic diversity in their group, this can result in epistemic benefits even if that diversity doesn’t involve cognitive differences. Here I critically discuss research advocating this proposal, introduce a distinction between two types of detection of demographic diversity, and apply this distinction to the theorizing on diversity (...) in science. Focusing on ‘invisible’ diversity, I argue that in one common kind of group in science, if group members have full insight into their group’s diversity, this is likely to create epistemic costs. These costs can be avoided and epistemic benefits gained if group members only partly detect their group’s diversity. There is thus an epistemic reason for context-dependent limitations on scientists’ insight into the diversity of their group. (shrink)

Ce qui a fait naitre l'emerveillement des premiers penseurs grecs, c'est qu'il y a quelque chose plutot que rien, et c'est la ce qui a donne le coup d'envoi a cette pensee de l'etre qui s'est developpee de Parmenide a Aristote et qui constitue le fondement de la philosophie occidentale. On trouve cependant, deja dans la pensee grecque, une denegation de la possibilite d'un discours sur l'etre, d'abord chez Gorgias, contemporain de Socrate, puis chez le fondateur de l'ecole sceptique, Pyrrhon. (...) Et a l'epoque meme ou Parmenide ecrivait son poeme, une pensee de la vacuite et du neant commencait a se developper en Orient dans le cadre du bouddhisme, laquelle met profondement en question la notion meme d'ontologie. Or c'est precisement cette pensee du rien qui resurgit en Occident a la fin de l'age classique, avec ce premier philosophe veritablement moderne qu'est Kant, dans son Essai sur les grandeurs negatives et cette Table du Rien qui, dans La Critique de la raison pure, clot l'Analytique transcendantale. Et c'est le concept de negativite qui va former chez Hegel la matrice meme de la pensee dialectique, alors que celui du neant constituera le coeur de la critique du nihilisme qu'entreprendra Nietzsche, avant de redevenir, avec le Heidegger de Qu'est-ce que la metaphysique?, avec le Sartre de L'Etre et le neant, et avec le dernier Merleau-Ponty, auteur de ce livre inacheve qu'est Le Visible et l'Invisible, un theme fondamental de la pensee de l'apparaitre.Une telle pensee du neant et de la negation traversant la frontiere qui separe l'Orient de l'Occident, il s'agira donc, en prenant comme reference majeure la pensee heideggerienne, d'en interroger les diverses figures, d'abord chez Gorgias et Pyrrhon, puis chez Nagarjuna, le plus grand penseur du bouddhisme indien (ii-iiie siecle) et chez Nishida (1870-1945), representant fameux de l'ecole de Kyoto et du bouddhisme zen, avant d'en venir a l'idealisme allemand avec Kant et Hegel, a la question du nihilisme europeen avec Schopenhauer, Nietzsche et Heidegger, puis a la phenomenologie avec Husserl, Sartre, Merleau-Ponty et Maldiney. (shrink)

Gerhard Richter's groundbreaking study argues that the concept of "afterness" is a key figure in the thought and aesthetics of modernity. It pursues questions such as: What does it mean for something to "follow" something else? Does that which follows mark a clear break with what came before it, or does it in fact tacitly perpetuate its predecessor as a consequence of its inevitable indebtedness to the terms and conditions of that from which it claims to have departed? Indeed, (...) is not the very act of breaking with, and then following upon, a way of retroactively constructing and fortifying that from which the break that set the movement of following into motion had occurred? The book explores the concept and movement of afterness as a privileged yet uncanny category through close readings of writers such as Kant, Kafka, Heidegger, Bloch, Benjamin, Brecht, Adorno, Arendt, Lyotard, and Derrida. It shows how the vexed concepts of afterness, following, and coming after shed new light on a constellation of modern preoccupations, including personal and cultural memory, translation, photography, hope, and the historical and conceptual specificity of what has been termed "after Auschwitz." The study's various analyses—across a heterogeneous collection of modern writers and thinkers, diverse historical moments of articulation, and a range of media—conspire to illuminate Lyotard's apodictic statement that "after philosophy comes philosophy. But it has been altered by the 'after.'" As Richter's intricate study demonstrates, much hinges on our interpretation of the "after." After all, our most fundamental assumptions concerning modern aesthetic representation, conceptual discourse, community, subjectivity, and politics are at stake. (shrink)

