Precursors Of Category Theory

[Jon Awbrey]

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

All,

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/

Regards,

Jon

cc: https://www.academia.edu/community/VWY2Qq
cc: https://mathstodon.xyz/@Inquiry/112502692087155139

[Jon Awbrey]

Precursors Of Category Theory • 2
• 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

All,

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 —

❝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’.❞

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/

Regards,

Jon

cc: https://www.academia.edu/community/5wbQ91
cc: https://mathstodon.xyz/@Inquiry/112502692087155139

[Jon Awbrey]

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/

Regards,

Jon

cc: https://www.academia.edu/community/LGJvkW
cc: https://mathstodon.xyz/@Inquiry/112502692087155139

[Jon Awbrey]

Precursors Of Category Theory • 4
• https://inquiryintoinquiry.com/2024/05/28/precursors-of-category-theory-4-a/

C.S. Peirce • “Prolegomena to an Apology for Pragmaticism” (1906)

❝I will now say a few words about what you have called Categories,
but for which I prefer the designation Predicaments, and which
you have explained as predicates of predicates.

❝That wonderful operation of hypostatic abstraction by which
we seem to create “entia rationis” that are, nevertheless,
sometimes real, furnishes us the means of turning predicates
from being signs that we think or think “through”, into being
subjects thought of. We thus think of the thought‑sign itself,
making it the object of another thought‑sign.

❝Thereupon, we can repeat the operation of hypostatic abstraction,
and from these second intentions derive third intentions. Does
this series proceed endlessly? I think not. What then are the
characters of its different members?

❝My thoughts on this subject are not yet harvested. I will only say
that the subject concerns Logic, but that the divisions so obtained
must not be confounded with the different Modes of Being: Actuality,
Possibility, Destiny (or Freedom from Destiny).

❝On the contrary, the succession of Predicates of Predicates is
different in the different Modes of Being. Meantime, it will be
proper that in our system of diagrammatization we should provide
for the division, whenever needed, of each of our three Universes
of modes of reality into “Realms” for the different Predicaments.❞
(CP 4.549).

The first thing to extract from the above passage is that Peirce's
Categories, for which he uses the technical term “Predicaments”, are
predicates of predicates. Considerations of the order Peirce undertakes
tend to generate hierarchies of subject matters, extending through what
is traditionally called the “logic of second intentions”, or what is
handled very roughly by “second order logic” in contemporary parlance,
and continuing onward through higher intentions, or higher order logic
and type theory.

Peirce arrived at his own system of three categories after a thoroughgoing
study of his predecessors, with special reference to the categories of
Aristotle, Kant, and Hegel. The names he used for his own categories
varied with context and occasion, but ranged from moderately intuitive
terms like “quality”, “reaction”, and “symbolization” to maximally
abstract terms like “firstness”, “secondness”, and “thirdness”.

Taken in full generality, k‑ness may be understood as referring to those
properties all k‑adic relations have in common. Peirce's distinctive claim
is that a type hierarchy of three levels is generative of all we need in logic.

Part of the justification for Peirce's claim that three categories
are necessary and sufficient appears to arise from mathematical facts
about the reducibility of k‑adic relations. With regard to necessity,
triadic relations cannot be completely analyzed in terms or monadic and
dyadic predicates. With regard to sufficiency, all higher arity k‑adic
relations can be analyzed in terms of triadic and lower arity relations.

Reference —

• Peirce, C.S. (1906), “Prolegomena to an Apology for Pragmaticism”,
The Monist 16, 492–546, CP 4.530–572.

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/

Regards,

Jon

cc: https://www.academia.edu/community/LY0pam
cc: https://mathstodon.xyz/@Inquiry/112502692087155139

[Jon Awbrey]

Precursors Of Category Theory • 5
• https://inquiryintoinquiry.com/2024/05/29/precursors-of-category-theory-5-a/

❝A demonstration rests in a finite number of steps.❞
— G. Spencer Brown • Laws of Form

David Hilbert • “On the Infinite” (1925)

❝Finally, let us recall our real subject and, so far as the
infinite is concerned, draw the balance of all our reflections.
The final result then is: nowhere is the infinite realized;
it is neither present in nature nor admissible as a foundation
in our rational thinking — a remarkable harmony between being
and thought. We gain a conviction that runs counter to the
earlier endeavors of Frege and Dedekind, the conviction that,
if scientific knowledge is to be possible, certain intuitive
conceptions [Vorstellungen] and insights are indispensable;
logic alone does not suffice. The right to operate with the
infinite can be secured only by means of the finite.

