Sorry if this is the wrong place to ask this, but I feel it's a question of vital importance to the future of math education.
I hear this listed as a "math joke" all the time and I've never got it. I've always thought that it was just dumb, but have started to wonder if I'm missing something. Is the joke literally that it's just unexpected to take $\varepsilon < 0$ since we usually let $\varepsilon$ represent a small positive number, or is there something else to it?
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In analysis it's almost reflexive to say "Let $\epsilon>0$ be given" at the beginning of an $\epsilon -\delta$ proof. It just seems so counter to how we've been trained to say "let $\epsilon<0$" that it almost seems repulsive.
Imagine if someone started a proof with "assume for verification", instead of "assume for contradiction". It's a similar circumstance.
I seem to recall seeing a proof given in a lecture course once where $\epsilon<0$ was an assumption, and I also recall the deepset uncomfortable feeling it gave in the pit of my stomach. I'm surprised there weren't more gasps of horror from the audience.
Everytime I see this explained I never see anyone mention the play on words between the mathematical 'let variable be whatever' versus the English protest 'let women vote!' Or whatever other such injustice.
The feigned importance is where I find the humor anyway. Not 'look I picked a value outside of canonical meaning isn't that funny', but I'm sick of epsilon always forced to be greater than 0 so I'm leading a rebel movement to convince people to let epsilon finally be something other than an arbitrary positive number.