Math 56 Homepage, Autumn 2020-2021

Proofs and Modern Mathematics

Instructor: András Vasy

Office: virtual

Phone: zoom

E-mail: andras "at" math.stanford.edu

Tentative office hours: TBA


Course Assistant: Naomi Kraushar

E-mail: naomilk "at" stanford.edu

Office: virtual

Office hours: TBA


Class location: TTh 2:30-3:50pm on zoom.



Textbook: `Linear Algebra Done Right' by Sheldon Axler (3rd edition, ISBN: 0387982582), `Understanding Analysis' by Stephen Abbott (2nd edition, ISBN: 1493927116), plus the instructor's lecture notes. The textbooks are electronically available via Stanford Libraries.

The syllabus may change somewhat, but should give an indication of the scope and speed of the course.


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 course, 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 two projects will involve 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 less intense 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 course develops it systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating algebraic notions such as groups, rings, and fields. This course may be paired with Math 51 although that course is not a pre- or co-requisite.

This course is a descendant of the freshman seminar, Math 83N, and is expected to take a similar format to the extent possible.

Satisfies Ways requirement for Formal Reasoning (FR).