Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.14
http://inquiryintoinquiry.com/2014/05/15/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-14/
Posted : May 15, 2014 at 1:48 am
Author : Jon Awbrey
Peircers,
Let's now look at a more homely example of a morphism J,
say, one of the mappings of reals into reals that are
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 ∙ v
and the defining formula J(L(u, v)) = K(Ju, Jv) becomes
J(u ∙ v) = J(u) + J(v), where ordinary multiplication
and addition are indicated by a dot (∙) and
a plus sign (+) respectively.
Figure 49 shows how the multiplication, addition, and logarithm functions fit
together.
Figure 49. Logarithm Arrow J : {+} ← {∙}
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-49.jpg
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:
• ''The image of the product is the sum of the images.''
• J(u ∙ v) = J(u) + J(v)
• J(L(u, v)) = K(Ju, Jv)
Regards,
Jon
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