Academic Program


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The Mathematics Tutorial

The college believes that mathematics is 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 the four years they study pure mathematics and the foundations of mathematical physics and astronomy. The goals of tutorial center on developing in rigor in thinking and appreciation of a reasoned account, as well as a spirit of inquiry. Tutorials have 1 tutor and 13 to 16 students. Tutorials meet three times a week.


Freshman: Greek Mathematics, Geometry and Astronomy.

The study of mathematics begins in freshman year with Euclid's Elements, concentrating on the geometrical books, with 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 account that articulates its presuppositions and proceeds by demonstration. The last seven to eight weeks of the year are devoted to Ptolemy's Almagest and primarily cover his account of the motion of the sun. Ptolemy's Almagest uses the geometrical understanding gained from Euclid and begins a new inquiry: how do heavenly bodies move? 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: Astronomy, Conic Sections, Transition to Modern Mathematics

Sophomore Mathematics examines two of the most fundamental transitions in the tradition of astronomy and mathematics. Much of the first semester continues the study of Ptolemy begun in the freshman year, and moves to Copernicus's revision of Ptolemy. The rest of the year is devoted to studying the conic sections as presented by Apollonius, followed by the study of Descartes' Geometry. By the end of sophomore year, students must demonstrate proficiency with basic algebra as a prerequisite for the more advanced Junior Mathematics tutorial.

 

Junior: Calculus, Mathematic Theory

The Junior Mathematics Tutorial 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 (Zeno, Aristotle, Galileo) leads to the primary text, Newton's Principia and a sweeping vision of the mechanical motions of the universe. The year concludes with Dedekind's Essays on the Theory of Numbers.

 

Senior: Non-Euclidean Geometry, Relativity, Topics in Modern Mathematics

The first term of Senior Mathematics begins with non-Euclidean geometry and Lobachevsky's Geometrical Researches on the Theory of Parallels. In the second semester, seniors begin a study of mathematics more tied to physical concerns by working through Einstein's special relativity and energy-mass papers, followed by excerpts from Minkowski's Space and Time. At the end of the second semester, as at the end of the first, there is a range of possibilities pursued by the tutors of each campus. 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'.

 

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