Earlier this year a middle schooler emailed me to ask me what my favorite mathematical symbol was. I didn’t have a good answer, because I’d never thought about it at length, and I ended up giving him the old standby answer: 0. The apparent contradictory nature of having something stand for nothing kept humanity from developing it for some time, and its existence allows numbers to be written in the modern base 10 (or base anything) form. Books have been written about the innovation, like this one, or this one.
Honestly, though, I don’t have much of a personal connection to zero. The symbol I find really compelling, though, is the arrow. Let me explain why.
First of all, it’s everywhere. I can’t think of a mathematical subfield that doesn’t involve drawing arrows (each with their own precise mathematical meaning). We call them morphisms, mappings, functions, edges of directed grahps, etc., but it seems like they’re always present. The idea of moving your problem from here to there is central in mathematics.
Second, I love the implication of movement. There’s a line about chess, that “the threat is more powerful than the move.” Understanding how to look at a chess board as a place where forces are exerting themselves represents a leap forward in one’s ability with the game. Similarly, feeling a mathematical situation as wanting to be represented elsewehre is a key part of a developed (or developing) mathematical intuition.
Third, it’s powerful. It’s shocking how much information is carried in arrowed diagrams. They allow you to ignore clutter and focus just on what is essential. Category theory is a dramatic example of this—everything is reduced to objects and the arrows between them.
So, to that middle schooler I led astray with an out-of-date, impersonal answer, I hope you find this post.