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The Kinds of Mathematical Objects is an exploration of the taxonomy of the mathematical realm and the metaphysics of mathematical objects. I defend antireductionism about cardinals and ordinals: the view that no cardinal number and no ordinal number is a set. Instead, I suggest, cardinals and ordinals are sui generis abstract objects, essentially linked to specific abstraction functors (higher-order functions corresponding to operators in abstraction principles). Sets, in contrast, are not essentially values of abstraction functors: the best explanation of the nature of sethood is given by a variation on the standard iterative account. I further defend the theses that no cardinal number is an ordinal number and that the natural numbers are, as Frege maintained, all and only the finite cardinal numbers. My case for these conclusions relies not on the well-known antireductionist argument developed by Paul Benacerraf, but on considerations about ontological dependence. I argue that, given generally accepted principles about the dependence of a set on its elements, ordinal and cardinal numbers have dependence profiles that are not compatible with any version of set-theoretic ontological reductionism. In addition, a formal framework for set theory with sui generis abstract objects is developed on a type-theoretical basis. I give a philosophical defence of the choice of type theory and discuss various questions relating to the nature of its models.

The primary concern of this thesis is to investigate the explicit philosophy of mathematics in the work of Henri Poincare. In particular, I argue that there is a well-founded doctrine which grounds both Poincare's negative thesis, which is based on constructivist sentiments, and his positive thesis, via which he retains a classical conception of the mathematical continuum. The doctrine which does so is one which is founded on the Kantian theory of synthetic a priori intuition. I begin, therefore, by outlining Kant's theory of the synthetic a priori, especially as it applies to mathematics. Then, in the main body of the thesis, I explain how the various central aspects of Poincare's philosophy of mathematics - e.g. his theory of induction; his theory of the continuum; his views on impredicativiti his theory of meaning - must, in general, be seen as an adaptation of Kant's position. My conclusion is that not only is there a well-founded philosophical core to Poincare's philosophy, but also that such a core provides a viable alternative in contemporary debates in the philosophy of mathematics. That is, Poincare's theory, which is secured by his doctrine of a priori intuitions, and which describes a position in between the two extremes of an "anti-realist" strict constructivism and a "realist" axiomatic set theory, may indeed be true.

Higher-order metaphysics is in full swing. One of its principle aims is to show that higher-order logic can be our foundational metaphysical theory. A foundational metaphysical theory would be a simple, powerful, systematic theory which would ground all of our metaphysical theories from modality, to grounding, to essence, and so on. A satisfactory account of its epistemology would in turn yield a satisfactory epistemology of these theories. And it would function as the final court of appeals for metaphysical questions. It would play the role for our metaphysical community that ZFC plays for the mathematical community. I think there is much promise in this project. There is clear value in having a shared foundational theory to which metaphysicians can appeal. And there is reason to think that higher-order logic can play this role. After all, it has long been known that one can do math in higher-order logic. And there is growing reason to think that one can do metaphysics in higher-order logic in much the same way. However, most of the research approaches higher-order logic from a monist perspective, according to which there is 'one true' higher-order logic. And in the midst of the enthusiasm, metaphysicians seem to have overlooked that this approach leaves the program susceptible to epistemological problems that plague monism about other areas, like set theory. The most significant of these is the Benacerraf Problem. This is the problem of explaining the reliability of our higher-order-logical beliefs. The problem is sufficiently serious that, in the set-theoretic case, it has led to a reconception of the foundations of mathematics, known as pluralism. In this dissertation I investigate a pluralist approach to higher-order metaphysics. The basic idea is that any higher-order logic which can play the role of our foundational metaphysical theory correctly describes the metaphysical structure of the world, in much the way that the set-theoretic pluralist maintains that any set theory which can play the role of our foundational mathematical theory is true of a mind-independent platonic universe of sets. I outline my view about what it takes for a higher-order logic to play this role, what it means for such a logic to correctly describe the metaphysical structure of the world, and how it is that different higher-order logics which seem to disagree with each other can meet both of these conditions. I conclude that higher-order logical pluralism is the most tenable version of the higher-order logic as metaphysics program. Higher-order logical pluralism constitutes a radical departure from conventional wisdom, requiring a significant reconception of the nature of validity, modality, and metaphysics in general. It renders moot some of the most central questions in these domains, such as: Is the law of excluded middle valid? Is it the case that necessarily everything is necessarily something? Is the grounding relation transitive? On this picture, these questions no longer have objective answers. They become like the question of whether the Continuum Hypothesis is true, according to the set-theoretic pluralist. The only significant question in the neighborhood of the aforementioned questions is: which metaphysical principles are best suited to the task at hand?

