[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 37

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Cybernetics List, Peirce List,

There was a wobbly construction in my last paragraph,
so let me try to shore it up in this re-construction:

| The suggested perspective on algebraic expressions allows us to understand
| the algebraic law as saying, in effect, that every one of the arithmetic
| expressions that fits the indicated algebraic pattern evaluates to the
| very same canonical expression for the end result of its evaluation.
| This is, as far as I can tell, just about as close as we can come
| to a conceptually and ontologically minimal way of understanding
| the relation between an algebra and its corresponding arithmetic.

Of course, it is not really necessary to consider every possible substitution
of arithmetic expressions for the algebraic variables, since only the value
of each arithmetic expression can make any difference to the end result.
Nevertheless, taking an algebraic expression as a syntactic mechanism
for singling out a particular subset of the primary arithmetic is
a move that suggests very fruitful directions of generalization.

In particular, this point of view helps us to sidestep many of the mysteries
that encumber particular mechanisms of substitution, which it takes all the
rigors of combinator calculus and lambda calculus even to begin clearing up,
and it also provides us with an alternative way of approaching the puzzles
of so-called "imaginary" values.

But I will have to leave it with those hints for now,
as there is still much to do at the elementary level.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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