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This thesis articulates the resonances between J.M. Coetzee's lifelong engagement with mathematics and his practice as a novelist, critic, and poet. Though the critical discourse surrounding Coetzee's literary work continues to flourish, and though the basic details of his background in mathematics are now widely acknowledged, his inheritance from that background has not yet been the subject of a comprehensive and mathematically- literate account. In providing such an account, I propose that these two strands of his intellectual trajectory not only developed in parallel, but together engendered several of the characteristic qualities of his finest work. The structure of the thesis is essentially thematic, but is also broadly chronological. Chapter 1 focuses on Coetzee's poetry, charting the increasing involvement of mathematical concepts and methods in his practice and poetics between 1958 and 1979. Chapter 2 situates his master's thesis alongside archival materials from the early stages of his academic career, and thus traces the development of his philosophical interest in the migration of quantificatory metaphors into other conceptual domains. Concentrating on his doctoral thesis and a series of contemporaneous reviews, essays, and lecture notes, Chapter 3 details the calculated ambivalence with which he therein articulates, adopts, and challenges various statistical methods designed to disclose objective truth. Chapter 4 explores the thematisation of several mathematical concepts in Dusklands and In the Heart of the Country. Chapter Five considers Waiting for the Barbarians and Foe in the context provided by Coetzee's interest in the attempts of Isaac Newton to bridge the gap between natural language and the supposedly transparent language of mathematics. Finally, Chapter 6 locates in Elizabeth Costello and Diary of a Bad Year a cognitive approach to the use of mathematical concepts in ethics, politics, and aesthetics, and, by analogy, a central aspect of the challenge Coetzee's late fiction poses to the contemporary literary landscape.

This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour.

Cette thèse se propose de déterminer les apports d’Oronce Fine (1494-1555) à la philosophie des mathématiques de la Renaissance. En tant que premier titulaire de la première chaire royale de mathématiques, ce mathématicien a joué un rôle important dans la revalorisation de l’enseignement des mathématiques dans la France du XVIe siècle. Dans cette mesure, sa conception des mathématiques permet de montrer l’évolution du statut épistémologique et institutionnel de ces disciplines dans le milieu académique parisien de cette période. Parmi les thèmes abordés par Fine dans sa définition du statut des mathématiques, nous avons choisi d’étudier, dans une première partie, la nature des objets du mathématicien, le statut épistémologique de l’astronomie, la nature des procédures démonstratives et des principes des mathématiques, ainsi que la fonction du quadrivium dans le processus éducatif. Dans une seconde partie, notre analyse de la pensée de Fine porte sur le statut des mathématiques pratiques et des disciplines subalternes des mathématiques, à savoir la perspective et la géométrie, ainsi que sur le profit qui peut être obtenu de l’apprentissage du quadrivium. / The aim of this study is to determine the contributions of Oronce Fine (1494-1555) to Renaissance philosophy of mathematics. As first Royal lecturer in mathematics, Fine played a major part in the reassertion of the value of mathematical teaching in sixteenth-century France. Thus, his thought concerning mathematics allows to set forth the evolution of the epistemological and institutional status of these sciences within the parisian academic context of the period. Among the questions tackled by Fine in his definition of the status of mathematics, we consider, in a first part, the ontological status of mathematical things, the epistemological status of astronomy, the nature of mathematical demonstrations and principles, as well as the function of the quadrivium in the educative process. In a second part, our analysis of Fine’s conception on mathematics deals with the status of practical mathematics and of the sciences which are subalternated to mathematics, that is optics and geography, concluding with the definition of the profit which may be obtained from learning mathematics.

The philosophy of linguistics is a rich philosophical domain which encompasses various disciplines. One of the aims of this thesis is to unite theoretical linguistics, the philosophy of language, the philosophy of science (particularly mathematics and modelling) and the ontology of language. Each part of the research presented here targets separate but related goals with the unified aim of bringing greater clarity to the foundations of linguistics from a philosophical perspective. Part I is devoted to the methodology of linguistics in terms of scientific modelling. I argue against both the Conceptualist and Platonist (as well as Pluralist) interpretations of linguistic theory by means of three grades of mathematical involvement for linguistic grammars. Part II explores the specific models of syntactic and semantics by an analogy with the harder sciences. In Part III, I develop a novel account of linguistic ontology and in the process comment on the type-token distinction, the role and connection with mathematics and the nature of linguistic objects. In this research, I offer a structural realist interpretation of linguistic methodology with a nuanced structuralist picture for its ontology. This proposal is informed by historical and current work in theoretical linguistics as well as philosophical views on ontology, scientific modelling and mathematics.

