Re: Precursors of Category Theory

Post   : Precursors of Category Theory : 3
URL    : http://inquiryintoinquiry.com/2014/01/03/precursors-of-category-theory-3/
Posted : January 3, 2014 at 12:00 pm
Author : Jon Awbrey

Cf: Precursors of Category Theory
At: http://intersci.ss.uci.edu/wiki/index.php/Precursors_of_Category_Theory

Peirce
-------

Cued by Kant’s idea on the function of concepts in general, Peirce locates his categories on the 
highest level of abstraction that affords a meaningful measure of traction in practice, reserving 
judgment on the absolute unity of perfect ambiguity and numerous dualisms that taken together may 
well converge on the same conception as Peirce’s trinity.

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)

</quote>

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

Selection 2
------------

<quote>

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.

</quote>

C.S. Peirce, CP 4.549, “Prolegomena to an Apology for Pragmaticism”, The Monist 16, 492–546 (1906), 
CP 4.530–572.

The first thing to extract from this passage is the fact that Peirce's Categories, or 
“Predicaments”, are predicates of predicates.  Considerations like these 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 
that 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, respectively.  Taken in full generality, k-ness may be 
understood as referring to those properties that all k-adic relations have in common.  Peirce's 
distinctive claim is that a type hierarchy of three levels is generative of all that we need in logic.

Part of the justification for Peirce's claim that three categories are both 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.

-- 

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