This thesis attempts to provide a response to the Access Problem by developing a naturalist account of our access to mathematical knowledge. On the basis of recent empirical research into the nature of mathematical cognition, it is argued that our most basic access to arithmetical content is mediated by perceptual processes. Moreover, in line with the theory of embodied cognition, arithmetical cognition is grounded in the perceptual systems responsible for these processes, as well as other perceptual and motor systems that are involved with our everyday interaction with the world. This motivates a response to the Access Problem according to which access to some mathematical content is on a par with our access to everyday objects of perception. Whilst the picture that emerges on the basis of this response is ontologically neutral, in the sense of being compatible with either a realist or anti-realist approach to mathematics, it places significant constraints on a naturalistically acceptable approach to the ontology of mathematics.

One project in the epistemology of mathematics is to find a defensible account of what passes for mathematical knowledge. This study contributes to this project by examining philosophical theories of mathematics governed by certain basic assumptions. Foremost amongst these is the "linguistic realism" of the title. Roughly put, this is the view that the semantics of mathematical sentences should be taken at face value. Two approaches to mathematics are considered, realist and fictionalist. Mathematical realism affirms the existence of mathematical objects, taking much of what passes for mathematical knowledge as knowledge of such things. It faces the challenge of explaining how such knowledge is possible. The main strategies here are to appeal to the faculty of reason, to a faculty of intuition or to the faculty of sense perception. Recent examples of each strategy are considered and it is argued that the prospects for a satisfactory mathematical realism are limited. Mathematical fictionalism does not affirm the existence of mathematical objects, claiming that mathematics is, or should be considered to be, a form of pretence. It faces the challenge of explaining how a form of pretence can discharge the roles mathematics has in empirical applications. Strategies here are to argue that mathematics is an eliminable convenience or, acknowledging that this may not be the case, that the roles played by mathematics in empirical applications are played in similar contexts by acknowledged forms of pretence. It is argued that the first strategy is not promising but that there is a version of the second that can be defended against objections. In closing, consequences of the conclusions reached are explored and directions for future research indicated.

This thesis defends the view that the role of applied mathematics is a representational one, and develops a mapping account of the applicability of mathematics that does justice to this representational conception. The first chapter outlines some philosophical problems of applicability and some of its history. In the second chapter I explain in detail what the mapping account is, examining mappings, and representation theorems, and give any account how mathematics can represent derived attributes and laws. In chapter three I argue against the possibility of genuine platonistic explanations of physical phenomena. This is necessary as if there were such explanations they would entail that platonistic mathematics is not extrinsic to what actually goes on in the physical world, and that a purely representational conception of the applicability of mathematics is either straightforwardly false or radically incomplete. In chapter four a positive proposal, based largely upon the work of Hartry Field, is given for showing how it is that we can state scientific theories in such a way that platonistic mathematics does not appear as part of scientific theories. This is essential, since although the previous chapter argued that such genuine explanations are impossible, it did not show positively how we could dispense with platonistic mathematics in scientific explanations. Chapter five concerns a philosophical problem of applicability, the ’descriptive problem of applicability’ which, it has been argued, goes beyond mere ’representational’ issues and poses a problem for the mapping account of applicability. I identify three species of descriptive problems and reject the possibility of anthropocentrism as a solution to the descriptive problem.

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Attempts to solve the problems of the applicability of mathematics have gen- erally originated from the acceptance of a particular mathematical ontology. In this thesis I argue that a proper approach to solving these problems comes from an ‘application first’ approach. If one attempts to form the problems and answer them from a position that is agnostic towards mathematical ontology, the difficulties surrounding these problems fall away. I argue that there are nine problems that require answering, and that the problems of representation are the most interesting questions to answer. The applied metaphysical problem can be answered by structural relations, which are adopted as the starting point for accounts of representation. The majority of the thesis concerns arguing in favour of structural accounts of representation, in particular deciding between the Inferential Conception of the Applicability of Mathematics and Pincock’s Mapping Account. Through the case study of the rainbow, I argue that the Inferential Conception is the more viable account. It is capable of answering all of the problems of the applicability of mathematics, while the methodology adopted by Pincock trivialises the answer it can supply to the vital question of how the faithfulness and usefulness of representations are related.

