According to the honorable Frank Calegari, two engines operate in mathematical proofs:

1. Intuition of what should be

2. Construction and manipulation of symbols

The ideal situation is a mathematician can work back and forth between the intuition and the symbol operations. Or in other words, a well-defined mathematical structure should reflect and enlighten our hopes of what should be; thus such structure can serve as a powerful tool for understanding reality.

Consider the famous delta-epsilon definition of the limit: For all epsilon > 0, there exists a delta > 0, such that when |x-a| is within delta, |f(x)-L| will be within epsilon. We say L is the limit of f(x) at x=a.

The point is, this definition is nearly impenetrable if we only consider the symbols. Several unsaid but definitely implied messages from this definition:

1. when we say epsilon > 0, we really mean very small epsilons. If for epsilon = 0.1 we can find delta, certainly for epsilon = 1 we can find delta (just apply the same delta). We really mean "infinitely small" epsilon or say "the smallest but non-zero number"; but these doesn't really exist; thus we say for all epsilon > 0.

2. the concept of limit we want to capture is: as x gets closer to a, f(x) gets closer to L. Very simple.

3. continue with 2, though, we must ask: how close? It is not rational to define close as "maybe just within 0.0000001 away". It is therefore reasonable that we say we really want "NO MATTER HOW close we want f(x) to be relative to L, we can always get that value of f(x)." Notice the sequence of the two clauses switched: now instead saying closer x closer f(x), we say first how close f(x) then find x. This is how we end up with the spell-like chant "for all epsilon > 0, exists a delta > 0......"