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'''Note.''' I was going to go with "The Fruit Of Our Purloins", but I reckon this is more succinct. The way that I normally start an inquiry like this is just to collect a sample of source materials that seem like they belong together. Still traveling, so this will be sporadic at first. —[[User:Jon Awbrey|JA]]
'''Note.''' I was going to go with ''The Fruit Of Our Purloins'', but I reckon this is more succinct. The way that I normally start an inquiry like this is just to collect a sample of source materials that seem like they belong together. Still traveling, so this will be sporadic at first. —[[User:Jon Awbrey|JA]]
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<p>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.</p>
<p>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 (<i>Logische Syntax der Sprache</i>), and "natural transformation" from then current informal parlance.</p>
<p>Saunders Mac Lane, ''Categories for the Working Mathematician'', 29–30.</p>
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==Aristotle==
===Selection 1===
# Aristotle #
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<p>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.</p>
<p>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.</p>
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<p>Aristotle, <i>Categories</i>, 1.1<sup>a</sup>1–12.</p>
<p>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 <i>animals</i> (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.</p>
<p>'''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).</p>
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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.
==Kant==
<p><b>Translator's Note.</b> “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).</p>
<math>\ldots</math>
==Peirce==
===Selection 1===
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===Selection 2===
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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.
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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==Hilbert==
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==Hilbert and Ackermann==
===Selection 1===
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===Selection 2===
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==References==
<pre>
* Aristotle, "The Categories", Harold P. Cooke (trans.), pp. 1–109 in _Aristotle, Volume 1_, Loeb Classical Library, William Heinemann, London, UK, 1938.
* Aristotle, "The Categories", Harold P. Cooke (trans.), pp. 1–109 in _Aristotle, Volume 1_, Loeb Classical Library, William Heinemann, London, UK, 1938.
* Carnap, _[The Logical Syntax of Language](http://books.google.com/books?id=Yf9R6WFFLhYC&printsec=frontcover)_, _cf._ “[Functor](http://books.google.com/books?id=Yf9R6WFFLhYC&printsec=frontcover#v=onepage&q=Functor&f=false)”
* Carnap, _[The Logical Syntax of Language](http://books.google.com/books?id=Yf9R6WFFLhYC&printsec=frontcover)_, _cf._ “[Functor](http://books.google.com/books?id=Yf9R6WFFLhYC&printsec=frontcover#v=onepage&q=Functor&f=false)”
</pre>
==Related Topics==
* [[continuous predicate]]
* [[continuous predicate]]
* [[hypostatic abstraction]]
* [[hypostatic abstraction]]
==Discussion==
<pre>
_The following discussion took place at [[category theory]], when the subject of this page had appeared as a brief section in stub form._
_The following discussion took place at [[category theory]], when the subject of this page had appeared as a brief section in stub form._
