Listed in: Philosophy, as PHIL-50
Mathematics is often thought to be the paragon of clarity and certainty. However, vexing problems arise almost immediately upon asking such seemingly straightforward questions as: “What is the number 1?” “Why can proofs be trusted?” “What is infinity?” “What is mathematics about?” During the first decades of the twentieth century, philosophers and mathematicians mounted a sustained effort to clarify the nature of mathematics. The result was three original and finely articulated programs that seek to view mathematics in the proper light: logicism, intuitionism, and finitism. The mathematical and philosophical work in these areas complement one another and indeed are, to an important extent, intertwined. For this reason, our exploration of these philosophies of mathematics will examine both the philosophical vision that animated them and the mathematical work that gave them content. In discussing logicism, we will focus primarily on the writings of Gottlob Frege. Some indication of how the goal of logicism--the reduction of mathematics to logic--was imagined to be achievable will also be given: introduction to the concepts and axioms of set theory, the set-theoretic definition of “natural number,” the Peano axioms and their derivation in set theory, reduction of the concepts of analysis to those in set theory, etc. Some of the set-theoretic paradoxes will be discussed as well as philosophical and mathematical responses to them. In the section on intuitionism, we will read papers by L.E.J. Brouwer and Michael Dummett, who argue that doing mathematics is more an act of creation than of discovery. This will proceed in tandem with an introduction to intuitionistic logic, which stands in contrast to the more commonly used classical logic. Finally, we will discuss finitism, as articulated in the writings of David Hilbert, who sought to reconcile logicism and intuitionism. Students will then be taken carefully through Gödel’s Incompleteness Theorems and their proofs. The course will conclude with an examination of the impact of Gödel’s work on Hilbert’s attempted reconciliation, as well as on more general philosophical questions about mathematics and mind.
Requisite: Philosophy 13 or Mathematics 34 or consent of the instructors. Spring semester. Professors A. George and Velleman.