The central question of this thesis is: what is mathematics about? The answer arrived at by the thesis is an unsettling and unsatisfying one. By examining two of the most promising contemporary accounts of the nature of mathematics, I conclude that neither is as yet capable of giving us a conclusive answer to our question. The conclusion is arrived at by a combination of historical and conceptual analysis. It begins with the historical fact that, since the middle of the nineteenth century, mathematics has undergone a radical transformation. This transformation occurred in most branches of mathematics, but was perhaps most apparent in geometry. Earlier images of geometry understood it as the science of space. In the wake of the emergence of multiple distinct geometries and the realization that non-Euclidean geometries might lay claim to the description of physical space, the old picture of Euclidean geometry as the sole correct description of physical space was no longer tenable. The first chapter of the dissertation provides an historical account of some of the forces which led to the destabilization of the traditional picture of geometry. The second chapter examines the debate between Gottlob Frege and David Hilbert regarding the nature of geometry and axiomatics, ending with an argument suggesting that Hilbert’s views are ultimately unsatisfying. The third chapter continues to probe the work of Frege and, again, finds his explanations of the nature of mathematics troublingly unsatisfying. The end result of the first three chapters is that the Frege-Hilbert debate leaves us with an impasse: the traditional understanding of mathematics cannot hold, but neither can the two most promising modern accounts. The fourth and final chapter of the thesis investigates mathematical structuralism—a more recent development in the philosophy of mathematics—in order to see whether it can move us beyond the impasse of the Frege-Hilbert debate. Ultimately, it is argued that the contemporary debate between ‘assertoric’ structuralists and ‘algebraic’ structuralists recapitulates a form of the Frege-Hilbert impasse. The ultimate claim of the thesis, then, is that neither of the two most promising contemporary accounts can offer us a satisfying philosophical answer to the question ‘what is mathematics about?’.

Ce mémoire porte sur la réponse leibnizienne à la question de l’utilité des mathématiques pour la connaissance de la nature, c’est-à-dire, en l’occurrence, pour la connaissance des phénomènes corporels et de leurs relations. Dans le premier chapitre, nous nous intéressons à la façon dont les notions abstraites mathématiques entrent dans la connaissance la plus immédiate des choses. à travers le mode par lequel nous apparaît l’individualité des phénomènes. Après avoir fourni des éclaircissements métaphysiques sur la conception leibnizienne de l’individuation, nous nous plongeons dans l’étude de la position spatiale à la lumière de l’analyse géométrique leibnizienne. Ce dernier prédicat fournit une manière de déterminer les individus qui ne sont pas bien distingués par nous au moyen de leurs qualités réelles. Considérés sous le seul angle de leur individuation spatiale, les phénomènes ont un caractère idéal et indéterminé qui les rend immédiatement susceptibles d’un traitement mathématique. Dans le second chapitre, nous nous intéressons à la question de savoir pourquoi les explications physiques qui font usage des mathématiques sont pour Leibniz préférables épistémologiquement. Nous nous tournons en conséquence vers ses raisons d’adhérer à la philosophie mécanique, qui contient une composante mathématique essentielle, afin d’étudier celle qui tient à la plus grande intelligibilité du mécanisme. Nous tentons de montrer que la composante mathématique du mécanisme contribue à cette intelligibilité parce que les mathématiques proposent une mode de raisonnement valide et expressément adapté à la situation épistémologique des esprits finis. Ce mode produit des raisonnements nécessaires aux moyens de notions incomplètes. Il suscite également la découverte de nouvelles vérités en offrant à l’imagination un support sensible, contrôlable et évident. / This thesis explores Leibniz’s solution to the problem of how mathematics are useful to our understanding of the world, i.e., to our understanding of corporeal phenomena and their relations. In the first chapter, it focuses on how abstract mathematical notions enter in our most immediate understanding of the world. Here, the aim is connecting the pervasiveness of mathematics to the peculiar way by which the individuality of phenomena manifests itself to us. After some metaphysical remarks on Leibniz’s conception of individuation, we study spatial position in the light of the new leibnizian geometrical analysis : Analysis Situs. Spatial position provides us with a way to further distinguish between individual phenomena whose qualities relevant to their real individuation remain ignored. In the sole light of spatial individuation, phenomena are ideal and indeterminate. This situation renders them susceptible to mathematical treatment without further elaboration. In the second chapter, we turn our attention to the question of why mathematical methods in philosophy of nature are epistemologically superior in Leibniz’s eyes. We explore Leibniz’s reason to espouse a mechanical philosophy which comprise indispensable mathematical notions. Leibniz believes that mechanical philosophy is the most intelligible explanation of nature and we mean to assess how mathematics enter this picture. We try to show that the mathematical aspects of mechanical philosophy make it more intelligible by virtue of mathematics’ peculiar mode of reasoning. This mode of reasoning is valid as well as most suited for our finite minds. It provides necessary arguments through incomplete notions. It also encourages the discovery by assisting the imagination with controlled and sensible support that makes knowledge more evident.

