This thesis examines possible philosophies to account for the practice of mathematics, exploring the metaphysical, ontological, and epistemological outcomes of each possible theory. Through a study of the two most probable ideas, mathematical platonism and fictionalism, I focus on the compelling argument for platonism given by an appeal to the sciences. The Indispensability Argument establishes the power of explanation seen in the relationship between mathematics and empirical science. Cases of this explanatory power illustrate how we might have reason to believe in the existence of mathematical entities present within our best scientific theories. The second half of this discussion surveys Newtonian Cosmology and other inconsistent theories as they pose issues that have received insignificant attention within the philosophy of mathematics. The application of these inconsistent theories raises questions about the effectiveness of mathematics to model physical systems.

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The Backward Induction strategy for the Centipede Game leads us to a counterfactual reasoning paradox, The Centipede Game paradox. The counterfactual reasoning proving the backward induction strategy for the game appears to rely on the players in the game not choosing that very same backward induction strategy. The paradox is a general paradox that applies to backward induction reasoning in sequential, perfect information games. Therefore, the paradox is not only problematic for the Centipede Game, but it also affects counterfactual reasoning solutions in games similar to the Centipede Game. The Centipede Game is a prime illustration of this paradox in counterfactual reasoning. As a result, this paper will use a material versus subjunctive/counterfactual conditional analysis to provide a theoretical resolution to the Centipede Game, with the hope that a similar solution can be applied to other areas where this paradox may appear. The solution involves delineating between the epistemic systems of the players and the game theorists.

In 1942, Paul C. Rosenbloom put out a definition of a Post algebra after Emil L. Post published a collection of systems of many–valued logic. Post algebras became easier to handle following George Epstein’s alternative definition. As conceived by Rosenbloom, Post algebras were meant to capture the algebraic properties of Post’s systems; this fact was not verified by Rosenbloom nor Epstein and has been assumed by others in the field. In this thesis, the long–awaited demonstration of this oft–asserted assertion is given. After an elemental history of many–valued logic and a review of basic Classical Propositional Logic, the systems given by Post are introduced. The definition of a Post algebra according to Rosenbloom together with an examination of the meaning of its notation in the context of Post’s systems are given. Epstein’s definition of a Post algebra follows the necessary concepts from lattice theory, making it possible to prove that Post’s systems of many–valued logic do in fact form a Post algebra.

<p>In what is supposed to have been a radical break with neo-Hegelian idealism, Bertrand Russell, alongside G.E Moore, advocated the analysis of propositions by their decomposition into constituent concepts and relations. Russell regarded this as a breakthrough for the analysis of the propositions of mathematics. However, it would seem that the decompositional-analytic approach is singularly unhelpful as a technique for the clarification of the concepts of mathematics. The aim of this thesis will be to clarify Russell’s early conception of the analysis of mathematical propositions and concepts in the light of the philosophical doctrines to which his conception of analysis answered, and the demands imposed by existing mathematics on Russell’s logicist program. Chapter 1 is concerned with the conception of analysis which emerged, rather gradually, out of Russell’s break with idealism and with the philosophical commitments thereby entrenched. Chapter 2 is concerned with Russell’s considered treatment of the significance of relations for analysis and the overturning of his “doctrine of internal relations” in his work on Leibniz. Chapter 3 is concerned with Russell’s discovery of Peano and the manner in which it informed the conception of analysis underlying Russell’s articulation of logicism for arithmetic and geometry in PoM. Chapter 4 is concerned with the philosophical and logical differences between Russell’s and Frege’s approaches to logical analysis in the logicist definition of number. Chapter 5 is concerned with connecting Russell’s attempt to secure a theory of denoting, crucial to mathematical definition, to his decompositional conception of the analysis of propositions.</p> / Doctor of Philosophy (PhD)

