Agree to Agree

In the course of trying to slodge through a new Quanta article about the importance of modular forms (what are they? Hey, I said "slodge" for a reason), I ran across a different article by about how mathematical proofs have a social dimension.

Quanta writer Jordana Cepelewicz interviews mathematician Andrew Granville of the University of Montreal about this in an August Q&A. The jumping off point is the claim by reclusive Japanese mathematician Shinichi Mochizuki to have created a proof solving something called the "ABC Conjecture" that has do to with a relationship between addition and multiplication. Mochizuki's 2012 proof was 500 pages long and pretty dense, even for a mathematical proof. After two other mathematics professors visited Mochizuki in 2018 and found out what seemed to be fatally flawed gaps in the proof, Mochizuki dismissed their claims by saying they did not understand his work.

While "I'm right and you're too dumb to know it" might work in conversations with politicians and many celebrity figures, it's not an acceptable way of discussing mathematical proofs. In order to be useful, they must be held to be valid by a large group of mathematically knowledgeable people, so that when those people rely on the proof in their own work it won't fall apart.

According to Granville, Mochizuki's response hit on a key feature of mathematics and proof writing. The only way a math person may prove a proof is by convincing other math people their answer is accurate and not missing anything. Now, the other math people obviously have knowledge everyday folks lack. Ask me to evaluate a complex proof and my answer will be a single word: Hellifino.

But even the math people start with understandings that might be different from the proof writer.  They are the people who can say, "You cannot use that squiggly line in this spot! It must go here instead, and if you don't see that you breathe through your mouth and your knuckles drag the ground when you walk." They may say that because they know better. Granville points out that they may also say that because that's how they learned or because that's how their equations work properly.

The upshot of his understanding is that a social factor among mathematicians plays a much larger role than anyone might have thought it would in this discipline seen as the realm of cold logic. Perhaps it once was, but work in the first part of the 20th century opened the door to letting the community of mathematicians put their thumbs on the scale. It could have been a detriment to their work, but Granville sees it as a way to build closer ties among different mathematical disciplines and ideas.