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categorytheory.md

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Category Theory
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The following version is addressed to an audience that knows something about math and doesn't just equate it with the bullshit done in school. There are some mathematical examples that aren't developed much, but a reader who doesn't know about them can just ignore them and get the points anyway.

Category Theory

There was a revolution in mathematics in the 20th Century. Here is a highly idealized and probably inaccurate sketch of the history of the conceptual steps involved:

  1. Constructivism. Mathematical objects must be defined in terms of a small set of "primitive concepts" in order to be known. Our intuition of how they behave is not enough. The essence of a mathematical object is its construction. For example, the real numbers are constructed by Dedekind cuts.

  2. Structuralism But there are often many different constructions that embody the same intuition! The real numbers, for instance, didn't have to be constructed with Dedekind cuts. Cauchy sequences could also have been used! How do we know which to choose? Really, it is not the particular way we constructed an object that is important, but rather its internal structure. The essence of a mathematical object is its structure.

    1. Universal Algebra How do we define a structure? We use a set of axioms that are satisfied by it. For example, the real numbers are the unique complete ordered field (each of these words, "complete", "ordered", and "field", indicates a set of axioms satisfied by the real numbers). The essence of a mathematical object is the axioms that are satisfied by it.

    2. Category Theory But we could have used different axioms. The real numbers are also the unique Archimedean ordered field. How can the set of axioms that an object uniquely satisfies be its essence when it uniquely satisfies multiple distinct sets of axioms? The reason that these axiom sets are equivalent is that the object satisfying them interacts the same way with other objects. In category theory, the essence of an object is how it interacts with other objects. (I don't think I have fully justified this. I will edit it as I learn more.) (I'm not sure how to do a categorical analysis of the real numbers. Apparently others aren't either.)

In category theory, there is a natural shift of focus from the structures of individual objects to the structures of the networks of interactions between objects. These networks are called categories. Categories themselves can be studied in general, and they go through the same conceptual progression detailed above. They are first axiomatized, and then it turns out that really their interactions with each other is what is important. So we end up talking about the category of categories. The category theorist Lawvere talked about the foundations of mathematics through the category of categories.

I don't know much about this, so I kind of feel embarrassed that I am writing this publicly, but I will revise it later and I suppose its good that everybody can looka at the version history and see that I'm revising things and not just innately an expert. For my thinking on this see here.

Social Sciences

We can go through an analogous dialectic to attain a social conception of self:

  1. People are just physical matter. Someone's identity is simply the collection of matter that makes up their body. How could it be anything else? This matter determines the person commpletely in the sense that two people cannot share the same physical matter.

  2. But the matter comprising us changes with time! We are constantly admitting new matter into our bodies (e.g. food, drink, air) and expelling old matter from our bodies (e.g. excrement, dead skin cells, air). It seems like our bodies are really just a structure that matter cascades in and out of. The particular atoms that make up our body at a particular time are no more essential to ourselves than the place we happen to be located at that time. Really, a person's essence is the structure of their body.

    1. Well, how do we define someone's structure?

In 1992, one of the founders of category theory, William Lawvere, wrote:

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.