Epsilon-Delta Proof Intuition

There’s a nice question on MathOverflow about the mental experience of mathematics. I tend to lean heavily on bodily sensations of motion and mental imagery when working on something mathematical: stretching, compressing, and movement. Proofs and algebraic manipulation often seem like a sort of flowing. (Something I’ve covered before in my post on reading math.)

But, anyways, here’s an excerpt from one of the responses that I think is particularly valuable. On epsilon-delta proof intuition and understanding:

One specific mental image that I can communicate easily with collaborators, but not always to more general audiences, is to think of quantifiers in game theoretic terms. Do we need to show that for every epsilon there exists a delta? Then imagine that you have a bag of deltas in your hand, but you can wait until your opponent (or some malicious force of nature) produces an epsilon to bother you, at which point you can reach into your bag and find the right delta to deal with the problem. Somehow, anthropomorphising the “enemy” (as well as one’s “allies”) can focus one’s thoughts quite well.