Mathematical Systems

Winter Open
Class Standing
Rachel Hastings

This program is built around intensive study of several fundamental areas of pure mathematics, including Abstract Algebra, Real Analysis, and other areas such as Geometry and Topology. The program will also include seminars on 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? In what sense do mathematical objects actually exist? What is current mathematical practice? 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. A strong emphasis will be placed on developing an inclusive learning community which fosters mutual support and success.


Minimum prerequisite is successful completion of 1 year of calculus.  Additional mathematical experience is recommended, for example Mathematical Methods in Fall quarter will provide helpful experience in mathematical thinking, as would other courses such as discrete mathematics. Please contact faculty with questions about preparation.

Prerequisite is proficiency in a single-variable calculus (successful completion of a full year of calculus). Additional mathematical experience is recommended, and students planning to take Math Systems should strongly consider taking Math Methods in Fall quarter to gain background in mathematical thinking. Contact faculty with any questions.

Course Reference Numbers
So - Sr (16): 20322
So - Sr (1 - 16): 20323
Signature Required

Significant background in proof-based mathematics will be needed for students entering the program in Spring quarter. Contact faculty by email ( to discuss your preparation.

Course Reference Numbers
So - Sr (16): 30093
So - Sr (1 - 16): 30094

Academic Details

Studies or careers in Mathematics, Physics, Math Education, Philosophy of Math, History of Math


Each quarter will consist of 3 math topics and a seminar, with each thread representing a 4-credit unit. Students wishing to register for 4 credits may take just one of these threads. Students may also opt for 8 credits (1 math subject + seminar) or 12 credits (2 math subjects + seminar).  Please contact faculty if interested in these part-time options.

4 - Abstract Algebra

4 - Real Analysis

4 - Geometry and Topology

4 - Seminar in History, Philosophy, and Culture of Math


Math and science textbooks tend to be expensive and students should expect to spend several hundred dollars on books. Used books or rentals may be available at lower cost.

Each quarter will consist of 3 upper division math topics and a seminar.  The math subjects are each 4-credit units, and students who are successful with the material at the level presented will earn upper division credits.  The seminar credits will normally be lower division, though students who want to engage in advanced mathematics in their seminar papers should discuss the possibility of upper division credit with faculty.  Thus each quarter of the program will consist of 12-16 credits of upper division work.


In Person (W)
In Person (S)

See definition of Hybrid, Remote, and In-Person instruction

Schedule Details
Purce Hall 6 - Classroom