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

My thesis is an exposition and defence of Aristotle's philosophy of mathematics. The first part of my thesis is an exposition of Aristotle's cryptic and challenging view on mathematics and is based on remarks scattered all over the corpus aristotelicum. The thesis' central focus is on Aristotle's view on numbers rather than on geometrical figures. In particular, number is understood as a countable plurality and is always a number of something. I show that as a consequence the related concept of counting is based on units. In the second part of my thesis, I verify Aristotle's view on number by applying it to his account of time. Time presents itself as a perfect test case for this project because Aristotle defines time as a kind of number but also considers it as a continuum. Since numbers and continuous things are mutually exclusive this observation seems to lead to an apparent contradiction. I show why a contradiction does not arise when we understand Aristotle properly. In the third part, I argue that the ontological status of mathematical objects, dubbed as materially [hulekos, ÍlekÀc] by Aristotle, can only be defended as an alternative to Platonism if mathematical objects exist potentially enmattered in physical objects. In the fourth part, I compare Aristotle's and Plato's views on how we obtain knowledge of mathematical objects. The fifth part is an extension of my comparison between Aristotle's and Plato's epistemological views to their respective ontological views regarding mathematics. In the last part of my thesis I bring Frege's view on numbers into play and engage with Plato, Aristotle and Frege equally while exploring their ontological commitments to mathematical objects. Specifically, I argue that Frege should not be mistaken for a historical Platonist and that we find surprisingly many similarities between Frege and Aristotle. After having acknowledged commonalities between Aristotle and Frege, I turn to the most significant differences in their views. Finally, I defend Aristotle's abstractionism in mathematics against Frege's counting block argument. This whole project sheds more light on Aristotle's view on mathematical objects and explains why it remains an attractive view in the philosophy of mathematics.

In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content the truth of which does not require mathematical objects to exist. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue that the role played by mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more than we otherwise would be able to about, or yielding a greater understanding of, the physical world. Mathematical objects to not need to exist for mathematical language to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences and show that they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which shows that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show the hermeneutic fictionalist position that emerges is preferable to any alternative which interprets mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects.

This thesis is concerned with the debate between absolutists and relativists about generality. Absolutists about quantification contend that we can quantify over absolutely everything; relativists deny this. The introduction motivates and elucidates the dispute. More familiar, restrictionist versions of relativism, according to which the range of quantifiers is always subject to restriction, are distinguished from the view defended in this thesis, an expansionist version of relativism, according to which the range of quantifiers is always open to expansion. The remainder of the thesis is split into three parts. Part I focuses on generality. Chapter 2 is concerned with the semantics of quantifiers. Unlike the restrictionist, the expansionist need not disagree with the absolutist about the semantics of quantifier domain restriction. It is argued that the threat of a certain form of semantic pessimism, used as an objection against restrictionism, also arises, in some cases, for absolutism, but is avoided by expansionism. Chapter 3 is primarily engaged in a defensive project, responding to a number of objections in the literature: the objection that the relativist is unable to coherently state her view, the objection that absolute generality is needed in logic and philosophy, and the objection that relativism is unable to accommodate ‘kind generalisations’. To meet these objections, suitable schematic and modal resources are introduced and relativism is given a precise formulation. Part II concerns issues in the philosophy of mathematics pertinent to the absolutism/relativism debate. Chapter 4 draws on the modal and schematic resources introduced in the previous chapter to regiment and generalise the key argument for relativism based on the set-theoretic paradoxes. Chapter 5 argues that relativism permits a natural motivation for Zermelo-Fraenkel set theory. A new, bi-modal axiomatisation of the iterative conception of set is presented. It is argued that such a theory improves on both its non-modal and modal rivals. Part III aims to meet a thus far unfulfilled explanatory burden facing expansionist relativism. The final chapter draws on principles from metasemantics to offer a positive account of how universes of discourse may be expanded, and assesses the prospects for a novel argument for relativism on this basis.