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Dans cette thèse, je propose une lecture renouvelée de l’itinéraire philosophique d’Hilary Putnam concernant la problématique du réalisme. Mon propos consiste essentiellement à défendre l’idée selon laquelle il y aurait beaucoup plus de continuité, voir une certaine permanence, dans la manière dont Putnam a envisagé la question du réalisme tout au long de sa carrière. Pour arriver à une telle interprétation de son oeuvre, j’ai essentiellement suivi deux filons. D’abord, dans un ouvrage du début des années 2000, Ethics without Ontology (2004), Putnam établit un parallèle entre sa conception de l’objectivité en philosophie des mathématiques et en éthique. Le deuxième filon vient d’une remarque qu’il fait, dans l’introduction du premier volume de ses Philosophical Papers (1975), affirmant que la forme de réalisme qu’il présupposait dans ses travaux des années 1960-1970 était la même que celle qu’il défendait en philosophie des mathématiques et qu’il souhaitait défendre ultérieurement en éthique. En suivant le premier filon, il est possible de mieux cerner la conception générale que se fait Putnam de l’objectivité, mais pour comprendre en quel sens une telle conception de l’objectivité n’est pas propre aux mathématiques, mais constitue en réalité une conception générale de l’objectivité, il faut suivre le second filon, selon lequel Putnam aurait endossé, durant les années 1960-1970, le même type de réalisme en philosophie des sciences et en éthique qu’en philosophie des mathématiques. Suivant cette voie, on se rend compte qu’il existe une similarité structurelle très forte entre le premier réalisme de Putnam et son réalisme interne. Après avoir établi la parenté entre le premier et le second réalisme de Putnam, je montre, en m’inspirant de commentaires du philosophe ainsi qu’en comparant le discours du réalisme interne au discours de son réalisme actuel (le réalisme naturel du commun des mortels), que, contrairement à l’interprétation répandue, il existe une grande unité au sein de sa conception du réalisme depuis les années 1960 à nos jours. Je termine la thèse en montrant comment mon interprétation renouvelée de l’itinéraire philosophique de Putnam permet de jeter un certain éclairage sur la forme de réalisme que Putnam souhaite défendre en éthique. / In this dissertation I propose a new reading of the philosophical itinerary of Hilary Putnam on the matter of realism. In essence, my purpose is to argue that there is much more continuity than is normally understood, and even a degree of permanence, in the way in which Putnam has viewed the question of realism throughout his career. To arrive at this interpretation of Putnam I essentially followed two veins in his work. First, in a volume published in the early 2000s entitled Ethics without Ontology (2004), Putnam establishes a parallel between his conception of objectivity in the philosophy of mathematics and in ethics. The second vein comes from a comment he made in the introduction to the first volume of his Philosophical Papers (1975) to the effect that the kind of realism he presupposed in his work of the 1960s and 70s was the same that he upheld in the philosophy of mathematics and wished to argue for at a later date in ethics. Following the first vein makes it possible to better grasp Putnam’s general conception of objectivity, but in order to understand how such a conception of objectivity is not unique to mathematics but is instead a general conception of objectivity one must follow the second vein. There, in the 1960s and 70s, Putnam adopted the same kind of realism in the philosophy of science and in ethics as he had in the philosophy of mathematics. Following this path, one realises that there exists a very strong structural similarity between Putnam’s first realism and his internal realism. After establishing this connection between Putnam’s first and second realism, I draw on Putnam’s remarks and compare the internal realism discourse to his current realism (the natural realism of ordinary people) to demonstrate, contrary to the prevalent interpretation, that there has been a great deal of consistency in his conception of realism from the 1960s to the present day. I conclude the dissertation by demonstrating how my new interpretation of Putnam’s philosophical itinerary makes it possible to shed light on the kind of realism he wishes to champion in ethics.