Precursors Of Category Theory

Jon Awbrey

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Sep 13, 2020, 1:30:59 PM9/13/20

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Cf: Precursors of Category Theory • Discussion 1
http://inquiryintoinquiry.com/2020/09/13/precursors-of-category-theory-discussion-1/

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JA: In the logic of Aristotle categories are adjuncts to reasoning
designed to resolve ambiguities and thus to prepare equivocal signs,
otherwise recalcitrant to being ruled by logic, for the application
of logical laws. The example of ζωον illustrates that we don't need
categories to *make* generalizations so much as we need them to *control*
generalizations, to reign in abstractions and analogies that are stretched
too far.

EB Aristotelian categories are “adjuncts to reasoning”?

I explain ...

I've been exploring a particular type of commonality or continuity in the
way categorical references function in various systems of categories from
Aristotle, through Kant and Peirce, to contemporary mathematical category
theory. The following posts give the background on that.

Precursors Of Category Theory
=============================
https://inquiryintoinquiry.com/2013/12/20/precursors-of-category-theory-1/
https://inquiryintoinquiry.com/2013/12/30/precursors-of-category-theory-2/
https://inquiryintoinquiry.com/2014/01/03/precursors-of-category-theory-3/

Following on the discussion of Excluded Middle and Non-Contradiction together with
Peirce's observations about their relationship to the General and the Vague, the
point is that elements of natural language, indeed, of all forms of representation
in the wild, do not as a rule obey the customary laws of logic, but violate them
in various ways. It is only after signs and symbols have been categorized, their
equivocations "driven down" or "reduced" by reference to the appropriate category,
that they become subject to logical laws.

References
==========

* Aristotle, "The Categories", Harold P. Cooke (trans.), pp. 1–109 in
Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

* Collection Of Source Materials https://oeis.org/wiki/User:Jon_Awbrey/EXCERPTS
* Determination https://oeis.org/wiki/User:Jon_Awbrey/EXCERPTS#Determination
* Peirce CP 5.447 and CP 5.448
https://oeis.org/wiki/User:Jon_Awbrey/EXCERPTS#Excerpt_5._Peirce_.28CP_5.447.29
https://oeis.org/wiki/User:Jon_Awbrey/EXCERPTS#Excerpt_6._Peirce_.28CP_5.448.29

Resources
=========

* Precursors Of Category Theory
https://oeis.org/wiki/Precursors_Of_Category_Theory

* Survey of Precursors Of Category Theory
https://inquiryintoinquiry.com/2015/05/15/survey-of-precursors-of-category-theory-1/

* Lane, R. (2001), "Principles of Excluded Middle and Contradiction",
in M. Bergman and J. Queiroz (eds.), The Commens Encyclopedia : The
Digital Encyclopedia of Peirce Studies New Edition, Pub. 140731-0107a.
http://www.commens.org/encyclopedia/article/lane-robert-principles-excluded-middle-and-contradiction

Regards,

Jon

Jon Awbrey

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Sep 20, 2020, 12:56:23 PM9/20/20

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Cf: Survey of Precursors Of Category Theory • 2
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All,

Here's an updated Survey of blog posts and wiki notes on
"Precursors of Category Theory", covering up through the
discussions of Robert Marty's contributions with respect
to Peirce's Categories this summer.

Wiki Notes
==========

* Precursors Of Category Theory

( https://oeis.org/wiki/Precursors_Of_Category_Theory )

* Propositions As Types Analogy
( https://oeis.org/wiki/Propositions_As_Types )

Blog Posts
==========

* Notes On Categories
http://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/

* Precursors Of Category Theory

Alex Shkotin

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Sep 21, 2020, 8:26:40 AM9/21/20

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Jon,

Interesting collection. And looking at "categories, or types" in https://oeis.org/wiki/Precursors_Of_Category_Theory#Hilbert_and_Ackermann what do you think if to say "Precursors Of Type Theory" as Category Theory is a math discipline?

https://en.wikipedia.org/wiki/Category_theory

By the way your Preamble is about this kind of Cat-theory.

It seems you collect for three topics: phil-cat, type theory, math cat-theory.

Regards,

Alex

вс, 20 сент. 2020 г. в 19:56, Jon Awbrey <jaw...@att.net>:

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Jon Awbrey

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Sep 21, 2020, 2:36:26 PM9/21/20

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Cf: Precursors of Category Theory • Discussion 2
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AS:
Looking at “categories, or types” in Precursors Of Category Theory • Hilbert and Ackermann
( https://oeis.org/wiki/Precursors_Of_Category_Theory#Hilbert_and_Ackermann ) what do you
think of to say “Precursors Of Type Theory” as Category Theory is a math discipline? […]
It seems you collect for three topics: phil‑cat, type theory, math cat‑theory.

Dear Alex,

When it comes to math, computer science, and their applications
to logic and linguistics I see categories and types as pretty much
the same things. No doubt the words are used differently in other
contexts but I am concerned with the above contexts at the moment.

