Who's afraid of epsilon and Delta?

"And what are these Fluxions? ... They are neither finite Quantities nor Quantities infinitely small, nor yet nothing.
May we not call them the Ghosts of departed Quantities?"

-Bishop George Berkely

It's no wonder it took so long to rigorously define the notion of a limit. You can articulate the rough idea very easily, for a function its basically "just look at it", for a a sequence its "where its going", but actually narrowing down the idea into a concrete whole is difficult.

It took a hell of changes, both philosophical and mathematical in nature, to get to our modern "epsilon-delta definition" and it's hard to appreciate from our radically different perspective. So instead of trying to imagine how hard it was to get to these useful definitions, imagine how easy it was to fuck it up.

You can be too strict: "There must be some point, beyond which the sequence equals its limit" you can be too loose: "For any point along the sequence, there must be some point after which equals its limit"

It's clear we're missing something in our understanding here, by using any notion of equivalence in the definition we end up getting conditions too strict to encapsulate the behaviour we want, or too broad to be useless. Combining notions of infinity and equivalence together ends up being a tricky thing, it's like the mathematical equivalent of staring directly at the sun. You can relate closely with the early notions of "fluxions", appealing to some middle ground which the finite and infinite can begin to talk on.

But fluxions are a poor patch for what requires a conceptual jump. What we need is an alternative way of expressing equivalence of two real numbers, something more "subtle".

The real insight ends up being a trivial result mathematically, but has a philosophical layer to it which - I think -is often unappreciated:

"Two real numbers are equal if and only if their difference is smaller than any arbitrary positive number" i.e x = y iff Ae > 0, |x-y| < e

It allows us to state the equality of two numbers without ever requiring a direct statement of equality. It creates this kind of shielding from having to directly use equality, which can be used directly to inspire the notion of the limit. Its interesting to note the definite non-constructive tendency weve found ourselves in, proving the above result may be done constructively one way, but must be done non-constructively the other.