Show threadJon AwbreyMay 27, 2024

Precursors Of Category Theory • 3
• https://inquiryintoinquiry.com/2024/05/27/precursors-of-category-theory-3-a/

❝Act only according to that maxim by which you can at the same time will that it should become a universal law.❞

— Immanuel Kant (1785)

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

❝§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).

Cued by Kant's idea regarding the function of concepts in general, Peirce locates his categories on the highest levels of abstraction able to provide a meaningful measure of traction in practice. Whether successive grades of conceptions converge to an absolute unity or not is a question to be pursued as inquiry progresses and need not be answered in order to begin.

Resources —

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

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Survey of Precursors Of Category Theory
• https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-theory-5/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 3

Act only according to that maxim by which you can at the same time will that it should become a universal law. Immanuel Kant (1785) C.S. Peirce • “On a New List of Categories” (186…

Inquiry Into Inquiry
Show threadJon AwbreyMay 26, 2024

Precursors Of Category Theory • 2.3
• https://inquiryintoinquiry.com/2024/05/26/precursors-of-category-theory-2-a/

In the logic of Aristotle categories are adjuncts to reasoning whose function is 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 the fact that we don't need categories to “make” generalizations so much as to “control” generalizations, to reign in abstractions and analogies which have been stretched too far.

References —

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

• Karpeles, Eric (2008), Paintings in Proust, Thames and Hudson, London, UK.

Resources —

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

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Survey of Precursors Of Category Theory
• https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-theory-5/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 2

Thanks to art, instead of seeing one world only, our own, we see that world multiply itself and we have at our disposal as many worlds as there are original artists … ☙ Marcel Proust W…

Inquiry Into Inquiry
Show threadJon AwbreyMay 26, 2024

Precursors Of Category Theory • 2.2
• https://inquiryintoinquiry.com/2024/05/26/precursors-of-category-theory-2-a/

Aristotle —

❝Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different. For instance, while a man and a portrait can properly both be called animals (ζωον), these are equivocally named. For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different. For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.

❝Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding. Thus a man and an ox are called animals. The name is the same in both cases; so also the statement of essence. For if you are asked what is meant by their both of them being called animals, you give that particular name in both cases the same definition.❞ (Aristotle, Categories, 1.1a1–12).

Translator's Note. ❝Ζωον in Greek had two meanings, that is to say, living creature, and, secondly, a figure or image in painting, embroidery, sculpture. We have no ambiguous noun. However, we use the word ‘living’ of portraits to mean ‘true to life’.❞

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 2

Thanks to art, instead of seeing one world only, our own, we see that world multiply itself and we have at our disposal as many worlds as there are original artists … ☙ Marcel Proust W…

Inquiry Into Inquiry
Show threadJon AwbreyMay 26, 2024

Precursors Of Category Theory • 2.1
• https://inquiryintoinquiry.com/2024/05/26/precursors-of-category-theory-2-a/

❝Thanks to art, instead of seeing one world only, our own, we see that world multiply itself and we have at our disposal as many worlds as there are original artists …❞

— Marcel Proust

When it comes to looking for the continuities of the category concept across different systems and systematizers, we don't expect to find their kinship in the names or numbers of categories, since those are legion and their divisions deployed on widely different planes of abstraction, but in their common function.

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 2

Thanks to art, instead of seeing one world only, our own, we see that world multiply itself and we have at our disposal as many worlds as there are original artists … ☙ Marcel Proust W…

Inquiry Into Inquiry
Jon AwbreyMay 25, 2024

Precursors Of Category Theory • 1
• https://inquiryintoinquiry.com/2024/05/25/precursors-of-category-theory-1-a/

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. My notes on the project are still very rough and incomplete but I find myself returning to them from time to time.

