\section{The place of rigor in a calculus course}

The level and form of rigor is one of the key decisions in a calculus course.
Rigor is the \emph{natural} language of mathematics, but it took
mathematicians centuries to find their voice.

\subsection{You can't read if you can't write}
If students are not asked to \emph{write} rigorous proofs, i.e., to produce
them on their own, then they will not be able to understand them if presented
in class or in the book; they may grasp them one step at a time, but they
won't retain them, nor understand the concepts therein.

Further, just asking students to repeat proofs they've seen (instead of
writing their own, using techniques they've seen) will result in them
memorizing them mantra-style, rather than learning the point.

If you want students to understand the proofs (or really understand the
theorems), you need a radical commitment to
rigor; if the ramifications are too objectionable, then ask yourself how much
understanding is worth.

\subsection{Newton wasn't rigorous}
Standards of proof have changed over the centuries, and much of calculus is
rooted in the informal 17th century, yet draws on the rigor of the 19th.

Physics reasoning is often an acceptable balance
