The Relativist’s Ontology

“We keep finding more systems for relating objects, but we never seem to find the objects!”

Category Theory has been called both the “glue of mathematics” and the “mathematics of mathematics,” with many equivalent descriptions that often promote it with a sense of overarching superiority or deeper elegance when compared to other branches of abstract research. The truth is that it can be as ugly and ad hoc as any overly convoluted formalism, but it nonetheless has a few strengths that make it uniquely powerful and expressive:

  1. It allows you to escape from the hidden assumptions baked into traditional notions of equality and identity.
  2. Its framework provides a natural setting where both reductionist and non-reductionist approaches can live side by side.
  3. It also provides a natural setting for both classical and intuitionist approaches, keeping both constructivists and non-constructivists happy.
  4. It lends itself to diagrammatic expression, which is much more intuitive and readable than traditional numeric or formulaic expressions.
  5. It acknowledges from the outset how the unavoidable complexity and hidden potential of objects cannot just be abstracted away. But if so desired, it also allows you to abstract away, as needed.
  6. It can be applied equally well to a surprisingly wide range of fields, including linguistics, logic, computer science, and quantum mechanics.
  7. The axiom of choice doesn’t always hold, which is a good sign 😜

From The Mathematical Specification of the Statebox Language, Fabrizio Genovese and Jelle Herold, Statebox Team (2019):

What is Category Theory?

That’s a great question – in many ways the answer deepens every day. Category theory is primarily a way of thinking, more than just a theory in the usual sense of the term. Probably the simplest idea of category theory is that everything is interrelated. This applies not only to mathematics, but also computation, physics, and other sciences which are just beginning to be elucidated and unified via the use of categories – and is precisely the reason why category theory has such natural real-world applications.

William LawvereCategories of space and quantity in J. Echeverria et al (eds.), The Space of mathematics , de Gruyter, Berlin, New York, pages 14-30, 1992.

It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.

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