Jerry, List,
With regards to the following statement:
JA: The ''structure'' that is preserved by a structure-preserving map
is just the structure that we all know and love as a triadic relation.
I am saying this in the context of a long-running discussion about
arrows between triadic relations, considered both abstractly and
in the instance of concrete examples like real number addition
and multiplication, so the remark is true in that context.
Of course there are other contexts in which we would be
dealing with arrows between other types of structures.
I an not conflating functions and relations.
Functions are special cases of relations,
that is, relations that are subject to
certain additional constraints.
For instance, the things usually called "binary operations",
say, real number addition + : R × R → R, are the same things
as "ternary" or "triadic" relations, say, [+] ⊆ R × R × R.
(Here I am using the device of enclosing the operation sign
in square brackets as a name for the associated relation.)
There is background material on relations in general and the
principal species of functional relations in this article:
Relation Theory
http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
Regards,
Jon
Jerry LR Chandler wrote:
List, Jon:
On May 15, 2014, at 11:22 PM, Jon Awbrey wrote:
"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."
I continue to struggle with the foundational notions that you are seeking to
communicate with
your rhetoric.
Your write:
The ''structure'' that is preserved by a structure-preserving map is just the
structure that we
all know and love as a triadic relation.
This assertion is generally denied by current mathematical thinking. You are
once again, as
typically is the case in your expositions, conflating the notion of a triadic
relation with the
notion of a function. These are well-separated concepts in category theory. The
adicity of a
function may be any number you choice. If you have an alternative argument to
my assertion,
please bring it forward.
Secondly, your introduction of the notion of "love" in this context is an overt
attempt to appeal
to social / political perspectives of the readers of this list. How do you
introduce the concept
of "love" into mathematics?
(BTW, the notion of "love" is intrinsic to chemical thinking in the direct form
of
"nucelio-philic" and "electro-philic" reactions.)
You further write:
"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."
What do you mean by "very typically"?
Certainly, mathematics is not merely about "very typical" inferences, or is
this assertion
foundational to your beliefs that mathematics is restricted to very typical
inferences? If so,
how would you frame this point of view in terms of group theory or any other
structural theory of
mathematics?
Clearly, you are struggling to make your philosophy of mathematics clear.
Perhaps it would be
helpful is you formulated a sharp paragraph or page on your underlying
philosophy of mathematics
so that I could attempt to decipher your encodings of mathematical symbols.
Cheers
Jerry
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