Abstract
The aim of the paper is to present the categorical notion of an adjoint functor as a key to formally capturing the philosophical notion of “universal” especially as it figures in relation to semantics and epistemology. In the first part (first section to seventh section) the relevance of category theory for the main topics of analytic philosophy is suggested, in opposition to a widespread conservative attitude towards the entrenched conjunction of logic and ∈-based set theory. In the second part (8th section to 16th section) the concept of an adjunction is introduced and shown to provide the framework of some fundamental examples of universality. The paper is of an introductory character, because it is addressed to a broad philosophical audience, in particular to philosophically-oriented logicians and logically-educated philosophers with no previous knowledge of category theory.
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Notes
- 1.
For any given formalised philosophical thesis, by its “idiosyncratic” character I mean one essentially relative to a chosen formal “base”, thus one that cannot remain stable under change of base.
- 2.
There is already an extensive literature on each of the aspects (a)–(d), but it presupposes familiarity with categorical language. Those who wish to learn more about how the notions introduced here (along with others) have been put to work could begin by consulting the following papers: [3, 10, 13, 24]; in particular, for categorical logic see [5, 6, 31].
- 3.
The phenomenology of these relationship is extensively described in [18].
- 4.
- 5.
Jim Lambek pioneered the investigation of categorial grammar in terms of category theory, see [9]. En passant, a purely lexical note: since the adjective “categorical” was already established in model theory as having a very specific meaning, one might opt for “categorial” in application to category theory. But unfortunately, “categorial” has a no less specific meaning within grammar. In the face of such an impasse, standard usage in the community in question is declared the winner.
- 6.
Readers who know Italian can find these aspects examined in [29].
- 7.
- 8.
These topics are covered by essays collected in [19].
- 9.
“Uniqueness” is always intended as up to isomorphism, in a sense that will be defined in what follows. The fact that a theory allows us to prove the existence only of those x’s that are unique (in satisfying a given condition) marks one of the distinguishing features of the categorical approach.
- 10.
There are also “systemic” approaches to semantics, but so far they are inadequate for the management of logical syntax and call for notable modifications as we move from one semantic field to another. Thus they lack the uniformity found in the categorical approach.
- 11.
The objective conditions for the possibility of exercising any typing activity are usually set aside in the construction of a type theory. Consequently, the entire philosophical discussion on the choice of one type theory rather than another is essentially pre-Kantian.
- 12.
The escape route provided by saying that anything (hence any such added constraint too) can be relationally conceived, though not all at the same time, is still waiting for a consistently relational justification.
- 13.
The uniqueness condition in this definition can be relaxed if one aims at an even more general framework.
- 14.
Passing from a category to its opposite (or “dualising”) highlights the fact that many mathematical notions, including those of logical interest in particular, are the mirror images of others. This provides a theoretical economy: if we have proved a proposition which is supposed to hold in general, its dual holds too (by a dual proof).
- 15.
As in the case of the set of all sets, care is needed with CAT too in order to avoid paradoxes. Such care usually takes the form of principles of size limitation and the most common method consists of dealing with locally small categories, i.e., such that for any two objects the collection of morphisms from one to the other is small (is a set). One might object that, by such a move, category theory demonstrates its conceptual dependence on set theory in a manner which runs counter to the claim that it can be viewed as an alternative to set theory for foundational purposes. There is more than one response to this objection and both the objection and the various responses to it are the subject of much debate. Since this paper was intended to avoid an excursion into such foundational controversies, I shall here simply refer the reader to the works of Colin McLarty for a detailed treatment. See, e.g., [19].
- 16.
In order for the interpretation to be functorial, the syntax too must be categorically reformulated. This step introduces intrinsic constraints on semantics, but it requires further details than those strictly needed for the aim of this paper and therefore will not be examined here.
- 17.
In the Preface to the first edition of [17], p. vii.
- 18.
Usually written “colimit”.
- 19.
The most extensive treatise on topos theory is [8].
- 20.
An adjunction-based approach to the theory of concepts was first explored by Ellerman in [7].
- 21.
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Peruzzi, A. (2016). Category Theory and the Search for Universals: A Very Short Guide for Philosophers. In: Abeles, F., Fuller, M. (eds) Modern Logic 1850-1950, East and West. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-24756-4_8
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