What are the philosophical implications of category theory?

Question

I have heard about topoi being the ideal entities to use for foundations of mathematics (since we are able to reasonably interpret our theories in them), so I imagine there might possibly be some reasons for using categories in (an intuitionistic analog of?) analytic philosophy or something related.

Since the term "category" is, as far as I know (at least that's what I remember seeing in Mac Lane's book), borrowed from Aristotle (and Kant used the same word in a slightly different meaning), I am wondering if there might be some kind of connection between the notion of category in mathematics and in philosophy? Or is the word used in mathematics merely because it sounds good?

Aside from this question, that is possibly somewhat historical in nature, I am also wondering: are there any interesting implications of mathematical category theory (or of categorial mode of thought) in contemporary philosophy?

Thanks in advance for any helpful suggestions on this.

Answer 1

The Stanford Encyclopedia of Philosophy has an extensive article on category theory and its philosophical implications. On the significance of the theory, it says that:

Category theory challenges philosophers in two ways, which are not necessarily mutually exclusive. On the one hand, it is certainly the task of philosophy to clarify the general epistemological and ontological status of categories and categorical methods, both in the practice of mathematics and in the foundational landscape. On the other hand, philosophers and philosophical logicians can employ category theory and categorical logic to explore philosophical and logical problems.

Thus, it's apparent that category theory is relevant to and has implications for both mathematics and philosophy, and is not just semantic. From the perspective of mathematics, category theory is very significant because "doing mathematics in a categorical framework is almost always radically different from doing it in a set-theoretical framework."

The article provides a list of philosophical results following from category theory, such as:

The hierarchy of categorical doctrines: regular categories, coherent categories, Heyting categories and Boolean categories; all these correspond to well-defined logical systems, together with deductive systems and completeness theorems; they suggest that logical notions, including quantifiers, arise naturally in a specific order and are not haphazardly organized.

And it also mentions that "category theory allowed for the development of methods that have changed and continue to change the face of mathematics."

These are only examples that begin to answer your questions; there is far too much in the (excellent) article to put here, but reading it should complete the answers well.

Answer 2

The following quote is from the wikipedia entry on Structuralism:

Structuralism is a theoretical paradigm that emphasizes that elements of culture must be understood in terms of their relationship to a larger, overarching system or "structure." In other words, Structuralism posits that discrete cultural elements are not explanatory in and of themselves, but rather form part of a meaningful system and are best understood with respect to their location within (and relationship to) the structure as a whole.

I think that that is very reminiscent of the philosophy of category theory (in solely mathematical terms) in as much it has one. In fact, I would go so far as saying, that category theory is an implementation of this general philosophical idea in the precise mode of mathematics, and has proven itself to be very successful there.

A specific example may show what I mean by this explicitly: Take the simplest mathematical operation we can think of, that of addition. Originally this was defined with respect to the integers, and it was seen that they followed certain rules (ie existence of identity & the associative law). Notice also to add two integers together we don't need the assistance of a third, its solely defined in terms of the two integers we have at hand.

With the birth of modern algebra, and the invention of new mathematical structures, say for simplicity - groups, each of which internally had a notion of addition (you could add two elements of a group to get a third, and in fact this characterises that group), and an external notion of addition (you could add two different groups together to get a third, you also had the existance of an identity & an associative rule, which is why its called addition).

It was noted that in other algebraic systems, the same pattern persisted: you had internal operations which characterised that individual instance of that kind of algebraic system, and an external addition. Now these external additions for each category of algebraic systems were different, and the problem was to come up with a uniform definition. Eventually it was noticed that defining this external addition with reference to every other algebraic system of its kind gave a uniform definition (it generally called a universal property/definition) and furthermore, the definition did not directly refer to the internal structure of any of these systems, but rather how they related to each other (called a morphism in category theory).

Finally, to complete the circle, by using this definition in the category of finite sets (which have no internal operations given), we find that we re-invent the integers.

In contemporary continental philosophy, I'm aware that Badiou uses Category Theory in the later development of his thought, though I can't say I understand what he's doing, so I'm not going to say anything about it. However, see 'Logics of Worlds'. The following is a quote from this site: http://theimmeasurableexcess.blogspot.com/2008/07/badiou-and-deleuze-brothers-in.html

'Badiou employs two different regimes: being/the ontological/set theory and appearing/logic/category theory.'

Answer 3

It seems that the mind uses category theory to systematize and navigate his knowledge:

http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000858

The paper demonstrate how systematicity, that is, the ability to generalize and extract knowledge without spurious conclussions that are a nighmare in Artificial Inteligence, for example: a car may be red, therefore red flowers may consume fuel, comes naturally from mental processes(behaviours) that follow the rules of category theory.

"Our minds are not the sum of some arbitrary collection of mental abilities. Instead, our mental abilities come in groups of related behaviours. This property of human cognition has substantial biological advantage in that the benefits afforded by a cognitive behaviour transfer to a related situation without any of the cost that came with acquiring that behaviour in the first place [..] systematicity emerges as a natural consequence of structural relationships between cognitive processes, rather than relying on the specific details of the cognitive representations on which those processes operate, and without relying on overly strong assumptions"

I think that the relation between categories of math and categories as mental reasoning of phylosophy and psychology are very close at last.

Because this has implications about the structure of the human mind, and thus, the structure of reality that we perceive and the way we think and act upon the perceived, it may give certain support for a Aristothelic-Tomistic philosophy on the essences, and a firm support for reasoning by analogy.

On analogy, from the wikipedia:

http://en.wikipedia.org/wiki/Analogy

Steven Phillips and William H. Wilson [18][19] uses category theory to mathematically demonstrate how the analogical reasoning in the human mind, that is free of the spurious inferences that plague conventional artificial intelligence models, (called systematicity), could arise naturally from the use of relationships between the internal arrows that keep the internal structures of the categories rather than the mere relationships between the objects. Thus, the mind may use analogies between domains whose internal structures fit according with a natural transformation and reject those that don´t

Answer 4

If the OP will allow me to rephrase the question, so that I can provide some kind of an answer: 'Is there any connection between category theory as practiced by mathematicians today and Aristotle's categories as seen in his ORGANON'?

Yes, there is a connection, but it is not direct. In PRIOR ANALYTICS we see the first systematic analysis of logic in the West, which takes the form of a two term propositional calculus. This relatively simple scheme admits a small number of categories that Aristotle used to classify propositions.

In the 19th century, Boole extended Aristotle's results to include an arbitrary number of terms making the prospect of a catalog impossible. In his LAWS OF THOUGHT, Boole proved that the number of categories is infinite.

In the 20th century, Stone proved that any algebra that is Boolean is also a set. This is the connection to category theory. Categories are used in evaluating sets using algebra.

So, that's the connection! Beginning with Aristotle's categories, continuing on with Boole's algebra, and ending with Stone's sets. All of the relevant material is in the public domain, if you care to read it.