Mathematics

Mathematics encompasses an area of human endeavor that aids our grasp, construction, and conveyance of notions such as quantity, ratio, change, measure, value, and space and time—all basic and integral to the human experience. This statement arguably rings truer than ever today, as much of our communication, information exchange, and transactions flow over the digital highway, which is built on and most readily described by discrete mathematics. Therefore, at DRBU students study mathematics as part of an integrated liberal arts curriculum and not as a separate specialty. In addition to gaining college-level mathematical skills, such as constructing geometric and analytic proofs and applying simple calculus operations, students will get to look into and reflect on one of the main human processes for gaining knowledge.

In class, students will read and prepare demonstrations on assigned materials, including selected primary texts, study guides, and supplements. Live demonstrations not only allow students to work through the materials, but also to think about how to communicate their rationales and insights to others. Demonstrations also bring students closer to the thinking and learning processes the authors of primary texts went through in struggling to advance new human knowledge.

Finally, developing in students the propensity to raise and ponder important questions and providing them with the learning tools to explore and address those questions are central to DRBU’s educational goals. The main activities of the Mathematics strand—reading the primary texts, performing demonstrations, and engaging in discussions around the materials—serve this central objective well by inviting students to explore and reflect on some of life’s deeper questions: What constitutes knowledge and what meaning does it have? How is such knowledge acquired? What assumptions, if any, are such knowledge contingent upon? Is the scope of knowledge limited? If so, what are the limits?

Freshman Year

The Mathematics strand of DRBU’s Bachelor of Arts in Liberal Arts program is comprised of two years (freshman and junior years) of three-hour weekly class sessions. Students will begin year one with “A point is that which has no part” from Euclid’s Elements and proceed to work through much of the thirteen books of this Greek classic on geometry. The last portion of the year is devoted to Conics by Apollonius. The study of this classic treatise leverages the skills in geometry that students gain from working through the Elements to further develop their understanding of the fundamental properties of curves generated from cutting an oblique circular cone. Knowledge of conic sections is essential to understanding planetary motion and anticipates Descartes’ work in analytical geometry.

Junior Year

Two transitions take place in year two of the Mathematics strand—from geometry to algebra and from figures and spaces to motions and changes. Year two begins with a second look at conic sections, this time using the analytical methods outlined in Descartes’ Geometry. Other texts students will encounter include Pascal’s Generation of Conic Sections and Viete’s Introduction to the Analytical Art. Descartes’ analytical geometry provides an algebraic description of geometry and lays the foundation for modern mathematics in general and calculus in particular.

The latter three quarters of the year are mostly devoted to reading Newton’s Principia Mathematica. Attention is focused on sections of the Principia devoted to Newton’s construction of calculus using geometric methods. Readings from the Principia are supplemented with writings from Taylor, Euler, and Leibniz. Here, students are introduced to calculus from both geometric and analytical approaches. They also develop skills in applying some simple calculus techniques. Year two concludes with the reading of Dedekind’s writings on number theory, an area that allows students to reflect on the nature of numbers and their structural and symbolic significance in the contemporary world.

Selection of authors and works explored in the Mathematics strand

  • Euclid, Element
  • Apollonius of Perga, Conics
  • Descartes, Geometry
  • Pascal, Generation of Conic Sections
  • Viète, Introduction to the Analytical Art
  • Newton, Principia Mathematica
  • Dedekind, Selected writings on the Theory of Numbers
  • Selected writings from Taylor, Euler, and Leibniz