Produced in honor of White's eightieth birthday, _Re-Figuring Hayden White_ testifies to the lasting importance of White's innovative work, which firmly reintegrates historical studies with literature and the humanities. The book is a major reconsideration of the historian's contributions and influence by an international group of leading scholars from a variety of disciplines. Individual essays address the key concepts of White's intellectual career, including tropes, narrative, figuralism, and the historical sublime while exploring the place of White's work in the philosophy (...) of history, postmodernism, and ethics. They also discuss his role as historian and teacher and apply his ideas to specific historical events. (shrink)

The chapter opens with a discussion of ACTORS ARE MOVERS, which is a subset of events are actions and projects stories of body motion onto action stories. Another special subset, ACTORS ARE MANIPULATORS, involves projecting stories of bodily manipulation onto other action stories. The two cases are compatible and even overlap to create another sub-pattern: ACTORS ARE MOVERS AND MANIPULATORS. These in turn lead to more specific patterns which include THE MIND IS A BODY MOVING THROUGH SPACE and a second (...) one involving a body manipulating cognitive instruments and conceptual tools: A THINKER IS A MOVER AND A MANIPULATOR. The subsets of the general pattern involving event stories are also identified and discussed and these include: EVENTS ARE BODY ACTORS, EVENTS ARE MOVERS, EVENTS ARE MANIPULATORS, and EVENTS ARE MOVERS AND MANIPULATORS. The remaining section discusses the projection of spatial stories without animate actors as well as nonspatial stories. (shrink)

Using Figurative Language presents results from a multidisciplinary decades-long study of figurative language that addresses the question, 'Why don't people just say what they mean?' This research empirically investigates goals speakers or writers have when speaking figuratively, and concomitantly, meaning effects wrought by figurative language usage. These 'pragmatic effects' arise from many kinds of figurative language including metaphors, verbal irony, idioms, proverbs and others. Reviewed studies explore mechanisms - linguistic, psychological, social and others - underlying pragmatic effects, some traced to (...) basic processes embedded in human sensory, perceptual, embodied, cognitive, social and schematic functioning. The book should interest readers, researchers and scholars in fields beyond psychology, linguistics and philosophy that share interests in figurative language - including language studies, communication, literary criticism, neuroscience, semiotics, rhetoric and anthropology. (shrink)

This article seeks to illuminate the figure of the child in democratic politics by arguing that children play a constitutive role as temporary outsiders who present both renewal and risk to the demos. Using Hannah Arendt’s concept of natality, we begin with an ontological account of children as new individuals that are central to renewing democratic freedom and plurality. In the second section, we explore how children can be conceived in terms of political risk by focussing on Arendt’s debate (...) with Ralph Ellison concerning the desegregation of American schools in the 1960s. Their arguments about whether children should appear in politics underscores the constitutive role that child–adult relationships play in debates about the normative fabric of democratic society. Finally, we use the radical democratic theory of Chantal Mouffe to argue that children can be characterised as an excluded group of potential adversaries that appear in political contests over claims to represent the demos. From this perspective, the article reveals that children are central figures in democratic politics because they are constituents of an unknown future polity. (shrink)

In this posthumous collection of writings, Cornelius Castoriadis (1922-1997) pursues his incisive analysis of modern society, the philosophical basis of our ability to change it, and the points of intersection between his many approaches to this theme. His main philosophical postulate, that the human subject and society are not predetermined, asserts the primacy of creation and the possibility of creative, autonomous activity in every domain. This argument is combined with penetrating political and social criticism, opening numerous avenues of critical thought (...) and action. The book’s wide-ranging topics include the core worldview of ancient Athens, where the idea of self-creation and self-limitation made democracy possible; the wealth of poetic resources; a deconstruction of the so-called rationality of capitalism and of the current conception of democracy, along with a discussion of what a radical, revolutionary project means today; the role of what he calls the radical imagination in the creation of both societal institutions and history; the roots of hate; a psychoanalytic view of human development torn between heteronomy and autonomy; the role of education in forming autonomous individuals; and notions of chaos, space, and number. (shrink)

Provides an overview and examples of what impossible figures are, and explains their interest to many different disciplines including philosophy, psychology, art and mathematics.

