Formalism

As much of what students do in calc is \emph{formal}
(meaning ``pushing symbols around''),
it's worth acknowledging it and pointing it out.

That is, the fact that the derivation rules are formal means that they
work over the complex numbers (and $p$-adics!) as well as the reals.

The notion of a ``formal power series'' confused me at first.
The idea is: ``it's a symbol, not representing a -function-''.

That is: we can add, subtract, multiply (but not quite divide) power
series just -as power series-,
without worrying about what this means in terms of functions,
ie., w/o plugging anything in.

Here's another EG: if you look at $x^2-x$ over $\bZ/2$ (easily
introduced as ``evens and odds''), we see that -as a function-, it's
identically $0$ (this is clear, as it factors as $x(x-1)$) -- but
it's clearly not $0$ -as a polynomial-.

That's essentially what we mean.

More formally (meaning ``not informal''), ``formal'' means:
follows from the axioms, and nothing more: nothing special about our
situation.

Thus derivative rules work in any metric field
(note that we don't use any ordering, for instance).