❝The role that remains to the infinite is, rather, merely that
of an idea — if, in accordance with Kant’s words, we understand
by an idea a concept of reason that transcends all experience and
through which the concrete is completed so as to form a totality —
an idea, moreover, in which we may have unhesitating confidence
within the framework furnished by the theory that I have sketched
and advocated here.❞ (p. 392).

References —

• Hilbert, D. (1925), “On the Infinite”, pp. 369–392
in Jean van Heijenoort (1967/1977).

• van Heijenoort, J. (1967/1977), From Frege to Gödel :
A Source Book in Mathematical Logic, 1879–1931, Harvard
University Press, Cambridge, MA, 1967. 2nd printing, 1972.
3rd printing, 1977.

• Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin,
London, p. 54.

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/
Regards,

Jon

cc: https://www.academia.edu/community/VrKAKZ
cc: https://mathstodon.xyz/@Inquiry/112502692087155139

[Jon Awbrey]

Precursors Of Category Theory • 6
• https://inquiryintoinquiry.com/2024/05/30/precursors-of-category-theory-6-a/

Hilbert and Ackermann • Principles of Mathematical Logic (1928)

❝For the intuitive interpretation on which we have hitherto
based the predicate calculus, it was essential that the
sentences and predicates should be sharply differentiated
from the individuals, which occur as the argument values
of the predicates. Now, however, there is nothing to prevent
us from “considering the predicates and sentences themselves
as individuals which may serve as arguments of predicates”.

❝Consider, for example, a logical expression of the form (x)(A → F(x)).
This may be interpreted as a predicate P(A, F) whose first argument
place is occupied by a sentence A, and whose second argument place
is occupied by a monadic predicate F.

❝A false sentence A is related to every F by the relation P(A, F);
a true sentence A only to those F for which (x)F(x) holds.

❝Further examples are given by the properties of “reflexivity”,
“symmetry”, and “transitivity” of dyadic predicates. To these
correspond three predicates: Ref(R), Sym(R), and Tr(R), whose
argument R is a dyadic predicate. These three properties are
expressed in symbols as follows:

• Ref(R) : (x)R(x, x),

• Sym(R) : (x)(y)(R(x, y) → R(y, x)),

• Tr(R) : (x)(y)(z)(R(x, y) & R(y, z) → R(x, z)).

❝All three properties are possessed by the predicate ≡(x, y)
(x is identical with y). The predicate <(x, y), on the other
hand, possesses only the property of transitivity. Thus the
formulas Ref(≡), Sym(≡), Tr(≡), and Tr(<) are true sentences,
whereas Ref(<) and Sym(<) are false.

❝Such “predicates of predicates” will be called
“predicates of second level”.❞ (p. 135).

❝We have, first, predicates of individuals, and these are classified
into predicates of different categories, or types, according to
the number of their argument places. Such predicates are called
“predicates of first level”.

❝By a “predicate of second level”, we understand one whose
argument places are occupied by names of individuals or by
predicates of first level, where a predicate of first level
must occur at least once as an argument. The categories, or
types, of predicates second level are differentiated according
to the number and kind of their argument places.❞ (p. 152).

Reference —

• Hilbert, D. and Ackermann, W., Principles of Mathematical Logic,
Robert E. Luce (trans.), Chelsea Publishing Company, New York, 1950.
1st published, Grundzüge der Theoretischen Logik, 1928. 2nd edition,
1938. English translation with revisions, corrections, and added notes
by Robert E. Luce, 1950.

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/

Regards,

Jon

cc: https://www.academia.edu/community/Lgrj0b
cc: https://mathstodon.xyz/@Inquiry/112502692087155139