<p> Category theory has been advocated as a replacement for set theory as the foundation for mathematics. It is claimed that as a foundation set theory is both inadequate and inappropriate. Set theory is considered inadequate because it cannot produce all of the mathematical objects of interest. Set theory is considered inappropriate because it provides a poor framework for mathematical research. In this dissertation, I argue that category theory is subject to exactly the same objections by considering the use of category theory for work in graph theory.</p>

Thesis (Ph.D.)--Indiana University, Dept. of Philosophy, 2008. / Title from PDF t.p. (viewed on May 13, 2009). Source: Dissertation Abstracts International, Volume: 69-08, Section: A, page: 3174. Adviser: David C. McCarty.

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.

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Thesis (PH.D.) -- Syracuse University, 2007. / "Publication number AAT 3267442"

This thesis is concerned with explaining how a subject can acquire a priori knowledge of arithmetic. Every account for arithmetical, and in general mathematical knowledge faces Benacerraf's well-known challenge, i.e. how to reconcile the truths of mathematics with what can be known by ordinary human thinkers. I suggest four requirements that jointly make up this challenge and discuss and reject four distinct solutions to it. This will motivate a broadly Fregean approach to our knowledge of arithmetic and mathematics in general. Pursuing this strategy appeals to the context principle which, it is proposed, underwrites a form of Platonism and explains how reference to and object-directed thought about abstract entities is, in principle, possible. I discuss this principle and defend it against different criticisms as put forth in recent literature. Moreover, I will offer a general framework for implicit definitions by means of which - without an appeal to a faculty of intuition or purely pragmatic considerations - a priori and non-inferential knowledge of basic mathematical principles can be acquired. In the course of this discussion, I will argue against various types of opposition to this general approach. Also, I will highlight crucial shortcomings in the explanation of how implicit definitions may underwrite a priori knowledge of basic principles in broadly similar conceptions. In the final part, I will offer a general account of how non-inferential mathematical knowledge resulting from implicit definitions is best conceived which avoids these shortcomings.

<p> There are many different ways to talk about the world. Some ways of talking are more expressive than others&mdash;that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power. In doing so, I show how these investigations help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language.</p><p> When we attempt to describe the world, we are trying to distinguish the way things are from all the many ways things could have been&mdash;in other words, we are trying to locate ourselves within a region of logical space. According to this picture, languages can be thought of as ways of carving logical space or, more formally, as maps from sentences to classes of models. For example, the language of first-order logic is just a mapping from first-order formulas to model-assignment pairs that satisfy those formulas. Almost all formal languages discussed in metaphysics and logic, as well as many of those discussed in natural language semantics, can be characterized in this way. </p><p> Using this picture of language, I analyze two different approaches to defining expressive power, each of which is motivated by different roles a language can play in a debate. One role a language can play is to divide and organize a shared conception of logical space. If two languages share the same conception of logical space (i.e., are defined over the same class of models), then one can compare the expressive power of these languages by comparing how finely they carve logical space. This is the approach commonly employed, for instance, in debates over tense and modality, such as the primitivism-reductionism debate.</p><p> But a second role languages can play in a debate is to advance a conception or theory of logical space itself. For example, consider the debate between perdurantism, which claims that objects persist through time by having temporal parts located throughout that time, and endurantism, which claims that objects persist through time by being wholly present at that time. A natural thought about this debate is that perdurantism and endurantism are simply alternative but equally good descriptions of the world rather than competing theories. Whenever the endurantist says, for instance, that an object is red at time <i> t</i>, the perdurantist can say that the object&rsquo;s temporal part at <i>t</i> is red. On this view, one should conceive of perdurantism and endurantism not as theories picking out disjoint regions of logical space, but as theories offering alternative conceptions of logical space: one in which persistence through time is analogous to location in space and one in which it is not. A similar distinction applies to other metaphysical debates, such as the mereological debate between universalism and nihilism.</p><p> If two theories propose incommensurable conceptions of logical space, we can still compare their expressive power utilizing the notion of a translation, which acts as a correlation between points in logical space that preserves the language&rsquo;s inferential connections. I build a formal theory of translation that explores different ways of making this notion precise. I then apply this theory to two metaphysical debates, viz., the debate over whether composite objects exist and the debate over how objects persist through time. This allows us to get a clearer picture of the sense in which these debates can be viewed as genuine.</p><p>

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