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Having only the a very cursory knowledge of Structuralism ( it's a movement generally held to have originated in linguistics, then moving on to philosophy & literature), there does appear to be some points of coincidence:

Structuralism (from wikipedia):

  1. Individual elements of culture must be placed within a System/Structure.
  2. The individual elements of culture must be understood by their inter-relationships within this System, and not by their individual identity, that is their identity is supressed.

and compare this with Category Theory:

  1. Individual objects of mathematical interest must be placed with a Category.
  2. Though these individual objects have their own character, this knowledge is supressed, and only their relationships (morphisms) have import.

There seems to me a clear correspondance here. Of course, it could mean that both paradigms evolved independently from some prior philosophy.

Some more evidence from Structuralism, by Sturrock:

'What is a structure, then, for Husserl, and 'in general'? The broadest definition is that a structure is an abstract model of organisation including a set of elements and the law of their composition...What stands out in a structure is that the relationships between the elements are more important than the intrinsic qualities of each element'.

and the definition of a category can be further elaborated as:

3.Morphisms between objects (i.e. the relationship) follow a law of composition.

Further, if I recall correctly Saunders MaClane remarks in the introduction to Categories for the Working Mathematician, that he purloined the term Category from Kant. I don't think, though, that Kant had any input or influence on Structuralism.

[I've asked this question on Philosophy.StackExchange but have no response from them]

Mozibur Ullah
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  • If there was such an influence, it was by way of Bourbaki. So the question can probably be reduced to how structuralism had an influence on Bourbaki. – Michael Greinecker Jun 17 '12 at 16:23
  • Structuralism in mathematics predates category theory. It is already evident in the abstraction of algebraic structures (groups, rings, fields, etc), e.g. see the last two paragraphs here. Such abstractions were motivated purely internally in mathematics. If there is any external influence then it is almost certainly in the other direction, i.e. from math to other fields. – Bill Dubuque Jun 17 '12 at 17:03
  • @Dubuque: Levi-Strauss, an advocate of Structuralism, asked I think Cartan to apply Group Theory to a problem in Kinship Selection. – Mozibur Ullah Jun 17 '12 at 20:17
  • @Greinecker: Theres a letter by Jean-Michel Kantor, Bourbakis Structure and Structuralism at http://www.springerlink.com/content/x60030547jl61071/that throws a great deal of light on this: – Mozibur Ullah Jun 17 '12 at 20:28
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    he says,'When I asked Claude Levi-Strauss about the origin of theword ‘‘structure’’ in his work, he answered (letter to theauthor, Nov. 16, 1990): ‘‘Ne croyez pas un instant queBourbaki m’ait emprunte´ le terme ‘‘ structure’’ ou le contraire, il me vient de la linguistique et plus pre´cise´ment de l’Ecole de Prague.’’ (Do not believe for one minute that Bourbaki borrowed the word ‘‘structure’’ from me, or the contrary; it came to me from linguistics, more precisely, from the School of Prague.'' ' – Mozibur Ullah Jun 17 '12 at 20:29

2 Answers2

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There is a link, but it is not as straightforward as what you describe. Insofar as there has been an influence at all between structuralism in the human and social sciences and the pre-eminence given to the study of structures in mathematics, this influence has predominantly been between the pre-categorical point of view on structures in mathematics and structuralism in the social sciences. Moreover, this influence, if any at all, has been of mathematics on social sciences more than of social sciences on mathematics, as the title of your question might suggest.

To put it more concretely, people like Levi-Strauss, Lacan, Foucault and especially Piaget (to take some emblematic examples of structuralism in anthropology, psycholoy, sociology and epistemology) might have been influenced by their proximity-social and intellectual-with the Bourbaki group and might have thus adapted part of their intellectual framework from the structuralist point of view of Bourbaki, in the sense of the slogan of Dieudonné that "each structures carries with it its own language". I write might because, with the exception of Piaget, it is unclear how much real influence there has actually been and whether this influence amounted to much more than similar choices in some terminological terms. It is very unlikely that there has been any influence in the other direction, and in particular, there is no evidence that I know of that there has been any influence from the social sciences in the foundation of the theory of categories.

Interestingly, apart from the case of Piaget, who really did organize his theory of learning in analogy with the logical organization of mathematics in the Bourbaki style, the deepest and most significant link that I can think of between structuralism in mathematics and in the social science is about Foucault, who was a student of Canguilhem, himself a close friend of Cavaillès, who was very influential (at the time, that it is to say the 1920s/1930s) in mathematics and philosophy at École Normale Supérieure.

Olivier
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You might find the article on mathematical structuralism at the internet encyclopedia of philosophy helpful.

Cameron Smith
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