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)

Research devoted to characterizing phenomena is underappreciated in philosophical accounts of scientific inquiry. This paper develops a diachronic analysis of research over 100 years that led to the recognition of two related electrophysiological phenomena, the membrane potential and the action potential. A diachronic perspective allows for reconciliation of two threads in philosophical discussions of phenomena—Hacking’s treatment of phenomena as manifest in laboratory settings and Bogen and Woodward’s construal of phenomena as regularities in the world. The diachronic analysis also reveals the (...) epistemic tasks that contribute to establishing phenomena, including the development of appropriate investigative techniques and concepts for characterizing them. (shrink)

Two key themes in contemporary social theory are particularly relevant to the interpretation and critique of figurational sociology. On the one hand, some recent critiques of the sociological tradition — Touraine's attempt to deconstruct the received image of society is the most important example — have called into question a dominant paradigm that underlies both Marxist and structural-functional theories. Norbert Elias has not only anticipated some of the most important criticisms but also suggested correctives to some of the currently fashionable (...) alternatives. More specifically, his relationship to the sociological traditions and to its contemporary offshoots can be described in terms of six antitheses: his approach is anti-economistic, anti-normativistic, anti-reductionistic, anti-functionalist, anti-structuralist and anti-individualistic. On the other hand, the impact of all these critical strategies is somewhat blunted by the one-sided emphasis on power. A more balanced version of figurational sociology would need a concept of culture to match and counter-balance Elias's insights into the problematic of power. Further exploration of this issue could draw both on the classics and on post-Parsonian debates about cultural power and their interconnections. Cultural interpretations of power are the most important link between those two dimensions of social life; although Elias has never explicitly thematized them, his recent writings touch upon some aspects of their specifically modern variants. (shrink)

This volume juxtaposes philosophical and psychoanalytic speculation with literary and artistic commentary in order to approach a set of questions concerning the human relation to language. The multifold writing of the volume takes the form of a 'triptych' (following the model of works by Francis Bacon) rather than that of a thesis. The central section of the volume contains an extended dialogue on two textual passages from works by Maurice Blanchot and Jacques Lacan. The first part of the volume's triptych (...) focuses on the work of Francis Bacon, taking the motif of crucifixion as a path toward understanding his violent realism. The third part, which juxtaposes a dialogue with a critical essay, concerns the work of Salvatore Puglia. Through Bacon and Puglia, the author seeks another approach to a figural imperative at the limits of language. (shrink)

Representationalists hold that the phenomenal character of a perceptual experience is identical with, or supervenes on, an aspect of its representational content. As such, representationalism could be disproved by a counter-example consisting of two experiences that have the same representational content but differ in phenomenal character. In this paper, I discuss two recently proposed counter-examples to representationalism that involve ambiguous or reversible figures. I pursue two goals. My first, and most important, goal is to show that the representationalist can offer (...) plausible responses to both counter-examples. My second goal is to show the implications of these responses for the nature of the spatial representational contents of perceptual experiences. (shrink)

