## Introduction to Proofs

Spring 2019 MW 1:30-2:45pm

Bloomberg 276

**SYLLABUS**
The syllabus can be found here.

**REFERENCES**

The main text will be How To Prove It: A Structured Approach by Daniel J Velleman.

You might also enjoy How to write proofs: a quick guide by Eugenia Cheng.

**PROBLEM SETS**

Problem sets are due in class on Mondays.

**COMPUTER ASSISTED PROOFS**
Here are some resources:

**SOLUTIONS**

**METRIC SPACES**

**DISCUSSION**
Here are some questions for discussion (feel free to add others):

- What is a proof? (see On proof and progress in mathematics, Mathematicians know how to admit they're wrong, What is a proof?)
- How should the mathematical community assess “non-rigorous” work of an autodidact like Ramanujan? (see Instinct, intuition and mathematics: the divine genius of Srinivasa Ramanujan, There's more to mathematics than rigour and proofs, In Praise Of Amateurs)
- If an expert mathematician writes up an argument that few others understand, does this constitute a proof? Did Mochizuki resolve the ABC conjecture? (see
The ABC conjecture has not been proved, The ABC conjecture has still not been proved, A crisis of identification)
- If you don’t believe that a certain axiom is true, are proofs that rely on that axiom invalid? Is Banach-Tarski a theorem or a paradox? (see The most controversial axiom of all time, The axiom of choice is wrong)
- If mathematicians reduce a problem to 730 cases and a computer verifies the claimed result in each case, does this constitute a proof? Did Appel and Haken prove the four-color theorem and did Ferguson and Hales prove the Kepler conjecture? (see The colorful life of the four-color theorem, Proof confirmed of 400-year-old fruit-stacking problem)
- Can a proof ever be too long to be a proof? (see
An enormous theorem: the classification of finite simple groups)
- Does an argument constitute a proof if it hasn’t been formally verified? (see The Origins and Motivations of Univalent Foundations)

**QUESTIONS?**
If you have any questions about the course, please get in touch. My contact info can be found on my personal website.