Gary R, list,
I expect we'll learn more about this at the Centennial conference this
summer, as Zalamea is one of the listed speakers; but here's a couple of
quotes from his book, Peirce's Logic of Continuity, to give a more specific
idea of what I was referring to (and incidentally to get us back to the
subject of pragmatism):
[[ In particular, we claim that an understanding of modern methods in
topological model theory and in category theory are extremely useful to
disentangle the riddle of Peirce's continuum. ] p. 35]
[[ . the mathematical theory of categories is the environment of
contemporary mathematics which better can be fused with Peirce's thought,
and where perhaps the greater number of tools and models can be found to
faithfully approach both Peirce's general architectonics and Peirce's
particular ideas. The continuum - vessel and bridge between the general and
the particular - is therefore specially well suited to be understood
categorically. The paradigm of the mathematical theory of categories -
"arrows, not elements"; synthesis, not analysis; relational, contextual,
external knowledge, not monolithic, isolated, internal knowledge - reflects
nicely Peirce's pragmatic maxim. In category theory the pragmatic dimension
becomes evident through diverse functorial readings ("interpretations")
between "concrete categories". As invariants of a generic functorial
back-and-forth emerge - solidly: theorematically - "real" universal notions,
definable in any "abstract category", beyond its eventual existence (or
non-existence) in given particular categories. Category Theory thus provides
the more sophisticated technical arsenal, available in the present state of
our culture, which can be used to prove that there do exist real universals,
vindicating forcefully the validity of Peirce's scholastic realism. ] p. 46]
As for other uses of Category Theory in Peircean studies (or in extending
Peirce's ideas), Jon probably knows more about that than I do. As to whether
the presence of another "category theory" within Peircean studies will cause
confusion, that's only a guess on my part, and maybe not a very educated
one. Anyway I do want to hear more about what you call "category theory,"
regardless of what you call it. (Maybe not in this thread, though.)
gary f.
From: Gary Richmond [mailto:[email protected]]
Sent: 26-Apr-14 4:08 PM
To: Gary Fuhrman
Cc: Peirce-L
Subject: Re: [PEIRCE-L] Re: de Waal Seminar: Chapter 7, Pragmatism
Gary F. List,
You wrote:
On the term "Category Theory", I guess I wasn't very clear, so let me try
again: It seems to be already an established term within Peircean studies in
the sense that mathematicians use it. As I understand it, Fernando Zalamea
(and others) are looking into mathematical and logical connections between
Peirce's work and Category Theory in that established sense, and finding
those connections fruitful. If that's the case, then I think it would cause
confusion among Peirceans to also use "Category Theory" as a term for a
subdivision of Peircean phenomenology. The case of mathematicians resisting
non-mathematical uses of the word "vector" seems to me a very different
issue
I don't agree that the consideration of a branch of phenomenology named
"Category Theory" would result in contusion "within Peircean studies" at
all. Contemporary mathematicians (and logicians,etc.) are certainly free to
use any of the tools of those disciplines (and others) developed since
Peirce's death. But it seems to me that there is a compelling case for a
third branch of phenomenology which might best be termed "Category Theory,"
and at least one prominent Peircean scholar, namely Joseph Ransdell, called
it exactly that. For when most Peirceans read 'category' in relation to
Peirce's work, they immediately think of Peirce's three universal
categories, not of mathematical category theory (to which, btw, I found only
a very few brief references within the Zalamea articles I looked at,
although I didn't research this deeply, and not the "others" you mentioned
at all).
And just how much emphasis on the possible fruits of applying mathematical
category theory to Peirce's writings on continuity and Existential graphs is
there in fact anyhow? For example, in this passage from Zalamea's
"Plasticity and Creativity in the Logic Notebook," it is mentioned but once.