This study takes an activity-theoretic approach to abstraction in context recently proposed by Hershkowitz, Schwarz and Dreyfus (2001, HSD hereafter). Key to HSD's theory of abstraction is the construction of new mathematical knowledge and consolidation of it. In this connection, this study aims to investigate three particular issues: (1) the construction of mathematical knowledge through scaffolding, (2) the nature of the consolidation process and (3) the validity ofHSD's abstraction theory. In order to investigate these issues, a qualitative research design methodology with explanatory and exploratory inquiry purposes was taken. This study employed multiple case study strategy with the purpose of literal and theoretical replications. A number of cases were designed with students working as pairs and individuals such that some of the students worked with the scaffolded help and others without. All participants worked on four days over four sequential tasks connected with the graphs of absolute value functions. Tasks were applied in paper-andpencil format. The data for this study was composed of the participant's written works and audio records of the sessions. In relation to the first issue, analysing the students' verbal data suggests certain causative relationships between the scaffolder's interventions and the students' developing constructions. It is also observed that the scaffolder's interventions mediate the students' constructions. Analysis of the data further suggests that construction through scaffolding is a subtle and intricate phenomenon which involves a complex set of social, cultural, historical, contextual and semiotic issues. It is argued, with examples, that scaffolded discourse involves many dynamics such as value judgements, individuals' personal histories, common cultural practices, individuals' emergent goals, voices of absent others and certain patterns of interaction. Regarding the second issue, the data suggest that newly formed constructions are fragile entities and in need of consolidation. In the course of consolidation, it is observed that earlier constructions are reconstructed, used in a flexible manner and expressed confidently with general mathematical statements. These observations lead to the argument that an abstraction is a consolidated construction that can be used to create new constructions. With regard to the final issue, on the basis of the students' verbal data, this study provides a critical evaluation of HSD's theory of abstraction by focusing on three key dimensions which characterise it: its epistemological and sociocultural principles, epistemic actions and genesis of an abstraction. Throughout this evaluation some clarifications and amendments are proposed to this theory.

My aim is to clarify Wittgenstein's foundational outlook. I shall argue that he was neither a strict fmitist, nor an intuitionist, but a finitist (Skolem and Goodstein.) In chapter I, I argue that Wittgenstein was a "revisionist" in philosophy of mathematics. In chapter II, I set up a distinction between Kronecker's divisor-theoretical approach to algebraic number theory and the set-theoretic style of Dedekind's ideal-theoretic approach, in order to show that Wittgenstein's remarks on existential proofs and the Axiom of Choice are in the constructivist tradition. In chapter in, I give an exposition of the logicist definitions of the natural numbers by Dedekind and Frege, and of the charge of impredicativity levelled against them by Poincaré, in order to show, in chapter IV, that Wittgenstein's definition of the natural number in the Tractatus Logico-Philosophicus was constructivist. I also discuss the notions of generality and quantification, and Wittgenstein's later criticisms of the notion of numerical equality. In chapter V, after discussing the current strict finitist literature, I reject the contention that Wittgenstein's remarks give support to such a programme, by showing that he adhered to a potentialist view of the infinite, and, moreover, that his "grammatical" approach provides him with an argument against strict finitism. In chapter VII, I also reject the identification of his remarks about "surveyability" with the strict finitist insistence on "feasibility." In chapter VI, I describe the Grundlagenstreit about the status of Π⁰₁ -statements. Wittgenstein views on generality, induction, and the quantifiers lead to a rejection of quantification theory which sets him apart from intuitionism, and closer to finitism. I also examine Wittgenstein's argument against the Law of Excluded Middle. In the last chapter, I discuss Wittgenstein's prescriptions for the formation of real numbers, showing that they imply a constructivization of the Cauchy sequences of the type of Bishop or of the finitist "recursive analysis", and the rejection of the intuitionistic notion of choice sequences.

