The problem is that as we understand them today, mathematical formulas like yours are void of meaningful semantics unless you have any objective abstract notion of "infinite set", which seems not to be the case because "everything is a set", and "everything" is not an abstraction.

So we have to deal with them as pure sequences of signs which are part of the set of deducible formulas.

Your "understanding" (or mine or Euler's) of the formula is most likely a metaphor (well I'd say an analogy in this case) and is what led to its proof.