philosophy

# Contents

## Idea

These many different things stand in essential reciprocal action via their properties; the property is this reciprocal relation itself and apart from it the thing is nothing; (WdL §1065a)

In philosophy, structuralism is the point of view which emphasises that the entities considered are parts of a system and that their very meaning and identity is defined according to its relation to the rest of the system. For instance, if one part of the system changes in time, all other parts change as well. The system has features which are not just the composition of features of its constituent parts.

### In the humanities

In the 20th century, structuralism in the humanities is associated with Emile Durkheim and Georg Simmel in sociology, Ferdinand de Saussure (and later Roman Jakobson) in linguistics, and Claude Lévi-Strauss in anthropology.

### In philosophy of mathematics

Structuralism is also the term given to a position in the philosophy of mathematics which holds that the entities treated in mathematics are simply structures. It is associated also with structural realism in the philosophy of science, which maintains either that all we can know of the world is its structure (epistemological structural realism) or that indeed the world just is structure (ontological structural realism).

One may argue that category theory and type theory serve as formalization of these intuitive notions, see below.

## Formalization in category theory and type theory

Since in a category, the nature of any object is determined only by the system of morphisms which relate this object to the other objects of the category, and not by how the object itself is presented (if it is at all), it may be argued that category theory provides a formalization of philosophical structuralism, at least in the philosophy of mathematics (Awodey 96, Awodey 03). [Yuri Manin]] says (citation?) that category theory regards objects as part of a “society”. (See also at objective logic for more on categorical logic and philosophical notions.)

Notice that the terminology morphism of course originates in homomorphism, since the morphisms in a category of mathematical structures (such as groups, modules, etc.) are precisely the maps of the underlying sets that preserve this structure.

This is essentially Bourbaki’s emphasis on mathematics as the science of abstract structures, where structures are important only up to isomorphism and the emphasis is on relations (including) functions which are part of the “structure” (e.g. Kantor 11).

However, a category need not be such a concrete category of sets with structure. Even when there is no such explicit structure on the objects, the same category-theoretic reasoning still applies and one may detect structure on objects from the morphisms between them.

For instance, while in general an object $X$ in a category does not have an underlying set, if the category has a terminal object $\ast$ one may still find a set of global elements of $X$ as the set of morphisms of the form $\ast \to X$.

Accordingly, in a category one only talks about properties invariant under isomorphisms; from such a point of view an object is determined by the morphisms to or from other objects.

This perspective of the nature of objects of a category being determined only by the morphisms between them is fully embodied by the concept of equivalence of categories: two categories are to be regarded as equivalent if there is a functor between them that is essentially surjective in that it does not omit any isomorphism class of objects and which otherwise is a bijection on morphisms (precisely: on the hom-sets between any ordered pair of objects, a fully faithful functor).

Given that type theories of sorts are the internal language of suitable kinds of categories, with morphisms now being terms of function types (see at relation between category theory and type theory), these comments also give that type theory offers some kind of formalization of structuralism (Awodey 15, Corfield 15).

Notably in homotopy type theory with univalence, which is the internal language of categories called (infinity,1)-toposes, the type universe reflects all the isomorphisms (equivalences) in the category in its identity type, which therefore may be thought of as providing a structuralist concept of identity.

(Notice however that the categories of mathematical structures such as groups, modules, etc., even when regarded as (infinity,1)-categories, i.e. of infinity-groups, (infinity,n)-modules etc.) are rarely (infinity,1)-toposes. The “structure” carried by an object in an $(\infty,1)$-topos is generically more a kind of geometric structure.

## References

• wikipedia structuralism

• Steve Awodey, Structure in mathematics and logic: a categorical perspective, Philosophia Mathematica (3), vol. 4, p. 209-237 (1996)

• Steve Awodey, An Answer to Hellman’s Question: “Does Category Theory Provide a Foundation for Mathematical Structuralism?, 2003 (pdf)

• Colin McLarty, 2004, Exploring Categorical Structuralism, Philosophia Mathematica, 12, 37–53 (pdf)

• Jean-Michel Kantor, Bourbaki’s Structures and Structuralism, The Mathematical Intelligencer 33:1 (2011), 1, doi

• Steve Awodey, Structuralism, Invariance and Univalence, 2014 (pdf)

• David Corfield, Expressing ‘The Structure of’ in Homotopy Type Theory, 2015 (pdf)

category: philosophy

Revised on March 10, 2016 05:26:18 by Urs Schreiber (31.55.28.91)