My goal in this site is to develop a rigorous acccount of undergraduate math
concepts, following four fundamental principles:
No magic lemmas. Theorems should be broken down into steps that are
motivated, that actually seem like a step towards the goal, before those
steps are proved. e.g. don't prove Rolle's theorem until after you
can explain what that has to do with Taylor polynomials.
No magic formulas. Before you even state the goal of a theorem or
lemma, the individual operations and formulas should themselves have
emerged naturally in the course of proving earlier theorems. e.g. don't
even state Taylor's theorem as a goal until you can explain why
Taylor polynomials are interesting and useful all on their own.
Respect common language. If academic language would allow a set to be
both open and closed, or for continuous functions to have jumps, then
academic language is wrong, not common language.
Emphasize computation and real-world inference. Differential and integral
calculus are used by real people to infer real facts about the world every
day; a rigorous account of mathematics should rigorously explain why
that is possible, not just why some theorems follow from some
arbitrary choice of axioms.
I have worked out lots of ideas and sketches following these principles, but
I haven't written many of them up. See
jarviscarroll.net/projects/derivations.html
for more information about this and related projects.