Politics, geometry, and foundations in general

Euclid's Elements provided a foundation for mathematics for over two millennia. During the Nineteenth Century mathematicians exposed some of the flaws in Euclid's system. Those flaws were repaired and many new foundations were provided for Euclidean and non-Euclidean geometry. One of the beautiful things about mathematics in general is that it provides rigorous foundations for its propositions along with its extreme usefulness for the sciences.

Can the tools of mathematics provide a foundation for philosophy and political philosophy in general? Imagine applying set theoretic or category theoretic ideas to political philosophy so as to achieve a mapping of systems, beliefs, and states of subjects' belief systems. It would seem that is what political philosophers have attempted, but without completely embedding their results in set theoretic or category theoretic language, at least not successfully so far.

One might decide immediately that political theory is not the kind of thing that fits itself into category theoretic terms. Yet category theory is abstract and general in the extreme. One should not a priori dismiss its usefulness for mapping the structure of the political realm.

One could try the reduction of the political to category theory starting with Plato and working one's way through the classics. This would replace the informal with the brutally formal. At the end of the exercise one might have a scheme exposing what fits with what.

One of the problems with the reducing political philosophy to category theoretic terms is that category theory is itself a product of the imagination and is based on basic analogies and metaphors just as are the other abstract branches of learning such as political philosophy. That admittedly is a strong statement. How many people actually believe that our higher reasonings are produced through metaphor? However, exploring how metaphor creates foundational ideas remains a seductive notion especially when brain science provides evidence for some relations between foundations and metaphor.

Instead of the trial and error method of starting with Plato and reducing political theory into category theoretic terms, one could investigate the foundations of foundational thinking to see if a category theoretic approach is apt for studying political philosophy.

At any rate, the multitude of questions arising from foundations in general seem nontrivial and important. Are there foundations for specific foundations or are foundations just not the kind of things that work that way?