The Paradox of the Proof is an essay by Caroline Chen about Shinichi Mochizuki‘s supposed proof of the abc conjecture. The article is accessible and very well written. I especially like that the article touches on the social aspect of mathematics. It seems a common conception about math that mathematical statements are either true or false, and that the truth or falsity is somehow “objective.” However, Mochizuki’s work shows that this is not the complete story. Mochizuki claims to have a proof of the abc conjecture, but literally no one (except presumably Mochizuki himself) understands the proof. So the truth of the abc conjecture still seems in limbo. This gives credence to my belief that a mathematical proof is not merely a mechanical verification that a statement is true. Rather, a proof is a narrative about why a statement is true — if there is no consensus in the mathematical community that a proof is valid, it may as well not exist. Chen’s article does a wonderful job of conveying this tension between the austere and rigorous abstractions of mathematics and its social nature that gives the field life.