In broadest terms, this thesis is concerned to answer the question of whether the view that arithmetic is analytic can be maintained consistently. Lest there be much suspense, I will conclude that it can. Those who disagree claim that accounts which defend the analyticity of arithmetic are either unable to give a satisfactory account of the foundations of mathematics due to the incompleteness theorems, or, if steps are taken to mitigate incompleteness, then the view loses the ability to account for the applicability of mathematics in the sciences. I will show that this criticism is not successful against every view whereby arithmetic is analytic by showing that the brand of "conventionalism" about mathematics that Rudolf Carnap advocated in the 1930s, especially in Logical Syntax of Language, does not suffer from these difficulties. There, Carnap develops an account of logic and mathematics that ensures the analyticity of both. It is based on his famous "Principle of Tolerance", and so the major focus of this thesis will to defend this principle from certain criticisms that have arisen in the 80 years since the book was published. I claim that these criticisms all share certain misunderstandings of the principle, and, because my diagnosis of the critiques is that they misunderstand Carnap, the defense I will give is of a primarily historical and exegetical nature. Again speaking broadly, the defense will be split into two parts: one primarily historical and the other argumentative. The historical section concerns the development of Carnap's views on logic and mathematics, from their beginnings in Frege's lectures up through the publication of Logical Syntax. Though this material is well-trod ground, it is necessary background for the second part. In part two we shift gears, and leave aside the historical development of Carnap's views to examine a certain family of critiques of it. We focus on the version due to Kurt Gödel, but also explore four others found in the literature. In the final chapter, I develop a reading of Carnap's Principle - the `wide' reading. It is one whereby there are no antecedent constraints on the construction of linguistic frameworks. I argue that this reading of the principle resolves the purported problems. Though this thesis is not a vindication of Carnap's view of logic and mathematics tout court, it does show that the view has more plausibility than is commonly thought.

Ce travail vise à identifier et définir une nouvelle tendance du platonisme mathématique que l'on propose d'appeler « platonisme sobre ». Comme le platonisme mathématiques classique, le platonisme sobre admet la fiabilité de la connaissance mathématique et l'existence d'objets mathématiques. Contrairement au platonisme mathématique classique, son engagement ontologique aux objets mathématiques est atténué par des arguments démontrant qu'un monde sans objets mathématiques ne serait pas cohérent. Quand bien même il le serait, on ne pourrait pas accepter de rejeter les mathématiques pour des raisons philosophiques. Le platonisme sobre suggère donc de concilier l'enquête philosophique avec la pratique mathématique. Dans le premier chapitre, on analyse le platonisme mathématique classique. Le deuxième, troisième, quatrième et cinquième chapitre sont respectivement dévoués à l'examen du platonisme pur-sang, du structuralisme ante rem, de la théorie de l'objet abstrait du trivialisme. Cette théories sont explicitement platoniciennes, mais seulement sobrement engagées dans l'existence d'objets mathématiques. Elles traitent l'existence d'objets mathématiques, la possibilité d'accéder à la connaissance mathématique, le sens des énoncés mathématiques et la référence de leur termes en tant que questions philosophiquement pertinentes. Cependant, elles sont dévouées à l'élaboration d'une description précise des mathématiques en tant que telles. Dans le dernier chapitre, le platonisme sobre est défini comme une description méthodologique de la façon dont les mathématiques sont réalisées, plutôt que comme une prescription normative de la façon dont les mathématiques doivent être réalisées. / This work aims at identifying and defining a new trend in mathematical platonism I propose to call “Sober Platonism”. As classical mathematical platonism, Sober Platonism acknowledges the reliability of mathematical knowledge and the existence of mathematical objects. But, contrary to classical mathematical Platonism, its ontological commitment with mathematical objects is softened by several arguments that demonstrate the claim that a world without mathematical abjects wouldn't be consistent. And even if it would be, rejecting mathematics for philosophical reasons wouldn't be acceptable. As a result, Sober Platonism suggests to lined up philosophical inquiry with mathematics as practiced. In the first chapter, I analyzed classical mathematical Platonism. The second, third, fourth and fifth chapters are devoted to the examination of full-blooded Platonism, ante rem Structuralism, Object Theory and Trivialism respectively. This theories are explicitly platonist, but only soberly committed with the existence of mathematical abjects. They take into account the existence of mathematical abjects, the possibility to access to mathematical knowledge, the meaning of mathematical statements and the reference of their terms as philosophically relevant questions. But they are firstly focused on providing an accurate description of mathematics by its own. In the last chapter, Sober Platonism is defined as a methodological description of how mathematics is performed, rather than as a normative prescription of how mathematics should be performed. In conclusion, Sober Platonism admittedly achieves the goal of providing both philosophy and mathematics with a proper domain of inquiry.