This program is built around intensive study of fundamental areas of pure mathematics. The two focus topics are Real Analysis and Abstract Algebra, which are core areas of study across most college math curricula. The program will also include a weekly seminar discussion on readings about the culture, history, and philosophy of mathematics.
The work in this advanced-level mathematics program is quite likely to differ from students' previous work in mathematics, including calculus, in a number of ways. We will emphasize the careful understanding of the definitions of mathematical terms and the statements and proofs of the theorems that capture the main conceptual landmarks in the areas we study. Hence, the largest portion of our work will involve the reading and writing of rigorous proofs, often in the context of axiomatic systems. These skills are valuable not only for continued study of mathematics but also in many areas of thought in which arguments are set forth according to strict criteria for logical deduction. Students will gain experience in articulating their evidence for claims and in expressing their ideas with precise and transparent reasoning.
In addition to work in core areas of advanced mathematics, we will devote seminar time to looking at our studies in a broader historical, philosophical, and cultural context, working toward answers to such critical questions as: Are mathematical systems discovered or created? How did the current mode of mathematical thinking come to be developed? What are the connections between mathematics and culture?
This program is designed for students who intend to pursue graduate studies or teach in mathematics and the sciences, as well as for those who want to develop a detailed understanding of mathematical thinking at a high level. Program activities will include lectures, workshops, seminar discussions and essays, weekly homework assignments, and regular quizzes and exams in each area of study. The material will be challenging and will require a very substantial time commitment outside of class in addition to consistent in-class participation. Students will be expected to complete several substantial homework assignments each week and to bring that learning to the classroom through active engagement with classmates. Most work will be low-tech: pencils, paper, and whiteboards for collaboration; academic technologies such as computers and calculators will not play an important role in our work. A strong emphasis will be placed on developing an inclusive learning community which fosters mutual support and success.
Anticipated credit equivalencies:
*6 - Abstract Algebra
*6 - Real Analysis
4 - History and Philosophy of Math
Registration
The minimum prerequisite for this program is a full year of calculus, including derivatives, integrals, and sequences and series. However, Math Systems' emphasis on theorems, proofs, logic, and the underpinnings of mathematical thought makes it quite different from most calculus classes. For this reason, any additional math experience beyond calculus (such as linear algebra or discrete math) will also provide valuable background for students joining Math Systems.
Academic Details
Upper division science credit may be awarded in our 12 credits of upper division pure mathematics. Students seeking to earn upper division credit in our 4-credit seminar may do so by working with faculty on identifying an appropriate seminar project at the start of the quarter.