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Math Trek

One of the most abstract fields in math finds application in the 'real' world

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Every pure mathematician has experienced that awkward moment when asked, “So what’s your research good for?” There are standard responses: a proud “Nothing!”; an explanation that mathematical research is an art form like, say, Olympic gymnastics (with a much smaller audience); or a stammered response that so much of pure math has ended up finding application that maybe, perhaps, someday, it will turn out to be useful.

That last possibility is now proving itself to be dramatically true in the case of category theory, perhaps the most abstract area in all of mathematics. Where math is an abstraction of the real world, category theory is an abstraction of mathematics: It describes the architectural structure of any mathematical field, independent of the specific kind of mathematical object being considered. Yet somehow, what is in a sense the purest of all pure math is now being used to describe areas throughout the sciences and beyond, in computer science, quantum physics, biology, music, linguistics and philosophy.

Samuel Eilenberg of Columbia University and Saunders Mac Lane of the University of Chicago developed category theory in the 1940s to build a bridge between abstract algebra (the generalization of high school algebra) and topology (the qualitative study of shapes, including those in very high dimensions). Very similar arguments repeatedly cropped up in the two fields in different contexts, so the mathematicians reasoned that some deeper structure must unite these situations.

They created an organizing framework that any field of mathematics could be put in. A “category” is a collection of mathematical objects together with arrows connecting them. So, for example, the natural numbers are the objects of a category, and one particular arrow in that category would connect each number to its double. Eilenberg and Mac Lane could then analyze maps between entire categories, and maps between those maps. This allowed the connections between different fields of mathematics to be formulated precisely.

Mathematicians sardonically dubbed the field “abstract nonsense.” Its extreme level of abstraction drains all the content out of the theory, since the objects can represent nearly anything. Draining the content, many expected, would also drain its power: What can anyone possibly say that will apply to essentially all mathematical objects?

Surprisingly, a lot. The recurrent arguments that had spurred the theory were ones that applied to all categories. Eilenberg and Mac Lane’s framework revealed an entire world of theorems that could be applied throughout mathematics.

Logicians started using category theory, viewing a deduction of one theorem from another as an arrow connecting the two. Then computer scientists carried category theory further still, viewing programs as maps connecting input of one category to output of another. A program that multiplies two numbers, for example, would go from the category of pairs of numbers (the numbers being multiplied) to the category of numbers (their product).

These connections turned out to be extraordinarily deep — indeed, the theory of programming languages and the field of logic can be seen as essentially identical to category theory. Computer scientist Robert Harper of Carnegie Mellon University jokingly calls this “computational trinitarianism,” imitating the Christian notion that God is a trinity of Father, Son and Holy Ghost.

“The central dogma of computational trinitarianism,” he wrote on his blog, “holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation.  There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives. Computational trinitarianism entails that any concept arising in one aspect should have meaning from the perspective of the other two.” Porting ideas between the fields has led to profound insights for all three.

Category theory’s spread has continued. Many results in quantum information theory turn out to follow directly from category theory. Category theory’s hierarchical structure has made it useful for modeling complex biological systems. Category theoretic models of language have outperformed conventional ones in distinguishing, for example, the meaning of “saw” in sentences like “I saw a man with a saw.” It’s even proving valuable in developing rigorous models of music theory.

David Spivak of MIT has perhaps the boldest vision for category theory’s potential. In a paper posted February 27 on arXiv.org, he argues that all scientific thought can be expressed in a structured way using category theory. Both ideas and the data supporting them can be encoded in the universal language of category theory, allowing scientists to present a database with their full work. Spivak even imagines a Facebook-like interface with people’s full thoughts and experiences presented in a category theoretic database that would connect people whose databases overlap.

“If people adopt the level of rigor of category theory,” he says, “it will provide a precise language for science as a whole, and it will help individual scientists to clarify their thinking. My ultimate dream is that communication problems would only happen because someone is trying to lie.” 

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