This shouldn’t be “memorized” as a “fact” though. As you point out, it’s the very definition of e. Which is to say, to really understand what the exponential function means implies building up a mental model about lim [n→∞] (1 + x/n)^n and its behavior, interacting with it, connecting it to other functions, seeing what happens when you combine it with other ideas: trying to integrate/differentiate it; noticing how it reacts to fourier transform; relating it to rotations, areas, imaginary numbers; writing it as a taylor expansion or continued fraction, or with a functional equation, or as the solution to a differential equation. Connecting it to derived functions such as e.g. the normal distribution, or sine, or hyperbolic sine. Generalizing it to an operation on complex numbers, or quaternions, or matrices. Thinking about what exponentiation in general really means. Coming up with algorithms for computing e’s decimal expansion or proving that e is irrational and transcendental. Solving problems with it, like to start with, continuously compounded interest (&c. &c. &c.).
No one who had really learned about exponentials in a deep way would easily forget that this is the definition of e, and that has nothing to do with lists of facts or rote memorization.
> Single-character symbols (and multiple character symbols) that not only hide tons of meaning through a dereference, which is naturally how language works, that also suggest a meaning themselves, are stupid and promote natural diseased human thinking.
I think you should try writing out some very difficult complex math proofs before you make this assertion (say, for instance, a proof . Things are bad enough when we pack ideas down. Start expanding them in the middle of the computation, and it the steps become almost impossible to see or reason about.
The whole point of assigning things short simple names is that it gives a clear analogy (to past experience w/ the conventions) that provide a shortcut to anticipating how particular types of structures and operations behave. Cramming a matrix or a multivariate function or an operator down into a single symbol helps us to treat that symbol as a black box for the purpose of the proof or problem, which helps us bring our existing mathematical tools and understandings to bear. Sometimes, we run afoul of false impressions, as when we apply intuition about the real numbers to matrices in ways that don’t quite apply, or intuition about metric spaces to general topological spaces, &c. But this is I think an unavoidable cost, and it’s why we strive to be careful and precise in making mathematical arguments.