“In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others.”
I return often to this quote from Bill Thurston. Here’s what I’m getting out of it today: the mathematical researcher is just looking for something new to share with others, a distinctive piece of the mathematical universe that they can teach the world. The search is for something new, but only because new ideas are something worth sharing.
Mathematical teachers, professionally speaking, are chasing the same goal via the opposite strategy. We don’t bother seek new mathematics, which after all is only one way among many to find something worth sharing. But what we do aim to share, we aim to share more widely than any researcher could.
I wonder whether there’s an alternate history of mathematics out there. Rather than focusing on those who were mathematically innovative, it would be a history of learners and sharers of mathematics par excellence. Would that be the same history? I don’t think it would.