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First there was arithmetic with numerical calculations (i.e., one unknown on one side of an equation). Then algebra with manipulations of variables (many unknowns anywhere in an equation). Then systems are studied that differ from ordinary arithmetic but share some of the same properties (equations where the unknowns represent all sorts of things - even functional equations) and then these properties are abstracted in abstract algebra and whole classes are studied such as groups and rings. Then category theory studies maps between structures (functorial equations), then n-category theory, then ...

Where do we go now? Is category theory the end of the road for the foreseeable future? Is the only way forward to go backwards and generalize in a different direction (like "generalized equations" of optimization or something)?

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    We can also go more general. The question is, how useful will it be?2010-09-03
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    Mathematical logic?2010-09-03
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    Can't this question be seen as equivalent to "What's a more general foundation of mathematics than category theory"? If you buy the idea that category theory is a suitable candidate for a foundation of mathematics, anyway. If so, this should maybe be tagged with foundations.2011-04-24
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    Shouldn't the answer be [higher category theory](http://en.wikipedia.org/wiki/N-category)? Of course, this falls under the classification of *category theory*, but I believe that [historically](http://plato.stanford.edu/entries/category-theory/) it was a significantly later development.2011-12-25

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