<http://www.pucsp.br/pragmatismo/dowloads/lectures_papers/zalamea-paper.pdf>
http://www.pucsp.br/pragmatismo/dowloads/lectures_papers/zalamea-paper.pdf
In fact, even if abstraction, order and visual harmony have been embodied,
for
example, in the paintings of Rothko or in the sculptures of Caro, Peirce's
heirs have still to understand that compelling mixture in mathematics. If
Category Theory confirms itself as an appropriate general topos for such an
encounter, if its technical expression turns out to be describable by the
logic of Sheaf Theory, and if sheaf logic situates finally at the "heart" of
a wider Synthetic Philosophy of Mathematics, then we could appreciate better
the extraordinary power of the LN [Logic Notebook] seeds.
Furthermore, I have seen mathematicians apply not only category theory but
vector analysis and other modern mathematical tools to aspects of Peirce's
work in continuity theory and EGs in particular, with no resultant confusion
within Peircean studies. So I think you may be fearing a 'confusion' which
is really highly unlikely to occur.
Yet at the outset of that same paper just referenced, Zalamea writes
something relating much more to the idea I have in mind for the use of
"category theory" within Peirce's classification of the sciences. He begins
the paper with this remark:
Peirce's architectonics, far from rigid, is bended by many plastic
transformations,
deriving from the cenopythagorean categories, the pragmaticist (modal)
maxim, the logic of abduction, the synechistic hypotheses and the triadic
classification of sciences, among many other tools capable of molding
knowledge
It is Peirce himself who held for a "triadic classification of sciences," at
least in his late work in the Science of Review, and the exceptional
sciences which aren't so divided, notably the physical and psychical wings
of the special, or idioscopic sciences, are themselves each trichotomically
divided, namely, into descriptive, classificatory, and nomonological
branches. The categories are a living presence in Peirce's classification.
So, when considering the movement from the phaneron to extracting something
from it for use in the sciences immediately following phenomenology, i.e.,
the normative sciences, and seeing that de Tienne had explored a possible
second phenomenological science, Iconoscopy, I began to see that even that
move, essential as I think it may be, doesn't take us far enough to in the
direction of extracting from the phaneron that which could be put to
cognitive use in the normative sciences.
So, reflecting on my couple of decades long work on Peirce's applied science
of Trichotomic, I began to imagine that what it "applied" were the findings
of an additonal theoretical science, the third branch of phenomenology,
namely, Category Theory. Now the work that de Tienne and I have been doing
is at best tentative. But I think Kees' question as to how we do extract
something from the phaneron for use in the normative sciences needs to be
addressed.
You continued:
It's been a long time since I read De Tienne's paper on "Iconoscopy", and I
only dimly remember the context in which we discussed that term before, so
I'll defer to your judgment on that. (I didn't even remember that your
proposal is to use "phaneroscopy" as only the first branch of
"phenomenology" . I'm still in the habit of using those terms synonymously
in reference to Peirce.) So I guess I shouldn't have ventured a comment on
that question.
Since you correctly, and following Peirce, note that the phaneron is one,
the analysis of it seems to require an expansion of phenomenology to include
other branches. And Peircean categoriality itself led me to posit that there
may be three, phaneroscopy, iconoscopy, and category theory.
We do seem to be using the Keesian phrase "extracting something from the
phaneron" in quite different senses. In my sense, it's not the elements that
are "extracted" from the phaneron for special attention but some phenomenal
ingredient of it; so the "essential elements" of the extracted idea (or
whatever we call it) would be completely different from the elements of the
phaneron (i.e. the "categories"). For one thing, they wouldn't be
indecomposable as the elements of the phaneron are. So again we're speaking
different dialects here, it seems.
I'm not exactly sure what you're aiming at in making this distinction and I
may be missing your point completely. But I would suggest that the
extraction of the indecomposable elements is primarily the work of
iconoscopy, and that the "essential elements" can be placed into trichotomic
relations, and that this is the work of category theory. But we indeed may
be "speaking different dialects here." So I think more work is needed in
phenomenology for getting from the phaneon to what might be usefully
extracted from it for the normative sciences? Absolutely. Therefore, I hope
we keep this conversation going and growing in the next few years. But I
certainly will continue to use 'category theory' as I have, and doubt that
many will be confused, or any for very long.
Now, it's probably time to return to pragmatism, from which we have strayed
pretty far, I think.
Best,
Gary,
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