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Precursors
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. —JA
Contents
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, 29–30. |
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, 1.1^{a}1–12. 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.
Kant
<math>\ldots</math>
Peirce
Selection 1
§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) C.S. Peirce, “On a New List of Categories” |
Selection 2
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. 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, <math>k</math>-ness may be understood as referring to those properties that all <math>k</math>-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 <math>k</math>-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 <math>k</math>-adic relations can be analyzed in terms of triadic and lower arity relations.
Hilbert
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). Hilbert (1925), “On the Infinite”, pp. 369–392 in Jean van Heijenoort (1967/1977). |
Hilbert and Ackermann
Selection 1
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 <math>(x)(A \rightarrow F(x)).</math> This may be interpreted as a predicate <math>P(A, F)</math> whose first argument place is occupied by a sentence <math>A,</math> and whose second argument place is occupied by a monadic predicate <math>F.</math> A false sentence <math>A</math> is related to every <math>F</math> by the relation <math>P(A, F);</math> a true sentence <math>A</math> only to those <math>F</math> for which <math>(x)F(x)</math> holds. Further examples are given by the properties of reflexivity, symmetry, and transitivity of dyadic predicates. To these correspond three predicates: <math>\operatorname{Ref}(R),</math> <math>\operatorname{Sym}(R),</math> and <math>\operatorname{Tr}(R),</math> whose argument <math>R</math> is a dyadic predicate. These three properties are expressed in symbols as follows:
All three properties are possessed by the predicate <math>\equiv(x, y)</math> (<math>x</math> is identical with <math>y</math>). The predicate <math><(x, y),</math> on the other hand, possesses only the property of transitivity. Thus the formulas <math>\operatorname{Ref}(\equiv),</math> <math>\operatorname{Sym}(\equiv),</math> <math>\operatorname{Tr}(\equiv),</math> and <math>\operatorname{Tr}(<)</math> are true sentences, whereas <math>\operatorname{Ref}(<)</math> and <math>\operatorname{Sym}(<)</math> are false. Such predicates of predicates will be called predicates of second level. (p. 135). |
Selection 2
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). Hilbert and Ackermann, Principles of Mathematical Logic, Robert E. Luce (trans.), Chelsea Publishing Company, New York, NY, 1950. First published, Grundzüge der Theoretischen Logik, 1928. Second edition, 1938. English translation with revisions, corrections, and added notes by Robert E. Luce, 1950. |
References
- Aristotle, “The Categories”, Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
- Aristotle, “Categories”, E.M. Edghill (trans.), eBooks@Adelaide, University of Adelaide, South Australia, 2007. Online.
- van Heijenoort, Jean (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.
- Hilbert, D., and Ackermann, W. (1938/1950), Principles of Mathematical Logic, Robert E. Luce (trans.), Chelsea Publishing Company, New York, NY, 1950. First published, Grundzüge der Theoretischen Logik, 1928. Second edition, 1938. English translation with revisions, corrections, and added notes by Robert E. Luce, 1950.
- Kant
- C.S. Peirce, “On a New List of Categories”, Online.
- Carnap, The Logical Syntax of Language, vide “Functor”