Re: Precursors of Category Theory

Post   : Precursors of Category Theory : 2
URL    : http://inquiryintoinquiry.com/2013/12/30/precursors-of-category-theory-2/
Posted : December 30, 2013 at 1:00 am
Author : Jon Awbrey

[[ 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 (Karpeles 2008)

Peircers,

When it comes to looking for the continuities of the category concept across different systems and 
systematizers, we don't expect to find them 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’.” ]] (H.P. Cooke).

In the logic of Aristotle categories are adjuncts to reasoning that are designed to resolve 
ambiguities and thus to prepare equivocal signs, that are 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 we need them to control generalizations, to reign 
in abstractions and analogies that are 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.

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