Peirce’s 1870 “Logic Of Relatives” • Comment 11.14

Let’s now look at a more homely example of a morphism $J,$ say, one of the mappings of reals into reals commonly known as logarithm functions, where you get to pick your favorite base.

In this case we have $K(r, s) = r + s$ and $L(u, v) = u \cdot v$ and the defining formula $J(L(u, v)) = K(Ju, Jv)$ becomes $J(u \cdot v) = J(u) + J(v),$ where ordinary multiplication and addition are indicated by a dot $(\cdot)$ and a plus sign $(+)$ respectively.

Figure 49 shows how the multiplication, addition, and logarithm functions fit together.

 (49)

Thus, where the image $J$ is the logarithm map, the compound $K$ is the numerical sum, and the ligature $L$ is the numerical product, one has the following rule of thumb:

$\textit{The image of the product is the sum of the images.}$

$\begin{array}{lll} J(u \cdot v) & = & J(u) + J(v) \\[12pt] J(L(u, v)) & = & K(Ju, Jv) \end{array}$

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