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philosophy (Rev #11) in nLab

Philosophy

Idea

Philosophical interest in n-categories may be characterised as belonging to one of two kinds.

  • Metaphysical: A new language which may prove to be as important for philosophy as predicate logic has been for Bertrand Russell and successive analytic philosophers.

  • Illustrative of mathematics as intellectual enquiry: Such a reconstitution of the fundamental language of mathematics reveals much about mathematics as a tradition of enquiry stretching back several millennia, for instance, the continued willingness to reconsider basic concepts.

  • Higher category theory refines the notion of sameness to allow more subtle variants.

  • There ought to be a categorified logic, or 2-logic. There are some suggestions that existing work on modal logic is relevant. Blog discussion: I, II, III, IV. Mike Shulman’s project: 2-categorical logic.

  • Higher category theory may provide the right tools to take physics forward. A Prehistory of n-Categorical Physics See also physics.

  • More speculatively, category theory may prove useful in biology. Blog post

Urs: Maybe there is an intermediate step which is less speculative? My impression is that the idea here is something like: noticing that category theory at first is a formalism of states and processes (directed arrows) and nn-category theory of processes of processes, etc., can we also naturally encode in its language structures of structures, i.e. hierarchical structures, which do not naturally or not manifestly have an interpretation as processes, in particular in that they are lacking the directionality of processes? Whatever the definition of hyperstructure really will be in the end, I think this question is what motivates them: a hyperstructure differes from an ∞\infty-category in that in degree nn it has cells ( bonds ) which bind (n−1)(n-1)-cells, but there is no directionality imposed on this, and not necessarily a notion of composition.

Now, biological structures are often of the complex hierarchical structure that one would imagine the concept of hyperstructure would describe to some extent, but if the notion of hyperstructure is good and natural, that should be just a very specific of a more general kind of applications which maybe should not be regarded as the archetypical application of the concept as such. In this respect it is maybe noteworthy that the idea of hyperstructure does not originate in a motivation from biologogy, but was originally conceived as a means to formalize extended cobordisms such as appear in the generalized tangle hypothesis.

David: It would be great to get some kind of grip on the range of formalisms out there used to model complex systems. As soon as you start to dig, you get swamped by an avalanche of ideas. E.g., look up Milner’s bigraphs and you’re directed to a massive bibliography. I’d like to see a very concrete piece of biology being well-modelled. Perhaps this.

Enquiry

“Mathematical wisdom, if not forgotten, lives as an invariant of all its (re)presentations in a permanently self–renewing discourse.” (Yuri Manin)

To categorify mathematical constructions properly, one must have understood their essential features. This leads us to consider what it is to get concepts ‘right’. Which kind of ‘realism’ is suitable for mathematics? Which virtues should a mathematical community possess to further its ends: a knowledge of its history, close attention to instruction and the sharing of knowledge, a willingness to admit to what is currently lacking in its programmes?

The Dangers of Category Theory

Gavin: This is a sensitive topic. I do not know if this is the right place for it, but I feel that nLab would be lacking balance if it did not have some part that attempted to address the fact Category Theory has been responsible for some pretty wretched mathematics. To go into too much detail might be discourteous. I was delighted by the beauty of category theory when I first met it (in the library of the Courant Institute in 1962, when I was supposed to be writing a thesis in Theoretical Physics). However, I soon realized that in the mathematical community of the UK, at least, at that time, too great an indulgence in abstraction was viewed with suspicion. It is a suspicion that I share, to some extent. I remember coming across an article, some years later, in a Physics journal, written by a well-respected physicist, in which large chunks of a paper by Applegate on the Tierney Tower of adjoint functors were repeated verbatim; and evidently with the corrupt intention of combombulating readers. Category theory is easily misused, either for unnecessary flights into the Empyrean, or for more evil El-Naschiean purposes. But I guess that most nLabians are perfectly aware of all this. Since those early days category theory has become a lingua franca for almost every branch of mathematics. I too love abstraction; but I am just a little bit wary of it.

Blog Discussion

  • Mathematical reality: I, II

Revision on July 1, 2009 at 13:49:10 by GavinWraith See the history of this page for a list of all contributions to it.