At St. John’s mathematics is considered an integral part of our understanding of the human intellect and of the world. The Mathematics Tutorial seeks to give students an insight into the fundamental nature and intention of mathematics and into the kind of reasoning that proceeds systematically from definitions and principles to necessary conclusions. During their four years at the college, all undergraduates study pure mathematics and the foundations of mathematical physics and astronomy. They develop rigor in thinking and appreciation of a reasoned account, as well as the spirit of inquiry the college attempts to cultivate across the curriculum. Tutorials meet three times a week, with one tutor and 13 to 16 students.
The study of mathematics begins with Euclid’s Elements, concentrating primarily on the geometrical books, but also giving some attention to Euclid’s treatment of number and to the relation between number and magnitude. The study of Euclid introduces students to a reasoned proof that articulates its presuppositions and proceeds by demonstration. The last eight weeks of the year are devoted to Ptolemy’s Almagest and focus mainly on his account of the motion of the sun. Ptolemy’s Almagest uses the geometrical understanding gained from Euclid and begins a new inquiry into the motion of heavenly bodies. Reading the Almagest also gives rise to questions that will recur over the four years, such as: What is meant by “giving an account” of how such bodies move?
Sophomore Mathematics examines two of the most fundamental transitions in the tradition of astronomy and mathematics. The first semester examines Ptolemy’s theory of the planets, and then considers Copernicus’ and Kepler’s revisions of the Ptolemaic account. The rest of the year is devoted to studying the conic sections as presented by Apollonius, followed by the study of Descartes’ Geometry, one of the foundational works of modern mathematics. Thinking through the differences between the Cartesian and ancient approaches occasions further reflection on the nature of mathematical objects and our efforts to understand them. By the end of sophomore year, students must demonstrate proficiency with basic algebra as a prerequisite for the more advanced work of the Junior Mathematics tutorial.
Junior Mathematics concerns itself with questions about the continuity of motion, the infinite, and the infinitesimal, which lead to a new form of mathematics, the calculus. The initial sequence of readings (Aristotle, Galileo, and Leibniz) leads to the primary text, Newton’s Principia, which offers a sweeping vision of the mechanical motions of the universe. The year concludes with Dedekind’s Essays on the Theory of Numbers, which attempts to establish the continuity of number and prompts students to revisit questions about the nature of the infinite and the infinitesimal. Some classes continue this inquiry with a brief study of Cantor at the close of the second semester.
Seniors study non-Euclidean geometry and Lobachevsky’s Geometrical Researches on the Theory of Parallels. They also study mathematics more closely tied to physical concerns by working through Einstein’s special relativity and energy-mass papers, as well as excerpts from Minkowski’s Space and Time. At the end of each semester, tutors choose from various possibilities: some classes explore general relativity; some read Poincaré; some re-read Kant on space and time; some read Einstein’s “Geometry and Experience,” or Lightman’s Einstein’s Dreams. A number of classes also study Gödel’s incompleteness theorems.