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This first half of this thesis investigates the epistemological foundations of mathematical theories, with special attention devoted to two questions: (1) how can we have a warrant for the satisfiability and consistency of mathematical theories, and (2) given we conceive of mathematical judgement as objective - as being concerned with a realm of abstract entities - can we have a warrant for thinking that such a realm of entities exists? In Chapter 2, two kinds of mathematical scepticism are developed. The regress sceptic argues that we can have a warrant for accepting neither the satisfiability nor the consistency of a mathematical theory. The I-II-III sceptic maintains that there can be no warrant for thinking that a realm of abstract entities exists if mathematical judgement is conceived as being objective. The notions of entitlement of cognitive project and entitlement of substance - recently introduced into the literature by Crispin Wright - are invoked to respond to the mathematical regress and I-II-III sceptic. This is done in Chapters 3 and 4. The distinctive feature of an entitlement is its non-evidential nature. What is relevant is not the presence of positive evidence, but rather the absence of sufficient countervailing evidence. The second half of the thesis explores and develops certain aspects of this proposal. Chapter 5 develops the notion of entitlement of cognitive project by investigating two of its three defining clauses. Chapter 6 draws a picture of a wider philosophical framework of which entitlement can be regarded an integrated part. In so doing entitlement is discussed in light of the internalism/externalism distinction and the distinction between monism and pluralism about epistemic value. Chapter 7 tables two fundamental challenges to the entitlement proposal - firstly, whether entitlement is an epistemic notion of warrant at all, and secondly, whether the notion of rationality associated with it is epistemic in nature or of some other kind?

Orientador: Irineu Bicudo / Banca: Carlos Roberto de Moraes / Banca: Henrique Lazari / Banca: Marcos Vieira Teixeira / Banca: Renata Cristina Geromel Meneghetti / Resumo: O presente trabalho exibe um estudo de caso sobre o desenvolvimento conceitual e metodológico da Matemática, por meio do exame e comparação das concepções de função de Gottlob Frege e Bertrand Russell. Em particular, são discutidos: a extensão fregiana da ideia matemática de função, a noção russelliana de função proposicional, os seus pressupostos filosóficos e as suas implicações. O presente estudo baseia-se em análises dos livros que os dois autores publicaram sobre os fundamentos da Matemática, e também de alguns outros escritos de sua autoria, entre eles, manuscritos que foram publicados postumamente. Conclui-se a partir dessas análises que a concepção compreensiva de função de Frege e a função proposicional de Russell são generalizações de uma importante aquisição do pensamento matemático, qual seja, a ideia de função, e que a conceitografia e as teorias dos tipos e das descrições, por sua vez, constituem a exploração metódica daquilo que essas generalizações acarretam. Conclui-se, finalmente, que embora existam diferenças expressivas entre as concepções de função de Frege e Russell, um padrão de rigor associado a reflexões mais amplas sobre a natureza do significado emerge em meio às investigações que empreenderam sobre a noção de função: a função fregiana e a função proposicional são as entidades que participam de suas respectivas relações de significado e cuja natureza é precisada no âmbito dessas relações / Abstract: This work presents a case study about the conceptual and methodological development of Mathematics by the examination and comparison of function conceptions in the thinking of Gottlob Frege and Bertrand Russell. Particularly, we discuss the fregean extension of mathematical idea of function, the russellian notion of propositional function and their philosophical assumptions and implications. The basis for this study is a analisys of the authors' books on the foundations of Mathematics and some other authors' writings, included among these some posthumous publications. From this analisys we conclude that the comprehensive function concept of Frege and the Russell's propositional function are both generalizations of an important acquisition of mathematical thought, namely the idea of function, and that the conceptography, the type theory and the theory of descriptions, in turn, constitute the methodical exploration of what these generalizations imply. Finally, we conclude that, though there are expressives differences between the function conceptions of Frege and Russell, a pattern of rigour associated with more wide reflections on the nature of meaning emerges from their investigations of the concept of function: the fregian function and the propositional function are the entities that participate of their respective meaning relations and whose nature is explained by these relations / Doutor

<p>The moduli space of Abelian differentials on Riemann surfaces admits a natural action by $\mathrm{SL}\left(2,\mathbb{R}\right)$.  This thesis is concerned with using the classification of invariant measures for this action due to Eskin and Mirzakhani, to study the growth of closed geodesics in the support of an invariant measure coming from the closure of an orbit for the $\mathrm{SL}\left(2,\mathbb{R}\right)$-action. These are always subvarieties of moduli space. For $0 \leq \theta \leq 1$, we obtain an exponential bound on the number of closed geodesics in the orbit closure, of length at most $R$, that have at least $\theta$-fraction of their length in a region with short saddle connections.</p>

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