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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? –  Casebash Sep 3 '10 at 9:48
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Mathematical logic? –  Martin Brandenburg Sep 3 '10 at 11:59
    
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. –  Robin Green Apr 24 '11 at 16:01
    
Shouldn't the answer be higher category theory? Of course, this falls under the classification of category theory, but I believe that historically it was a significantly later development. –  Peter Shor Dec 25 '11 at 14:09

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I don't really see a coherent logical progression in the branches of mathematics you're putting forward. Mathematics isn't just about abstraction and generalizing, making things more and more general. It's most often about solving particular problems. Category theory was born out of algebraic topology, in many ways as a notational convenience, to make sense of the tremendous and difficult-to-follow messes algebraic topologists were producing.

If you've ever programmed in a language like "C" you know the concept of a "macro". This is an idea that is re-usable in many different contexts. You plug in different objects and the macro continues to make sense. That's much of the point of category theory, as there are so many ideas that are duplicated over and over again in mathematics that it's confusing to give them special names. So we call these ideas by generic names that make sense in a wide-array of contexts, like "the co-product (or whatever) in the category C (name your category)", etc. It saves time and energy. Moreover, once you've reduced the "bulk" of your notation sufficiently, there is a phenomena where the concepts are lighter and easier to play with. So by using category theory you sometimes "lighten the load" a little, making other discoveries perhaps a little easier (if you're lucky).

In that regard what gets called "category theory" I think of as more of an attempt to find the natural language for certain types of ideas. The general idea being that certain types of problems become easy when using appropriate notation. Not all, but some. Some problems are just hard -- like the Poincare conjecture, or the Schoenflies problem, the classification of finite-simple groups, or existence and uniqueness of solutions to Navier-Stokes (and if you look at the work that's been done on these problems you will see almost no category theory at all, just a tiny little bit on the high-dimensional Poincare and Schoenflies problems). In programming category theory might be analogous to the study of data types, and how one structures memory efficiently.

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While I completely concur with the thrust of your post, the last paragraph raises some doubts. There is no doubt in my mind that category theory occupies a central place in mathematics because it structures patterns and ideas common to many parts of it. But on the other hand, category theory is itself a field of mathematics, much like group theory or differential geometry are, with its own problems and techniques. In that sense, it is as much a "natural language" as group theory or differential geometry are "natural languages". –  G. Rodrigues Sep 3 '10 at 16:41
    
If you measure a fields connections to other fields by citation rates, then Mathematical physics, PDEs, differential geometry, geometric topology, algebraic geometry, representation theory, combinatorics and probability would perhaps be the best-qualified to be called central. Click on the link Andy supplies in this thread: mathoverflow.net/questions/2259/… but perhaps I'm not sure what you mean by "central". –  Ryan Budney Sep 3 '10 at 18:07

Generalization goes where problems lead it to.

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A minor thought about directions beyond category theory as we know it would be that in its present form it is hard to model probabilistic, or optimisation theoretic problems, yet there seem to be some instances for instance in modelling networks, where there are both elementary category theoretic ideas and 'optimisational' approaches needed but they, as yet, interact badly. Some of the discussions on the n-Cat café have gone slightly in that direction. The problems are there but the ideas on how to go into that area are probably not yet 'ripe'.

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Robin Green is right, I think, in saying that "this should ... be tagged with foundations".

Tim Porter hit on an essential part of the equation for new discovery in the sciences, maths included: "ripeness" or "readiness". As a young child passes through the various necessary developmental stages, so it builds a repertoire of active concepts that both inform and manage its interactions with the external world. At some stage it begins producing language, not just hearing it. Once it's done that, the child can proceed to holding conversations, to understanding stories that involve dialogue, to reading, and to writing. But each stage is necessary to the subsequent ones; not even a "genius" (*) can completely skip all the intervening stages to pass from hearing language directly to writing it.


(*) A digression on what we mean by "genius": In fact, what we consider "genius" is often nothing more than a degree of insight which enables its possessor to move so swiftly through intervening stages that it may appear as though those stages were skipped. Look, for example, at Évariste Galois, and his amazing insights into group theory: he was clearly (and fortunately for us) "before his time"; yet he could not have produced his results without first passing - rapidly! - through the necessary intervening stages that included the current state of play in group theory; nor without insightfully applying those ideas to new ground.


Similarly, as we study the mathematical aspects of the external world, we can only develop concepts when a suitable foundation is in place. In this sense, abstract algebra is foundational to category theory, differential equations to analysis, and arithmetic to algebra - not the other ("logical") way round. So when we're looking for foundational ideas in maths, let's remember that a suitable foundation for deducing other ideas is usually only arrived at very late in our process of understanding those other ideas, by a considerable amount of abstraction and much sweat spent on proving that that foundation (or axiom schema, or theory) suffices and is necessary (only logically!) for those deductions.


In short, I'm saying to user1613 that discovering something more "fundamental" to maths than category theory (CT) will probably only happen after we've had plenty more experience with using CT and applying it in places not covered by the ideas that generated it (i.e. outside algebraic topology). Or, if we're lucky, it may happen sooner than otherwise "normally" to be expected if some genius happens along with amazing (but not impossible) insights that connect CT to hitherto unsuspected areas of study. Care to combine CT and molecular biology, anyone? ;-)

And of course, the genius that does happen along will be an "outsider", in other words, somebody who's not too close to the problem to begin with. Which will put many dedicated researchers' individual and collective noses out of joint. Remember David Hilbert's famous Hilbert Programme? Everybody believed that here were a century's worth of serious problems needing elucidation. But it only lasted about 30 years, when along came a Young Kurt (Gödel to be precise), with some devastating news about undecidable problems in proof systems such as Principia Mathematica, Bertrand Russell's crowning achievement of research into mathematical foundations. Ironic, that last ...

One more point, and I'm done. And that is that fundamental advances are usually disruptive, not the result of an assembly-line, production-oriented approach to making progress; by definition, such a "steady as she goes" mindset can only ever produce more of the same. User1613, you're looking for a big idea in the grand, unifying tradition of CT; but CT itself sprang from the difficulties of making sense doing algebraic topology, not from the impetus to produce some sort of Unified Field Theory for maths. I'm willing to bet that if you, for example, dedicate yourself to finding "the thing more abstract than CT", you'll only make any progress by first making several totally unexpected discoveries about apparently unrelated things.

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