Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.15
http://inquiryintoinquiry.com/2014/05/15/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-15/
Posted : May 15, 2014 at 4:32 pm
Author : Jon Awbrey
Peircers,
I'm going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving
maps, as a modest amount of extra work at this point will repay ample dividends when it comes time
to revisit Peirce’s “number of” function on logical terms.
The ''structure'' that is preserved by a structure-preserving map is just the structure that we all
know and love as a triadic relation. Very typically, it will be the type of triadic relation that
defines the type of binary operation that obeys the rules of a mathematical structure that is known
as a group, that is, a structure that satisfies the axioms for closure, associativity, identities,
and inverses.
For example, in the case of the logarithm map J and the real domain R we have
the following data:
• J : R ← R (properly restricted)
• K : R ← R × R where K(r, s) = r + s
• L : R ← R × R where L(u, v) = u · v
Real number addition and real number multiplication (suitably restricted) are examples of group
operations. If we write the sign of each operation in brackets as a name for the triadic relation
that defines the corresponding group, we have the following set-up:
• J : [+] ← [·]
• [+] ⊆ R × R × R
• [·] ⊆ R × R × R
It often happens that both group operations are indicated by the same sign, usually one from the set
{ · , * , + } or simple concatenation, but they remain in general distinct whether considered as
operations or as relations, no matter what signs of operation are used. In such a setting, our
chiasmatic theme may run a bit like one of the following two variants:
• ''The image of the sum is the sum of the images.''
• ''The image of the product is the sum of the images.''
Figure 50 presents a generic picture for groups G and H.
Figure 50. Group Homomorphism J : G ← H
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-50.jpg
In a setting where both groups are written with a plus sign, perhaps even
constituting
the same group, the defining formula of a morphism, J(L(u, v)) = K(Ju, Jv),
takes on
the shape J(u + v) = Ju + Jv, which looks analogous to the distributive
multiplication
of a factor J over a sum (u + v). This is why morphisms are regarded as
generalizations
of ''linear'' functions and are frequently referred to in these terms.
Regards,
Jon
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