Dear Irving and List:
Thank you for your post on the relation of category theory as an
abstract mathematical structure and lattice theory.
I want to study it before offering comments, if any. I was
delighted that the paper includes references that are of interest to
me!
Irving, you post of a week or two ago on the notion of the term
"category" and language usage has occupied my attention. I suspect
that we are using the term in very slightly different senses.
I note that in the early history of human language development, the
general public language was not separated into technical disciplines.
In particular, one notes that both chemical language and mathematical
language developed as separate sub-species of natural language with
specific definitions.
Why should one believe that the concept of "number" used by chemistry
are defined exactly as the numbers used in mathematics?
Numbers originated in natural language and the conceptual basis for
usage emerged over many centuries. It is my belief that the usages
of terms diverged.
With these sentences as background, I return to your post. I suspect
that my philosophy of mathematics is radically different than yours.
In light of the Rosen / Ehresman exchanges, I gradually adopted views
similar to those of Frederick Wasserman, a personal friend of
Wittgenstein. (see reference: Conceptual Foundations of
Mathematics?) This view of mathematics places priority on the role
of numerical calculation. Numerical calculations with atomic numbers
and associated properties are one of the foundation of chemistry.
As a simple one to one correspondence between the integers and the
identities of chemical elements exists,
my views on the relationships between mathematical philosophy and
chemical philosophy are NATURALLY SELF-CONSISTENT in terms of
calculations. This provides guidance in aiding the understanding of
the biological role of DNA in emergence and cellular metabolism.
Of course, I recognize that the symbol systems used in mathematics
and physics do not always correspond with exact calculations as we
force upon chemical thought (for example, chaos theory and many many
physical theories). The necessity for approximations in both
mathematical and physical computation is so common that the
distinction between exact and inexact calculations are often ignored.
Finally, I would note that in contrast to both abstract physical and
mathematics theories, chemical abstractions, in the form of the
identities of distinguishable structures, are tightly linked to
natural language and everyday events in biology an medicine. Thus,
"reality checks" are built into the logic of chemistry in a
profoundly different way than in either generalized logic or
mathematics.
Irving, perhaps you would give us a feel for your personal philosophy
of mathematics such that it motivated your post? I would also be
interested on how you relate your views of Mathematics to Firstness,
Secondness and Thirdness.
(The recent post of GaryR suggests that later in life, Peirce
completely separated the concepts of chemical valence from his
justification of exactly three categories.)
An open question:
What are you views on the relations between para-consistent logics
and category theory?
Cheers
Jerry
On May 17, 2006, at 1:06 AM, Peirce Discussion Forum digest wrote:
Subject: algebraic logic and category theory
From: "Irving Anellis" <[EMAIL PROTECTED]>
Date: Tue, 16 May 2006 11:37:31 -0500
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Not long ago we considered the question of [mathematical] category
theory
as it might relate to CSP's work and possible interest in logic.
A new article has just come up on arXiv.org using category theory to
discuss the problem of distributivity. The question of whether all
lattices are distributive was a major issue between Peirce,
Schr=F6der, and
Huntington, among others, and the historical and mathematical
background
and Peirce's treatment of the problem is available in Nathan Houser's
1985 Ph.D. thesis for the University of Waterloo, Peirce=92s
Algebra of
Logic and the Law of Distribution and in his article "Peirce and
the Law
of Distribution", in Thomas Drucker (ed.), Perspectives on the
History of
Mathematical Logic (Boston/Basel/Berlin: Birkh=E4user, 1991), 10-32.
The abstract of the new article applying category theory to study
distributivity for many-valued algebraic structures follows:
---------------------------------------------------------
Many-valued complete distributivity
Authors: Hongliang Lai, Dexue Zhang
Comments: 35 pages
Subj-class: Category Theory; Logic
MSC-class: 03G25,06D10,06F07,18B35,18D20,68Q55
Categories enriched over a commutative unital quantale can be
studied as
generalized, or many-valued, ordered structures. Because many
concepts,
such as complete distributivity, in lattice theory can be
characterized
by existence of certain adjunctions, they can be reformulated in the
many-valued setting in terms of categorical postulations. So, it is
possible, by aid of categorical machineries, to establish theories of
many-valued complete lattices, many-valued completely distributive
lattices, and so on. This paper presents a systematical
investigation of
many-valued complete distributivity, including the topics: (1)
subalgebras and quotient algebras of many-valued completely
distributive
lattices; (2) categories of (left adjoint) functors; and (3) the
relationship between many-valued complete distributivity and
properties
of the quantale of truth values. The results show that enriched
category
theory is a very useful tool in the study of many-valued versions of
order-related mathematical entities.
Full-text: PostScript, PDF, or Other formats
References and citations for this submission:
----------------------------------------------------------------------
----
The abstract is available at: http://arxiv.org/abs/math/0603590
and the full article can be downloaded there in several formats
including
pdf(http://arxiv.org/PS_cache/math/pdf/0603/0603590.pdf)
Irving H. Anellis
[EMAIL PROTECTED] ; [EMAIL PROTECTED] ;
[EMAIL PROTECTED]
http://www.peircepublishing.com
http://www.abebooks.com/home/PEIRCEPUBLISHING
--=20
Jerry LR Chandler
Research Professor
Krasnow Institute for Advanced Study
George Mason University
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