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

Posted on May 15, 2014 by Jon Awbrey

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

Let’s now look at a concrete 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.

Here we have K(r, s) = r + s and L(u, v) = u \cdot v and the 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 operations fit together.

Logarithm Arrow J : {+} ← {⋅}
\text{Figure 49. Logarithm Arrow}~ J : \{ + \} \gets \{ \cdot \}

In short, where the image operation J is the logarithm map, the source operation is the numerical product, and the target operation is the numerical sum, we have the following rule of thumb.

The image of the product is the sum of the images.

\begin{array}{lll} J(u \cdot v) & = & J(u) + J(v) \end{array}

Resources

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