Math 56: Proofs and Modern Mathematics
Spring 2026
Lectures: Tue, Th 10:30am-11:50am, room 380Y.
Professor: Eleny Ionel, office 383L, ionel "at" math "dot" stanford "dot" edu
Office Hours: Monday, 1-2pm, Tuesday 2-3pm and by appointment.
Course Assistant: Zehan Hu, office 381M, zehanhu "at" stanford "dot" edu
Office Hours: Tuesday 6pm-7:30pm and Wednesday 1:30pm to 3pm.
Course Description: 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. This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but at a pace and level of abstraction not as intense as Math 61CM/DM, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is strongly recommended at least at the AB level of AP Calculus. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces.
Learning Objectives: This course aims to support students as they develop skills in, and an appreciation for, mathematical proof in practice. Although mathematical proof can be studied in the abstract by studying formal languages and rules for logical reasoning, this course takes a more practical approach, exploring mathematical proof as it is practiced in two important fields of mathematics: linear algebra and real analysis. Aimed at students with little or no experience in proof-intensive courses, we have the following objectives:
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To generate an appreciation for, and a comfort with, mathematical proofs.
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To prepare students for further study in proof-intensive math courses.
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To further develop logical reasoning and abstract thinking skills.
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To provide an introduction to linear algebra and real analysis, two very important fields of mathematics.
Textbooks: There are two textbooks for this course
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Linear Algebra Done Right, Sheldon Axler, Third Edition, available online through Stanford Searchworks.
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Understanding Analysis by Stephen Abbott, Second Edition, available online through Stanford Searchworks.
Students are expected to read the relevant sections of the textbook(s) (listed below) and any handouts (posted on Canvas) before coming to class each day.
Other recommended reading (as needed): Book of Proof by Richard Hammack. For a nice quick guide on the basics of writing proofs, see Introduction to mathematical arguments by Michael Hutchings.
Course Policies: Please see https://goto.stanford.edu/mathcoursepolicies for important course policies on exam conflicts, academic accommodations, AI guidance and taking exams. It is your responsibility to thoroughly read this information.
Accommodations & Flexibility Form: http://goto.stanford.edu/math56oae
Course Logistics: Course announcements, homework and other course materials will be posted on Canvas. Weekly homework will be due and graded on Gradescope.
Homework Policy: Weekly homework assignments given out on Wednesday, and due the following Wednesday at 11:59PM on Gradescope (unless otherwise noted). No late submissions will be accepted under any circumstances. (This is as much a courtesy to the grader as an incentive to stay current with the course and not fall behind.) However, to accommodate such as a serious illness or anything else that may arise, your homework score will be multiplied by 10/9 (not to exceed 100%) at the end of the quarter.
We encourage you to form study groups to discuss and work together on the homework (ideally after thinking about it by yourself). However, you must write up your own solutions individually and in your own words, and indicate the names of any collaborators or sources of outside help that you received. Keep in mind that simply copying solutions from another student or from other sources (such as internet or AI generated) and then submitting it for credit will be considered a violation of the Stanford Honor Code.
Exams: There will be two in-person, proctored exams: a midterm and a final exam.
Grading: The course grade is based on the following components:
Homework: 15%
Midterm Exam: 35%
Final Exam: 50%
Important dates:
- Final Study List deadline: Friday, April 17, 5pm.
- Midterm Exam: Tuesday, May 5, in class.
- Final Exam: Tuesday, June 9, 2026, 3:30pm - 6:30pm.
Tentative Schedule: (which may be adjusted as the quarter goes on)
- Week 1: Brief intro: Sets, Logic, Functions and Proof Techniques (Preliminaries notes; see also Abbott §1.1, §1.2 and §1.5)
- Week 2: Vector spaces and Subspaces (Axler 1.B-C)
- Week 3: Vector spaces, Span and Linear Independence; Bases and Dimension (Axler 2.A-C)
- Week 4: Linear maps; Null Spaces and Ranges (Axler 3.A-B)
- Week 5: Linear maps and Matrices; Isomorphisms (Axler, 3.C-D)
- Week 6: Midterm on Tuesday; Real numbers (Abbott Chapter 1)
- Week 7: Sequences and series (Abbott Chapter 2)
- Week 8: Basic topology of the reals (Abbott Chapter 3)
- Week 9: Continuity (Abbott Chapter 4 )
- Week 10: Some interesting examples.
Academic Integrity:
The Honor Code articulates Stanford University's expectations of students and faculty in establishing and maintaining the highest standards in academic work. Its purpose is to uphold a culture of academic honesty. Students will support this culture of academic honesty by neither giving nor accepting unpermitted academic aid in any work that serves as a component of grading or evaluation, including assignments, examinations, and research. Examples of conduct that have been regarded as being in violation of the Honor Code (and are most relevant for this course) include copying from another student's work (or other sources such as the internet, AI generated etc) or allowing another student to copy from your own work; plagiarism; representing as one's own work the work of another. Please visit the OCS website for more information on the Honor Code, and please see link above for the Math Department AI Policy.