The diversity of categorical systems across different disciplines and
theorists is obvious to all observers. But when we examine how systems
of categories operate in grammatical, logical, or more generally semiotic
frameworks we can detect a common function all the more useful systems share.
The semiotic framework is already well marked in Aristotle's founding text
on interpretation and the function of category references as go-betweens from
unruly language to the rule of logic is clearly delineated in his treatise on
categories. It is that order of function which is preserved from Aristotle's
categories to our current mathematical variety.

Regards,

Jon

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Alex Shkotin

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Sep 22, 2020, 5:14:13 AM9/22/20

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Dear Jon,

You are the boss! I like Kant's criticism of Aristotle versions of categories.

Kant: "Finding these basic concepts - such a proposal was worthy of such an astute thinker like Aristotle. But since he did not have any principle, he picked them up as they came across to him, and first typed ten concepts, which he called categories (predicates). Then it seemed to him that he found five more such concepts, which he added to the previous ones under the name of post-predicate. However, his table was still insufficient."

Kant, Critique of Pure Reason, §10. About pure rational concepts, or categories.

Regards,

Alex

пн, 21 сент. 2020 г. в 21:36, Jon Awbrey <jaw...@att.net>:

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Jon Awbrey

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Sep 25, 2020, 12:25:40 PM9/25/20

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Cf: Precursors Of Category Theory • Discussion 3
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| Take your place on The Great Mandala
| As it moves through your brief moment of time.
| Win or lose now you must choose now
| And if you lose you're only losing your life.
|
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AS: I like Kant's criticism of Aristotle versions of categories.

<QUOTE>

“Finding these basic concepts — such a proposal was worthy of such an

astute thinker like Aristotle. But since he did not have any principle,
he picked them up as they came across to him, and first typed ten concepts,
which he called categories (predicates). Then it seemed to him that he found
five more such concepts, which he added to the previous ones under the name of

post-predicate. However, his table was still insufficient.”

(Kant, Critique of Pure Reason, §10. About Pure Rational Concepts, or Categories).

</QUOTE>

Dear Alex,

My sketch on Precursors Of Category Theory shows a big ellipsis
under the heading for Kant. I meant to get back to him, as I used
to do every half-decade or so, but it's been a long time since I kept
to that schedule. Kant is a lodestar in the Peircean constellation —
Peirce's “New List of Categories” invokes his guidance on the function
of concepts just as he tries his own hand at the wheel. I quoted that
passage in my selections from Peirce:

https://inquiryintoinquiry.com/2014/01/03/precursors-of-category-theory-3/

Selection 1
===========

<QUOTE>

§1. This paper is based upon the theory already established, that the
function of conceptions is to reduce the manifold of sensuous impressions
to unity, and that the validity of a conception consists in the impossibility
of reducing the content of consciousness to unity without the introduction of it.
(CP 1.545).

§2. This theory gives rise to a conception of gradation among those conceptions
which are universal. For one such conception may unite the manifold of sense and
yet another may be required to unite the conception and the manifold to which it is
applied; and so on. (CP 1.546).

C.S. Peirce, “On a New List of Categories” (1867)

</QUOTE>

To be continued …

Regards,

Jon

Alex Shkotin

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Sep 26, 2020, 6:30:13 AM9/26/20

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Dear Jon,

Let me cite my favorite from Hegel,

"Diese Kategorien, die nur instinktmäßig als Triebe wirksam sind und zunächst vereinzelt, damit veränderlich und sich verwirrend in das Bewußtsein des Geistes gebracht [sind] und ihm so eine vereinzelte und unsichere Wirklichkeit gewähren, zu reinigen und ihn damit in ihnen zur Freiheit und Wahrheit zu erheben, dies ist also das höhere logische Geschäft."

from http://www.zeno.org/Philosophie/M/Hegel,+Georg+Wilhelm+Friedrich/Wissenschaft+der+Logik/Erster+Teil.+Die+objektive+Logik/Vorrede+zur+zweiten+Ausgabe

I am sure in English it should be great as in Russian it's marvelous

"Следовательно, высшая задача логики — очистить категории, действующие лишь инстинктивно как влечения и осознаваемые духом прежде всего разрозненно, тем самым как изменчивые и путающие друг друга, доставляющие ему таким образом разрозненную и сомнительную действительность, и этим очищением возвысить его в них к свободе и истине."

http://philosophy.ru/library/nauka-logiki-tom-1/

Regards,

Alex

пт, 25 сент. 2020 г. в 19:25, Jon Awbrey <jaw...@att.net>:

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Alex Shkotin

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Sep 26, 2020, 7:08:26 AM9/26/20

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Jon,

Luckily I have found English translation "As impulses the categories do their work only instinctively; they

are brought to consciousness one by one and so are variable and mutually
confusing, thus affording to spirit only fragmentary and uncertain actuality.
To purify these categories and in them to elevate spirit to truth and freedom,

this is therefore the loftier business of logic."

https://academiaanalitica.files.wordpress.com/2016/10/georg_wilhelm_friedrich_hegel__the_science_of_logic.pdf

Sorry for delay,

Alex

пт, 25 сент. 2020 г. в 19:25, Jon Awbrey <jaw...@att.net>:

Cf: Precursors Of Category Theory • Discussion 3

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