Categories?
Adventures in Abstraction
I was recently invited to consider taking a course in “Categorical Foundations for Mathematical Systems Theory” for my Systems Science Master’s program.
My first reaction was to dismiss the idea as unrealistic.
While my friends at organizations like BlockScience and the ISSS have made Category Theory sound exciting, and even revolutionary, it’s always seemed out of reach for me. Like something I might only have time to learn about in the very distant future.
But, at the suggestion of the professor offering the course, I’ve taken some time to consider the possibility that I could actually handle it. To that end, I’ve started reading The Joy of Abstraction: An Exploration of Math, Category Theory, and Life by mathematician and educator, Eugenia Cheng.1
Cheng does an incredible job of explaining math in a way that makes it seem fun, relevant, and accessible. For her, math is a beautiful discipline which combines philosophy’s rigorous logic with art’s intuitive abstractions. She describes category theory as “the mathematics of mathematics” and argues that it provides a secure framework that can be used for performing the often ambiguous task of abstraction.
She also firmly believes (and has demonstrated through her teaching) that category theory can be taught effectively to students with no background in pure mathematics. In her experience, certain learners — like her art students — actually benefit from interfacing with math at higher, rather than lower levels of abstraction.
Reading the book has inspired me to learn more about why category theory was created, what it aims to accomplish, and how it might relate to my current research on systems.
A few things I’ve learned so far:
Systems theorist and biologist Robert Rosen was (allegedly) the first to apply category theory to the natural sciences starting in the 1950s — bringing it outside of the realm of pure mathematics.2
Recent efforts in applied category theory aim to advance general systems theory by developing formal methods for focusing on how systems interact with each other and their environment.3
A mathematician exploring categorical sociology believes that category theory “lends itself to axiomatic approaches to social science, such as that attempted in Austrian economics.” 4
Category theory can help blockchain researchers and engineers reason formally and systematically about compositionality, “the study of how systems compose to give life to more complex systems” 5
Systems theorist and biophysicist Martin Zwick believes category theory might compete with graph theory and set theory for the role of serving as the core mathematical foundation for systems theories. 6
I’m not sure whether or not I’ll commit to taking the course, but I do feel inspired by the work Cheng has done to help me feel truly excited about math.
At this point, I’m convinced that category theory, along with the theories of sets and graphs, will be a useful framework for me to add to my systems toolkit at some point in my life.
Cheng, E. (n.d.). The Joy of Abstraction. Retrieved August 7, 2024, from https://www.cambridge.org/core/books/joy-of-abstraction/00D9AFD3046A406CB85D1AFF5450E657
Lennox, J. B. (2024). Robert Rosen and Relational System Theory: An Overview (Vol. 8). Springer Nature Switzerland. https://doi.org/10.1007/978-3-031-51116-5
Myers, D. J. (n.d.). Categorical Systems Theory. http://davidjaz.com/Papers/DynamicalBook.pdf
Joncas, G. (n.d.). Topos-theoretic Social Science. Oneironomics. Retrieved August 7, 2024, from https://gjoncas.github.io/posts/2020-12-16-topos-theoretic-social-science.html
Genovese, F. (Director). (n.d.). Compositionality: The 10x Engineer Secret Sauce [Video recording]. Retrieved August 7, 2024, from https://archive.devcon.org/archive/watch/6/compositionality-the-10x-engineer-secret-sauce
Zwick, M. (2023). Elements and Relations: Aspects of a Scientific Metaphysics (Vol. 35). Springer International Publishing. https://doi.org/10.1007/978-3-030-99403-7