The more I read of the introductory chapter of Category Theory for the Sciences by David Spivak, the more I'm sure there is a firm and solid foundation for potentially making mathematics a major pillar of Big History.
The structural scheme for category theory within mathematics sounds almost like that of the structure of Big History as a whole. If nothing else it screams convergence while making very complex structures much simpler. (Big History in reverse perhaps?)
I can't wait to delve further in. Spivak's text, though very mathematical in nature seems to have all the basic set theory (at a high school level), and the introduction indicates that he expects readers to have some facility with linear algebra, but there are no other highly daunting prerequisites.
A few excerpts:
This book extols the virtues of a new branch of mathematics, category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of analogous theorems in another category. A functor is like a conductor of mathematical truth.
We build scientific understanding by developing models, and category theory is the study of basic conceptual building blocks and how they cleanly fit together to make such models. Certain structures and conceptual frameworks show up again and again in our understanding of reality. No one would dispute that vector spaces are ubiquitous throughout the sciences. But so are hierarchies, symmetries, actions of agents on objects, data models, global behavior emerging as the aggregate of local behavior, self-similarity, and the effect of methodological context.
Hierarchies are partial orders, symmetries are group elements, data models are categories, agent actions are monoid actions, local-to-global principles are sheaves, self-similarity is modeled by operads, context can be modeled by monads.
The paradigm shift brought on by Einstein’s theory of relativity led to a widespread realization that there is no single perspective from which to view the world. There is no background framework that we need to find; there are infinitely many different frameworks and perspectives, and the real power lies in being able to translate between them. It is in this historical context that category theory got its start.
However, in 1957 Alexander Grothendieck used category theory to build new mathematical machinery (new cohomology theories) that granted unprecedented insight into the behavior of algebraic equations. Since that time, categories have been built specifically to zoom in on particular features of mathematical subjects and study them with a level of acuity that is unavailable elsewhere.
Bill Lawvere saw category theory as a new foundation for all mathematical thought. Mathematicians had been searching for foundations in the nineteenth century and were reasonably satisfied with set theory as the foundation. But Lawvere showed that the category of sets is simply one category with certain nice properties, not necessarily the center of the mathematical universe. He explained how whole algebraic theories can be viewed as examples of a single system. He and others went on to show that higherorder logic was beautifully captured in the setting of category theory...
In 1980, Joachim Lambek showed that the types and programs used in computer science form a specific kind of category. This provided a new semantics for talking about programs, allowing people to investigate how programs combine and compose to create other programs, without caring about the specifics of implementation. Eugenio Moggi brought the category-theoretic notion of monads into computer science to encapsulate ideas that up to that point were considered outside the realm of such theory.
It is difficult to explain the clarity and beauty brought to category theory by people like Daniel Kan and Andr´e Joyal. They have each repeatedly extracted the essence of a whole mathematical subject to reveal and formalize a stunningly simple yet extremely powerful pattern of thinking, revolutionizing how mathematics is done.
All this time, however, category theory was consistently seen by much of the mathematical community as ridiculously abstract. But in the twenty-first century it has finally come to find healthy respect within the larger community of pure mathematics. It is the language of choice for graduate-level algebra and topology courses, and in my opinion will continue to establish itself as the basic framework in which to think about and express mathematical ideas.
As mentioned, category theory has branched out into certain areas of science as well. Baez and Dolan [6] have shown its value in making sense of quantum physics, it is well established in computer science, and it has found proponents in several other fields as well.
But to my mind, we are at the very beginning of its venture into scientific methodology. Category theory was invented as a bridge, and it will continue to serve in that role.
