Look for the catalog of 2020-21IntroSems to roll out here in August.
Introductory Seminars for First-Year Students
Proofs and Modern Mathematics
How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why. Through this seminar, students will prove some mathematical statements with which they may already be familiar; introduce new concepts and ways of thinking that illuminate known facts and help us to explore further; and learn how to carry out careful logical arguments and write proofs. The final project will be writing up, in stages, a logically more complicated proof or sequence of proofs.
This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but hope to do so at a more manageable pace and level of abstraction than Math 61CM/DM offers, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is useful since a significant part of the seminar develops systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces. This seminar may be paired with Math 51 although that course is not a pre- or co-requisite.