Arguing in order to think

In this episode of The Economist Asks a guest comes with the thesis that people argue not to convince others, but to help oneself to reason.

Can we learn to disagree better? (Adam Grant)

I think this is a somewhat fascinating idea, in a sense, because mathematicians have known that this is true for a very long time!

Ok, maybe not exactly in the very precise sense that Adam Grant means this claim. But argumentation is the cornerstone of modern matheamtics. Any modern math textbook has MUCH more English prose than it has equations and graphs. Usually on a given page, about two lines are given to defining some object, and four lines are given to some examples, and about another few lines are given for the statement of some result. And then the next 100 lines are dedicated to the purpose of proving that statement.

That is to say, measuring by line numbers, modern mathematics is almost entirely dedicated to proofs.

If we measured by time spent in class lectures at universities, or if we measured by proportion of exercise problems in advanced undergraduate and graduate ourses, the result would be the same. Most people think mathematics is about computing, but that's actually for computer science. Mathematicians compute a bit, but mostly we prove.

And I think just about every mathematician has had the experience of understanding a subject better BY giving a proof to someone else. And there really needs to be the "someone else". It's totally different if you give a proof only to yourself. Giving the proof to someone else truly does something to your psychology which makes you more attenuated to the details that you otherwise ignore.

There is even this phenomenon called "rubber ducking" which is common in lots of technical fields, especially programming. This is the phenomon in which you have a problem you are trying to solve, and cannot work out the solution on your own. So you call a collegue over to look at your code, and you start explaining the problem. In the middle of explaining the problem to someone else, you suddenly realize the solution! It seems that the act of explaining to someone else just changes how you think, and in general makes you think about 10 times better than how you were thinking without explaining your thoughts to someone else. In any case, the colleague might as well have been a rubber duck because in some sense she wasn't really needed at all.

I think mathematicians also have this common experience when visiting StackExchange or MathOverflow. At least I often have the experience of writing up a question about a problem I can't solve for myself, only to realize about halfway through the description of the problem, what the solution is. This happens, I would estimate, about once a week for me. It is actually a lot more common that I realize the solution, than it is for me to reach the end and still need to hit the "submit" button to actually ask my question.

As a final note on this idea, as a tutor I find that teaching a subject causes me to understand it so much more profoundly than I ever could have understood it just be reading and doing exercises. In fact, I have come to the belief that the single most important part of learning is explaining. Of course you must first read or listen, and then do exercises, but these are merely the preliminaries. The most comprehensive step in learning anything is to teach it to someone else.