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Publications:
Salvaging La Geometrie in Meteorite Student Journal of Philosophy, V1(4), 2004 Abstract: After investigating various mathematical, historical and philosophical sources concerning René Descartes' La Géométrie, I am convinced that this work is not understood in accordance with the original intent of its author. By using Descartes' philosophical works, I'll offer a different interpretation of this publication, and explain how poor interpretations have resulted in paradoxically beneficial impacts on mathematical knowledge. My aim is to connect early 17th Century Western intellectual discourse with the intent of Descartes in publishing this mathematical work, while looking beyond any mere categorical relation between Descartes and the church, or Descartes and the skeptics. Specifically, my goal is not to denounce these relations, but to broadly consider the interpersonal social and scientific discourse of Descartes' day, along with his personal experiences and philosophy, while attempting to construct a more accurate interpretation of La Géométrie. Such an interpretation is made possible by utilizing a method that both mathematicians and philosophers can easily follow, and this merely requires being familiar with certain key aspects of Descartes' Discourse on the Method of Reasoning Well and Seeking Truth in the Sciences. Papers (available here soon for download) On the Distinction of Psychologism and Logicism Abstract: Instead of stressing the usual logicism - formalism - intuitionism distinctions that are commonly taken as given in most texts on the history of the foundations of mathematics, my goal here is to briefly introduce the reader to a slightly different perspective on the naissance of formal logicism in relation to its starkest opponent of the turn of last century: psychologism. In fact, in Husserl's introduction to the first volume of the "Logical Investigations" of 1900, he made clear that of the three trends in logic (formal, psychologistic, metaphysical), formal logic, what was soon to become a major force in mathematics, was loosing acceptance and was weak in comparison to other trends. After this introduction, a different understanding of the nature of logicism, formalism, and specifically intuitionism begin to take shape. References are made primarily to Cantor, Husserl, Frege and Russell, with Husserl receiving the greatest amount of attention. Secondary mentions are made of Hilbert and Kant. The Mathematician's Introduction to Foucauldian Archaeology: An Analysis of the History and Philosophy of Mathematics Abstract: After studying the archaeological works of Michel Foucault, I have realized that Foucauldian archaeology can say many important things about the history and philosophy of mathematics. My hypothesis is that Foucault's perspective on methods used to conduct historical investigations are involved in the same discourse as much of mathematics after the revolution of formalization. The problem is, this revolution leaves us with an unclear view of the past. How can Foucault's theories help us to better understand historically what has happened in mathematics, what mathematics itself is, how it is done, and so on? I consider these questions, and at the same time introduce and utilize the methods of Foucauldian archaeology. My claim is that mathematicians are already familiar with many of the statements that Foucauldian archaeology makes. Syntax and Semantics: From Frege to Husserl and From Lacan to Derrida, Understanding How Discourse "is" Mathematics. Soon available. Recent Advancements in Axiomatic and Algebraic Geometry Abstract: In this paper, I communicate in greater detail the topics discussed in my seminar on November the 25th, 2003 at the University of Michigan - Flint. Specifically, I discuss here several ideas and results pertaining to recent studies in axiomatic and algebraic systems related to the existence of asymptotic parallels in hyperbolic geometry. We shall also consider several philosophical notions related to non-Euclidean geometries and the negation or removal of continuity axioms. Prerequisites include an understanding of non-Euclidean geometries with algebraic embeddings. |
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