First Proof Class

When the entirety of a lecture involves a renowned mathematician doodling around the letter “T”, you know you’re in one of those “once in a lifetime” classes.

It is rare in the context of a humanities class to talk about infinity, prove the famous Delphian problem (why you can never square a circle using a straightedge and compass), or use set theory to prove why two types of infinities exist. Rarer is that such discussions occur weekly in a class of 15 frosh, a fantastic philosophy and writing instructor (Ruth Starkman) and a brilliant mathematician (Ravi Vakil). Professor Vakil has a knack for teaching and a hunger for “experiments” when it comes to education. Part of Stanford’s newest Education as a Self Fashioning program, his class not only fulfills humanities requirements but might be one of the only classes in the world that finds Plato’s Meno in the same syllabus as Terence Tao’s Einstein Series Lecture, Godel Escher Bach, and lecture series by famous historians and public intellectuals including Anthony Grafton, Alexander Nehamas, and Peter Galison.

_{Professor Vakil in Action (photo credit: Ruth Starkman)}

What do we learn? If you peek at our notes and problem sets, we are learning proof based mathematics, basic number and set theory, and how using basic principles of math, one can prove a claim to be true or false. We’re also debating philosophy and researching on any topic of our choosing (as long as it is related to “Godel Escher Bach”). The class is called “Rigorous and Precise Thinking In Mathematics” and we’re learning - or relearning - precisely that. How to think.

What intrigues me about this class is how beautiful proofs are and how we can use language to express truths about itself. More than one discussion has ended with a “mind blown” feeling - something that makes any class worth the sweat and blood.

One might be frightened by the math prerequisites for this class, because there aren’t any. We started from the absolute basics. Professor Vakil asked, “What is a number?” And someone replied, “well, the number 1 is like - 1 apple.” Another added that the absence of an apple creates all of the negative integers and we went from there. Our proofs have included proving why square root of 2 is irrational, why we can think of 1 as .99999 repeating, and why you can never map the set of integers to the set of real numbers (proving that two different types of infinities indeed do exist).

The type of thinking is trhilling - defined, precise, from step A to step B, and coupled with the risky jumps one takes when trying to crack at a proof. (Something both Jordan Ellenberg and Ravi Vakil relate to mountain climbing).

There is magic in the conversations we have with Prof. Vakil and TA Otis Chodosh. The lectures are dialectics and explorations. Our curiosity is piqued and rather than show us the beauty of math, Prof. Vakil has us discover it for ourselves. I knew nothing of proof based mathematics coming in and I happily find myself asking questions about things that I had normally stopped asking questions about by the time I had turned 6.

Why does the world work this way? And how can I prove it?

Thanks for a great class. To see the problem sets, check out the wordpress site.