The advent of techniques of sex selection that rely on assisted reproduction led to a questioning of whether sex selection should be deemed always and everywhere unethical. While China and India are normally associated with condemned practices, they are also implicated in processes that constitute globally stratified sex selection inclusive of its more valued form, often referred to as family balancing. Through an application of Ong and Collier’s concept of global assemblage, I demonstrate how family balancing, which has taken on (...) a “global form,” is tied to an “assemblage” of factors related to the anti-natal, population control contexts that have been pervasive in Asia. Three simultaneously occurring processes since the mid-1990s constituted stratified sex selection: the surfacing of China and India as figurative counter examples in deliberations of ethics on new techniques; active national surveillance of sex ratios as well as denunciation and criminalization of sex selective abortion in China and India; and the role of China and India in neoliberalizing population control and developing globalized markets in reproduction. Accounting for globally stratified sex selection requires a more robust interpretation of ethics that rethinks disciplinary approaches just as much as relativist ones in which respect for individual autonomy tends to overtake all other concerns. (shrink)

In this updated edition of his brief, engaging book, Robert J. Fogelin examines figures of speech that concern meaning-irony, hyperbole, understatement, similes, metaphors, and others-to show how they work and to explain their attraction.

We think and speak in figures. This is key to our creativity. We re-imagine one thing as another, pretend ourself to be another, do one thing in order to achieve another, or say one thing to mean another. This comes easily because of our abilities both to work out meaning in context and re-purpose words. Figures of speech are tools for this re-purposing. Whether we use metaphor, simile, irony, hyperbole, and litotes individually, or as compound figures, the uses are all (...) rooted in literal meanings. These uses invite us to explore the context to find new meanings, new purposes, beyond the literal. -/- This special issue of Philosophical Studies brings together eight papers on various aspects and applications of figurative speech including: David Hills, Mitch Green, Liz Camp, Ofra Magidor, Larry Horn, Ken Walton, Stephen Barker, and Mihaela Popa-Wyatt. (shrink)

This book offers a much-needed new political theory of an old phenomenon. The last decade alone has marked the highest number of migrations in recorded history. Constrained by environmental, economic, and political instability, scores of people are on the move. But other sorts of changes—from global tourism to undocumented labor—have led to the fact that to some extent, we are all becoming migrants. The migrant has become the political figure of our time. Rather than viewing migration as the exception (...) to the rule of political fixity and citizenship, Thomas Nail reinterprets the history of political power from the perspective of the movement that defines the migrant in the first place. Applying his "kinopolitics" to several major historical conditions and figures of migration, he provides fresh tools for the analysis of contemporary migration. (shrink)

Rarely has the history or philosophy of mathematics been written about by mathematicians, and the analysis of mathematical texts themselves has been an area almost entirely unexplored. _Figures of Thought_ looks at ways in which mathematical works can be read as texts, examines their textual strategies and demonstrates that such readings provide a rich source of philosophical issues regarding mathematics: issues which traditional approaches to the history and philosophy of mathematics have neglected. David Reed, a professional mathematician himself, offers the (...) first sustained and critical attempt to find a consistent argument or narrative thread in mathematical texts. In doing so he develops new and fascinating interpretations of mathematicians' work throughout history, from an in-depth analysis of Euclid's Elements, to the mathematics of Descartes and right up to the work of contemporary mathematicians such as Grothendeick. He also traces the implications of this approach to the understanding of the history and development of mathematics. (shrink)

L'objectif est ici de déterminer la mesure dans laquelle nous serions aujourd'hui sortis des catégories conceptuelles sur lesquelles la pensée politique moderne s'est construite.

This book offers a much-needed new political theory of an old phenomenon. The last decade alone has marked the highest number of migrations in recorded history. Constrained by environmental, economic, and political instability, scores of people are on the move. But other sorts of changes—from global tourism to undocumented labor—have led to the fact that to some extent, we are all becoming migrants. The migrant has become the political figure of our time. Rather than viewing migration as the exception (...) to the rule of political fixity and citizenship, Thomas Nail reinterprets the history of political power from the perspective of the movement that defines the migrant in the first place. Applying his "kinopolitics" to several major historical conditions (territorial, political, juridical, and economic) and figures of migration (the nomad, the barbarian, the vagabond, and the proletariat), he provides fresh tools for the analysis of contemporary migration. (shrink)