Preamble —

❝Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap (“Logische Syntax der Sprache”), and “natural transformation” from then current informal parlance.❞

— Saunders Mac Lane • “Categories for the Working Mathematician”

Resources —

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

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Survey of Precursors Of Category Theory
• https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-theory-5/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 1

Years ago I began a sketch on the Precursors of Category Theory, tracing a continuity of function in the use of categories from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to cont…

Inquiry Into Inquiry
Jon AwbreyMay 24, 2024

Survey of Precursors Of Category Theory • 5
• https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-theory-5/

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

Background —

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

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Blog Series —

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

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

Precursors Of Category Theory • Discussion
1. https://inquiryintoinquiry.com/2020/09/13/precursors-of-category-theory-discussion-1/
2. https://inquiryintoinquiry.com/2020/09/21/precursors-of-category-theory-discussion-2/
3. https://inquiryintoinquiry.com/2020/09/25/precursors-of-category-theory-discussion-3/

Categories à la Peirce —

C.S. Peirce • A Guess at the Riddle
• https://inquiryintoinquiry.com/2012/03/21/c-s-peirce-a-guess-at-the-riddle/

Peirce's Categories
1. https://inquiryintoinquiry.com/2015/10/30/peirces-categories-1/
2. https://inquiryintoinquiry.com/2015/10/31/peirces-categories-2/
3. https://inquiryintoinquiry.com/2015/11/04/peirces-categories-3/
•••
19. https://inquiryintoinquiry.com/2020/05/13/peirces-categories-19/
20. https://inquiryintoinquiry.com/2020/05/14/peirces-categories-20/
21. https://inquiryintoinquiry.com/2020/06/25/peirces-categories-21/

C.S. Peirce and Category Theory
1. https://inquiryintoinquiry.com/2021/06/23/c-s-peirce-and-category-theory-1/
2. https://inquiryintoinquiry.com/2021/06/24/c-s-peirce-and-category-theory-2/
3. https://inquiryintoinquiry.com/2021/06/27/c-s-peirce-and-category-theory-3/
4. https://inquiryintoinquiry.com/2021/06/28/c-s-peirce-and-category-theory-4/
5. https://inquiryintoinquiry.com/2021/06/29/c-s-peirce-and-category-theory-5/
6. https://inquiryintoinquiry.com/2021/06/30/c-s-peirce-and-category-theory-6/
7. https://inquiryintoinquiry.com/2021/07/01/c-s-peirce-and-category-theory-7/
8. https://inquiryintoinquiry.com/2021/07/02/c-s-peirce-and-category-theory-8/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Survey of Precursors Of Category Theory • 5

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackerma…

Inquiry Into Inquiry
Jon AwbreyMar 23, 2024

Survey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/

In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Please follow the above link for the full set of resources.
A few basic articles are linked below.

Relation Theory
• https://oeis.org/wiki/Relation_theory

Relation Composition
• https://oeis.org/wiki/Relation_composition

Relation Construction
• https://oeis.org/wiki/Relation_construction

Relation Reduction
• https://oeis.org/wiki/Relation_reduction

Relative Term
• https://oeis.org/wiki/Relative_term

Sign Relation
• https://oeis.org/wiki/Sign_relation

Triadic Relation
• https://oeis.org/wiki/Triadic_relation

Six Ways of Looking at a Triadic Relation ⌬ 1
• https://inquiryintoinquiry.com/2015/02/04/six-ways-of-looking-at-a-triadic-relation-%E2%8C%AC-1/

Mathematical Demonstration and the Doctrine of Individuals
• https://inquiryintoinquiry.com/2023/05/16/mathematical-demonstration-and-the-doctrine-of-individuals-1-a/
• https://inquiryintoinquiry.com/2023/05/16/mathematical-demonstration-and-the-doctrine-of-individuals-2-a/

Peirce's 1870 “Logic of Relatives” —
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

#Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
#MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
#PredicateCalculus #ContinuousPredicate #HypostaticAbstraction
#RelationComposition #RelationConstruction #RelationReduction

Survey of Relation Theory • 8

In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special at…

Inquiry Into Inquiry
Jon AwbreyAug 1, 2023

Survey of Precursors Of Category Theory • 4
• http://inquiryintoinquiry.com/2023/08/01/survey-of-precursors-of-category-theory-4/

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

Background —

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

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Blog Series —

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

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

Precursors Of Category Theory • Discussion
1. https://inquiryintoinquiry.com/2020/09/13/precursors-of-category-theory-discussion-1/
2. https://inquiryintoinquiry.com/2020/09/21/precursors-of-category-theory-discussion-2/
3. https://inquiryintoinquiry.com/2020/09/25/precursors-of-category-theory-discussion-3/

