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Having only the a very cursory knowledge of Structuralism, there does appear to be some points of coincidence:

Structuralism:

  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 (called 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. I'm thinking of Leibniz rather than Kant, from whom Saunders MacLane, one of the two cofounders of Category Theory, purloined the word 'Category' for his own uses. He also studied at Gottingen, which from my limited knowledge of German philosophical history, was a centre of philosophy, presumably due to Kant.

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.

  • Minor note: Kant was based in Königsberg (today Kaliningrad, Russia) and not in Gottingen. – Artem Kaznatcheev Jun 13 '12 at 18:12
  • I've always understood him to be a German philosopher, does this mean he had Slavic roots? – Mozibur Ullah Jun 14 '12 at 3:00
  • "Another element relevant to this discussion would be Set Theory, its relation with Structuralism and Category Theory, as the definitions of Structure and Category also resemble the definition of a Set." ‌–Tames – stoicfury Jun 18 '12 at 2:57
  • @stoicfury: Can you expand on that, as that seems to me to be taking this question in a different but interesting direction: Category Theory, in its incarnation of Topos Theory, was proposed as an alternative foundation to the usual Set Theory, and it also destabalises the notion of a universal & unique Set Theory (one of the motivating drives for formalising the theory) as Toposes come in very different shapes & sizez. One way of looking at Category Theory vs Set Theory, is that it eliminates (the primary) membership relation in favour of the (derivative) functional relationship. – Mozibur Ullah Jun 18 '12 at 3:12
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    I'm a she! what can I do for you @MoziburUllah? I do not have much knowledge to answer this question, that's why I was hoping someone did. I expected something more theory oriented than historical (who said what first), although it is a useful information. Possibly all this matters relate to formal thought, as relations are more important than content. – Tames Jun 18 '12 at 21:30
3

Consult this paper by Andrei Rodin for an interpretation of category theory without structures.

His central claim (from the abstract) is that while structuralism in the philosophy of mathematics studies "invariant form" (for instance, the sentences of a categorical theory are invariant across isomorphic models), categorical mathematics studies covariant and contravariant transformations, which in general have no such invariants.

If his reading is plausible, it provides reason to think Category Theory needn't be wed to structuralism. If his stronger claim is correct, then Category Theory shouldn't be wed to structuralism.

I'm not sure that connecting category theory to structuralism in the humanities is of much interest. The similarities, though they are present, are at a vague and broad enough level to be virtually meaningless.

  • His point of departure may be Maclanes comment that they invented category theory not to study categories (aka structures) but natural transformations. I found his terminology a little confusing since at one point he calls a functor a transformation. No, I don't think there is much interest in connecting the two, but I just wanted point out that there was some similarities - vague as though it is. What would interest me is to see how Structuralism has been applied in the humanities. – Mozibur Ullah Apr 13 '13 at 19:15
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    He also seems to be gesturing towards Topos theory - although he doesn't mention the name. The cue comes from his starting point the category of all categories which is a topos. He then develops Catgegory theory internally by specifying that this topos has certain features like a generator which will allow him to specify morphisms which in this context will be functors. – Mozibur Ullah Apr 13 '13 at 19:21
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    "What would interest me is to see how Structuralism has been applied in the humanities." Well then I think it's best to distance it from these views about the philosophy of mathematics and even mathematics itself and just study the relevant literature directly. – Dennis Apr 13 '13 at 19:25
  • I'm not also sure about his starting point. He quotes Awodey in 'the subject matter of pure mathematics is invariant form...not logical atoms'. Obviously this is robust polemic for mathematical structuralism as against Set Theory (personally I'm agnostic about the IS - I see each as a useful organising perspective). But although its true that useful categories are found by abstracting structure from certain mathematical theories. For example the theory of groups to the category of groups, where objects are abstract groups and morphisms are group homomorphisms. – Mozibur Ullah Apr 13 '13 at 19:41
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    Yes. I've read the article. I was hoping for some illuminating examples of the theory in practise for the layman, without having to having to read some books. Perhaps more focussed questions might be appropriate like how was structuralism is useful in architecture etc. – Mozibur Ullah Apr 13 '13 at 20:32
1