This paper is a systematic investigation of motion expressions in programmatic music description. To address issues with defining the Source MOTION and the Target MUSIC, we utilize Gestalt models (Figure-Ground and Source-Path-Goal) while also critically examining the ontological complexity of the Target MUSIC. We also investigate music motion descriptions considering the role of the describer’s perspective and communicative goals. As previous research has demonstrated, an attentional Goal-bias is common in physical motion description, yet this has been found also to (...) lessen due to audience accommodation effects. We investigate whether this also occurs in music description. Using cognitive linguistic frameworks, we conducted an analysis of 21 English speakers’ written descriptions of dynamic orchestral excerpts. All participants gave a description of one excerpt reporting their own personal experiences and the other excerpt reporting the events of the excerpt for a fictional future participant. We find that addressee accommodation shapes the choice of the ontological types of Figures used from being more subjective and creative in describing music for oneself versus being more objective in describing music for others. However, our investigation does not find sufficient evidence for a Goal-bias in music like there is in physical motion event descriptions. (shrink)

Ambiguous figures pose a problem for representationalists, particularly for representationalists who believe that the content of perceptual experience is non-conceptual (MacPherson in Nous 40(1):82–117, 2006). This is because, in viewing ambiguous figures, subjects have perceptual experiences that differ in phenomenal properties without differing in non-conceptual content. In this paper, I argue that ambiguous figures pose no problem for non-conceptual representationalists. I argue that aspect shifts do not presuppose or require the possession of sophisticated conceptual resources and that, although viewing ambiguous (...) figures often causes a change in phenomenal properties, this change is accompanied by a change in non-conceptual content. I illustrate the case by considering specific examples. (shrink)

Compound figures are a rich, and under-explored area for tackling fundamental issues in philosophy of language. This paper explores new ideas about how to explain some features of such figures. We start with an observation from Stern that in ironic-metaphor, metaphor is logically prior to irony in the structure of what is communicated. Call this thesis Logical-MPT. We argue that a speech-act-based explanation of Logical-MPT is to be preferred to a content-based explanation. To create this explanation we draw on Barker’s (...) expressivist speech-act theory, in which speech-acts build on other speech-acts to achieve the desired communicative effects. In particular, we show how Barker’s general ideas explain metaphor as an assertive-act, and irony as a ridiculing-act. We use Barker’s notion of proto-illocutionary-acts to show how metaphorical-acts and ironic-acts can build one on the other. Finally, we show that while an ironic-act can build on a metaphorical-act, a metaphorical-act cannot build on an ironic-act. This restriction on how they can be composed establishes Logical-MPT via a different route. (shrink)

From the seventeenth through the nineteenth centuries, scientists such as Kekule, Mendel, Lavoisier and Harvey argued for insights that depended critically on antithetical expressions and reasoning. The heuristic and persuasive use of devices like the antithesis has roots in the in combined grammatical, rhetorical and dialectical training established during the early modern educational reforms of the humanists. While the entire array of figures includes devices which inscribe all the rhetorical appeals, the set of devices derived from parallel phrasing illustrates how (...) certain figures of speech express lines of reasoning iconically. But the continued use of such devices invites a general rationale for their persuasiveness based on the importance of pattern completion in language processing. (shrink)

Rarely has the history or philosophy of mathematics been written about by mathematicians, and the analysis of mathematical texts themselves has been an area almost entirely unexplored. _Figures of Thought_ looks at ways in which mathematical works can be read as texts, examines their textual strategies and demonstrates that such readings provide a rich source of philosophical issues regarding mathematics: issues which traditional approaches to the history and philosophy of mathematics have neglected. David Reed, a professional mathematician himself, offers the (...) first sustained and critical attempt to find a consistent argument or narrative thread in mathematical texts. In doing so he develops new and fascinating interpretations of mathematicians' work throughout history, from an in-depth analysis of Euclid's Elements, to the mathematics of Descartes and right up to the work of contemporary mathematicians such as Grothendeick. He also traces the implications of this approach to the understanding of the history and development of mathematics. (shrink)