Categories à la Peirce —

C.S. Peirce • A Guess at the Riddle
• https://inquiryintoinquiry.com/2012/03/21/c-s-peirce-a-guess-at-the-riddle/

Peirce's Categories
1. https://inquiryintoinquiry.com/2015/10/30/peirces-categories-1/
2. https://inquiryintoinquiry.com/2015/10/31/peirces-categories-2/
3. https://inquiryintoinquiry.com/2015/11/04/peirces-categories-3/
•••
19. https://inquiryintoinquiry.com/2020/05/13/peirces-categories-19/
20. https://inquiryintoinquiry.com/2020/05/14/peirces-categories-20/
21. https://inquiryintoinquiry.com/2020/06/25/peirces-categories-21/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Survey of Precursors Of Category Theory • 4

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackerma…

Inquiry Into Inquiry
Show threadJon AwbreyMay 28, 2023

Logic Syllabus • 4
• https://inquiryintoinquiry.com/logic-syllabus/

Relational Concepts
• https://oeis.org/wiki/Logic_Syllabus#Relational_concepts

Continuous Predicate • https://oeis.org/wiki/Continuous_predicate
Hypostatic Abstraction • https://oeis.org/wiki/Hypostatic_abstraction
Logic of Relatives • https://oeis.org/wiki/Logic_of_relatives
Logical Matrix • https://oeis.org/wiki/Logical_matrix
Relation • https://oeis.org/wiki/Relation
Relation Composition • https://oeis.org/wiki/Relation_composition
Relation Construction • https://oeis.org/wiki/Relation_construction
Relation Reduction • https://oeis.org/wiki/Relation_reduction
Relation Theory • https://oeis.org/wiki/Relation_theory
Relative Term • https://oeis.org/wiki/Relative_term
Sign Relation • https://oeis.org/wiki/Sign_relation
Triadic Relation • https://oeis.org/wiki/Triadic_relation

#Logic #LogicSyllabus #ContinuousPredicate #HypostaticAbstraction #LogicOfRelatives
#LogicalMatrix #Relation #RelationComposition #RelationConstruction #RelationReduction
#RelationTheory #RelativeTerm #SignRelation #TriadicRelation

Logic Syllabus

This page serves as a focal node for a collection of related resources. Logical Operators Exclusive Disjunction Logical Implication Logical Conjunction Logical NAND Logical Disjunction Logical NNOR…

Inquiry Into Inquiry
Jon AwbreyApr 16, 2023

Survey of Relation Theory
• https://inquiryintoinquiry.com/2023/04/01/survey-of-relation-theory-6/

In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Please follow the above link for the full set of resources.
A few basic articles are linked below.

Elements —
• Relation Theory ( https://oeis.org/wiki/Relation_theory )

Relational Concepts —
• Relation Construction ( https://oeis.org/wiki/Relation_construction )
• Relation Composition ( https://oeis.org/wiki/Relation_composition )
• Relation Reduction ( https://oeis.org/wiki/Relation_reduction )
• Relative Term ( https://oeis.org/wiki/Relative_term )
• Sign Relation ( https://oeis.org/wiki/Sign_relation )
• Triadic Relation ( https://oeis.org/wiki/Triadic_relation )
• Logic of Relatives ( https://oeis.org/wiki/Logic_of_relatives )
• Hypostatic Abstraction ( https://oeis.org/wiki/Hypostatic_abstraction )
• Continuous Predicate ( https://oeis.org/wiki/Continuous_predicate )

Illustrations —

Six Ways of Looking at a Triadic Relation ⌬ 1
• https://inquiryintoinquiry.com/2015/02/04/six-ways-of-looking-at-a-triadic-relation-%E2%8C%AC-1/

Peirce's 1870 “Logic of Relatives” —

Overview
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

Preliminaries
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

#Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
#MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
#PredicateCalculus #ContinuousPredicate #HypostaticAbstraction
#RelationComposition #RelationConstruction #RelationReduction

Survey of Relation Theory • 6

Inquiry Into Inquiry