Introduction to Proofs

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.

• Problem Set 1
• Problem Set 2
• Problem Set 3
• Problem Set 4
• Problem Set 5
• Problem Set 6

Here are some resources:

• collacoq, an online theorem prover
• lecture notes
• source code

• Solutions to Problem Set 1
• Solutions to Problem Set 2
• Solutions to Problem Set 3

• Script 1
• Script 2
• Script 3
• Script 4
• Script 5
• Script 6
• Script 7

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)

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