Cette thèse essaie de reconstituer l’histoire de la réception du calcul leibnizien dans les milieux savants français (1690-1706). Nous repérons deux lieux : d’abord au sein d’un groupe autour de Malebranche, initié au calcul par Jean Bernoulli, puis à l’Académie des sciences. Dans les deux cas nous mettons en avant les horizons d’attente des acteurs. Alors que cet épisode a été beaucoup étudié en termes de rupture, nous insistons, par une analyse des sources primaires – dont plusieurs inédites – sur le fait que l’appropriation du calcul s’effectue aussi grandement sur le fond de pratiques en usage. Dans la première partie, nous examinons l’héritage mathématique à partir duquel est reçu le calcul de Leibniz par le groupe autour de Malebranche. Cette analyse nous permet de montrer que leur appropriation s’appuie sur des pratiques partagées et non sur un terrain vierge comme on l’a trop souvent supposé. Nos mathématiciens réalisent que l’algorithme différentiel permet de donner une étoffe nouvelle à des notions déjà impliquées dans les méthodes d’invention précédentes. Dans la seconde partie, nous étudions la genèse et la structuration du premier ouvrage de calcul différentiel écrit par l’Hospital et publié en 1696 sous le titre Analyse des infiniment petits pour l’intelligence des courbes. Après cette publication, le calcul devient très présent à l’Académie. Une crise y éclate entre partisans et adversaires du calcul. L’examen de leurs discours, objet de notre troisième partie, permet de préciser les notions telles que celle de différentielle ou de courbe, ainsi que la manière dont il est possible d’interpréter géométriquement les résultats issus des calculs. / This thesis is an attempt to reconstruct the reception history of Leibnizian calculus in French learned milieux (1690-1706). Two areas have been located: first among members of Malebranche’s circle, introduced to calculus by Jean Bernoulli, then the Académie des Sciences. In either case, the purpose is to highlight the horizon of expectation of the participants. Whereas this episode has been widely studied in terms of disruption, it is argued, through an analysis of primary sources –some of which un-edited– that calculus was greatly appropriated against a background of practices in use. The first chapter examines the mathematical heritage from which calculus was received by Malebranche’s circle. This analysis enables me to show that their appropriation rested on shared practices, and was not a virgin land, as has often been supposed. Our mathematicians realized that the differential algoritm fleshed out notions already involved in previous invention methods. The second chapter studies the genesis and construction of the first book of differential calculus written by L’Hospital and published in 1696, entitled Analyse des infiniment petits pour l’intelligence des courbes [Analysis of the infinitely small for the intelligence of curbs]. After this publication, calculus became very present at the Académie. A crisis arose between supporters and detractors of calculus. A close examination of their discourses –the object of my third chapter– helps clarify such notions as those of differential and curb, as well as the way it is possible to geometrically interpret the results from calculus.

Using ethnographic observations and interview based research I document the production of research mathematics in four European research institutes, interviewing 45 mathematicians from three areas of pure mathematics: topology, algebraic geometry and differential geometry. I use Bourdieu's notions of habitus, field and practice to explore how mathematicians come to perceive and interact with abstract mathematical spaces and constructions. Perception of mathematical reality, I explain, depends upon enculturation within a mathematical discipline. This process of socialisation involves positioning an individual within a field of production. Within a field mathematicians acquire certain structured sets of dispositions which constitute habitus, and these habitus then provide both perspectives and perceptual lenses through which to construe mathematical objects and spaces. I describe how mathematical perception is built up through interactions within three domains of experience: physical spaces, conceptual spaces and discourse spaces. These domains share analogous structuring schemas, which are related through Lakoff and Johnson's notions of metaphorical mappings and image schemas. Such schemas are mobilised during problem solving and proof construction, in order to guide mathematicians' intuitions; and are utilised during communicative acts, in order to create common ground and common reference frames. However, different structuring principles are utilised according to the contexts in which the act of knowledge production or communication take place. The degree of formality, privacy or competitiveness of environments affects the presentation of mathematicians' selves and ideas. Goffman's concept of interaction frame, front-stage and backstage are therefore used to explain how certain positions in the field shape dispositions, and lead to the realisation of different structuring schemas or scripts. I use Sewell's qualifications of Bourdieu's theories to explore the multiplicity of schemas present within mathematicians' habitus, and detail how they are given expression through craftwork and bricolage. I argue that mathematicians' perception of mathematical phenomena are dependent upon their positions and relations. I develop the notion of social space, providing definitions of such spaces and how they are generated, how positions are determined, and how individuals reposition within space through acquisition of capital.

Bibliography: leaves 263-266. / v, 266 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This thesis investigates two related aspects of a dualisation program for the intuitionist logic in categories. The dualisation program has as its end the presentation of closed set logic in place of the usual open set logic found in association with toposes. The study is concerned especially with Brouwerian algebras in categories as the duals of the usual Heyting algebras. Defines the notion of a sheaf over the closed sets of a topological space. Investigates the sheaves for their algebric properties in relation to base space topologies. / Thesis (Ph.D.)--University of Adelaide, Dept. of Philosophy, 1996