Differential Logic • 18

Tangent and Remainder Maps

If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition $f = pq : X \to \mathbb{B}$ in the following way.

The next venn diagram shows the differential proposition $\mathrm{d}f = \mathrm{d}(pq) : \mathrm{E}X \to \mathbb{B}$ we get by extracting the linear approximation to the difference map $\mathrm{D}f = \mathrm{D}(pq) : \mathrm{E}X \to \mathbb{B}$ at each cell or point of the universe $X.$ What results is the logical analogue of what would ordinarily be called the differential of $pq$ but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for $\mathrm{d}f$ whenever it’s necessary to single it out.

$\text{Tangent Map}~ \mathrm{d}(pq) : \mathrm{E}X \to \mathbb{B}$

To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.

$\begin{array}{rcccccc} \mathrm{d}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \mathrm{d}p \texttt{,} \mathrm{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \mathrm{d}p \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0 \end{array}$

To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.

$\begin{matrix} \texttt{(} \mathrm{d}p \texttt{,} \mathrm{d}q \texttt{)} & = & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} & + & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \\[4pt] dp & = & \texttt{~} \mathrm{d}p \texttt{~} \texttt{~} \mathrm{d}q \texttt{~} & + & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} \\[4pt] \mathrm{d}q & = & \texttt{~} \mathrm{d}p \texttt{~} \texttt{~} \mathrm{d}q \texttt{~} & + & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \end{matrix}$

Capping the analysis of the proposition $pq$ in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map $\mathrm{r}(pq) : \mathrm{E}X \to \mathbb{B},$ which happens to be linear in pairs of variables.

$\text{Remainder}~ \mathrm{r}(pq) : \mathrm{E}X \to \mathbb{B}$

Reading the arrows off the map produces the following data.

$\begin{array}{rcccccc} \mathrm{r}(pq) & = & p & \cdot & q & \cdot & \mathrm{d}p ~ \mathrm{d}q \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p ~ \mathrm{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \mathrm{d}p ~ \mathrm{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p ~ \mathrm{d}q \end{array}$