A letter by Jean-Michel Kantor, Bourbakis Structure and Structuralism at springerlink.com/content/x60030547jl61071/that throws a great deal of light on this:

he says,'When I asked Claude Levi-Strauss about the origin of the word ‘‘structure’’ in his work, he answered (letter to the author, Nov. 16, 1990): ‘‘Ne croyez pas un instant que Bourbaki 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.'' '

Bourbaki was a society of mathematicians that were intent on putting the whole of mathematics on a rigourous foundation. The idea of Structure pervaded their thinking, in particular the idea of homomorphism=structure preserving map.

It looks like that the idea of Structure becoming pervasive and prominent in two unrelated fields at roughly the same time can be put down to the workings of the Zeitgeist.

  • At the same time the French are very keen on mathematics (I know I did my baccaleaureat there), and the growth of structuralims in mathematics cannot have not been noticed by philosophers there... – Henry Story Dec 23 '18 at 9:05
0

I think there is a "philosophy of abstract algebra," or "structuralist philosophy," if you prefer, that guides us in asking the right questions. It says, "It's not about what a real number is, its about how real numbers relate to one another; it's about plus, times, less-than, etc."

I think category theory takes this philosophy to its logical conclusion. Let me illustrate.

If we have a metric space, we can ask, "What is an open set?" A good answer would be, "The open sets are precisely those whose every point has enough space for a little ball around it." From this definition, we can prove some facts about open sets, such as: their union is an open set, finite intersections are open sets, and many other things, too!

After a while of proving things, you realize that many of the interesting statements about open sets follow from just two observations, namely that their union is an open set, and that finite intersections are open sets.

So you have an epiphany. You realize: It's not about what an open set is. Its about how they relate. So you define a topological space. You tell the reader what a topological space is, rather than telling them what an open set is. This is the philosophy of algebra in action.

After a while, you start thinking about continuous maps between topological spaces, homeomorphisms, Cartesian products, etc. And at some point, you have another epiphany. You realize: it's not about what a topological space is. It's about how they relate. So you define the category of topological spaces.

In conclusion, I think category theory takes the philosophy of abstract algebra (or "structuralism", if you prefer) to its logical conclusion.

  • I think you're right. Category theory is a codification of structuralist ideas in abstract algebra. – Mozibur Ullah Apr 5 '13 at 14:04
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    @MoziburUllah: 'is' is a bit strong. CT may -look- like such a codification, and your notice of the similarities is compelling, but if anything the linguists of the 50's and 60's (and so then, by Levi-Strauss, the anthropologists) borrowed the term superficially to label their movement. What else is there to math but structure. Structuralism as a philosophical/cultural movement is only one small part of the humanities. – Mitch Apr 6 '13 at 1:32
  • @Mitch: When I say structuralist in the sentence above I'm not referring to saussures ideas; I just mean that that the categorical ideas of structure (and this here is a very specific technical term - for example morphism, functor, and natural transformation) were a crystallisation of similar ideas in abstract algebra where they were implicit. There is more to mathematics than structure; but I understand the angle you're coming from, and again the structure I'm referring to is only a small corner of the vast range of structures that mathematics deals with. – Mozibur Ullah Apr 6 '13 at 17:28
  • @Mitch: My superficial understanding of structuralism in the humanities leads me to think that they are in fact quite different. I'd be interested to see more examples of where & how structuralism has been used there. – Mozibur Ullah Apr 6 '13 at 17:29
  • @MoziburUllah for humanities see simply en.wikipedia.org/wiki/Structuralism. For math, 'structure' is not a particularly technical term (unlike, group, set, field, etc). In the crossover field of philosophy of mathematics, structuralism has little to do, other than superficially, with either the structuralism of humanities or category theory (meaning there is little literature connecting them). Of course you may have found an interesting unknown connection. – Mitch Apr 6 '13 at 17:52
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In my knowledge of mathematics it is rather difficult to find something that is not a Category. It will not surprise me that Structuralism is one. In other words: to say of Structuralism that it is a Category is to say very little.

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