<p>The purpose of this dissertation is to provide a proper treatment for two groups of logical paradoxes: semantic paradoxes and set-theoretic paradoxes. My main thesis is that the two different groups of paradoxes need different kinds of solution. Based on the analysis of the diagonal method and truth-gap theory, I propose a functional-deflationary interpretation for semantic notions such as ‘heterological’, ‘true’, ‘denote’, and ‘define’, and argue that the contradictions in semantic paradoxes are due to a misunderstanding of the non-representational nature of these semantic notions. Thus, they all can be solved by clarifying the relevant confusion: the liar sentence and the heterological sentence do not have truth values, and phrases generating paradoxes of definability (such as that in Berry’s paradox) do not denote an object. I also argue against three other leading approaches to the semantic paradoxes: the Tarskian hierarchy, contextualism, and the paraconsistent approach. I show that they fail to meet one or more criteria for a satisfactory solution to the semantic paradoxes. For the set-theoretic paradoxes, I argue that the criterion for a successful solution in the realm of set theory is mathematical usefulness. Since the standard solution, i.e. the axiomatic solution, meets this requirement, it should be accepted as a successful solution to the set-theoretic paradoxes.</p> / Doctor of Philosophy (PhD)

<p>This sentence is whatever truth is <em>not</em>. The subject of this master’s thesis is the power, influence, and solvability of the liar paradox. This paradox can be constructed through the application of a standard conception of truth and rules of inference are applied to sentences such as the first sentence of this abstract. The liar has been a powerful problem of philosophy for thousands of years, from its ancient origin (examined in Chapter One) to a particularly intensive period in the twentieth century featuring many ingenious but ultimately unsuccessful solutions from brilliant logicians, mathematicians and philosophers (examined in Chapter Two, Chapter Three, and Chapter Four). Most of these solutions were unsuccessful because of a recurring problem known as the liar’s revenge; whatever truth is <em>not</em> includes, as it turns out, not <em>just</em> falsity, but also meaninglessness, ungroundedness, gappyness, and so on. The aim of this master’s thesis is to prove that we should not consign ourselves to the admission that the liar is and always will just be a paradox, and thus unsolvable. Rather, I argue that the liar <em>is</em> solvable; I propose and defend a novel solution which is examined in detail in the latter half of Chapter Two, and throughout Chapter Three. The alternative solution I examine and endorse (in Chapter Four) is not my own, owing its origin and energetic support to Graham Priest. I argue, however, for a more qualified version of Priest’s solution. I show that, even if we accept a very select few true contradictions, it should <em>not</em> be assumed that inconsistency inevitably spreads throughout other sets of sentences used to describe everyday phenomena such as motion, change, and vague predicates in the empirical world.</p> / Master of Arts (MA)

A philosophic account of quantum physics. The thesis is divided into two parts. Part I is dedicated to laying the groundwork of quantum physics, and explaining some of the primary difficulties. Subjects of interest will include the principle of locality, the quantum uncertainty principle, and Einstein's criterion for reality. Quantum dilemmas discussed include the double-slit experiment, observations of spin and polarization, EPR, and Bell's theorem. The first part will argue that mathematical-physical descriptions of the world fall short of explaining the experimental observations of quantum phenomenon. The problem, as will be argued, is framework of the physical descriptive schema. Part I includes in-depth discussions of mathematical principles. Part II will discuss the Copenhagen interpretation as put forth by its founders. The Copenhagen interpretation will be expressed as a paradox: The classical physical language cannot describe quantum phenomenon completely and with certainty, yet this language is the only possible method of articulating the physical world. The paradox of Copenhagen will segway into Kant's critique of metaphysics. Kant's understanding of causality, things-in-themselves, and a priori synthetic metaphysics. The thesis will end with a conclusion of the quantum paradox by juxtaposing anti-materialist Martin Heidegger with quantum founder Werner Heisenberg. Our conclusion will be primarily a discussion of how we understand the world, and specifically how our understanding of the world creates potential for truth.