The focus on function is a complete inversion from the approach taken in set theory, where one begins with individual, self-identical “things” – otherwise uncharacterized by any properties beyond their pure self-identity – and gathers these together as members of a “set.” Functions, to a set theorist, are an entirely parasitic notion about relating “things” and set-theoretic gatherings of “things” to other “things” and gatherings. To a process metaphysician, the set-theoretic approach simply begs the question. And here I am using “begs the question” correctly to mean that it is engaged in a form of circular reasoning in which it (set theory) is presupposing its conclusion as to the priority of self-identical “things” as a premise. (Persons using “begs the question” when they mean “invites” or “suggests” the additional question is quite enough to get me frothing at the mouth like a rabid dingo that hasn’t eaten a baby in days.) This category theoretic approach to mathematical structures has now, for quite a few years, been explored as an alternative approach to characterizing the “foundations” of mathematics to that of set theory (which is by far the standard approach.) Quite aside from the formal difficulties (which, as Jaakko Hintikka has shown, are legion) of situating set theory at the “foundations” of mathematics, the central characteristic of set theory is as barely mathematical as it can be. Once again, set theory springs from objects, and looks at combinations of membership. But mathematics is about structures and transformations, a phrase that ought still be at the forefront of everyone’s mind since that is the central feature of category theory.
It should come as no surprise that, while set theory has illuminated few if any insights into anything other than set theory, category theory has demonstrated surprising connections across multiple areas of mathematical inquiry. (You can read any text, including Goldblatt’s, for a list. So I’ll not rehearse it here.) One of those connections is with formal logic. It turns out that a deduction is a kind of transformation. Well … how could it not be? What else are you doing when you perform a logical deduction except transforming the “surfaces” of one body of information into a more structurally revealing system of information? The category theoretic structure that provides a functional and structural representation of formal deduction is called a “topos” (plural, topoi.)
I have long (20+ years) recognized that there was at least sympathetic threads of potential interaction between process philosophy and category theory, if on no other basis than that both shift the focus from substantive “things” to processes (operations, transformations, functions). This is a realization that is at least occasionally recognized. Epperson and Zafiris make good use of these notions in their book applying Whiteheadian ideas to the foundations of quantum physics. Others have been less careful, if rather more flamboyant. I was roundly disappointed by one such presentation at an International Whitehead conference I attended. Rather than offering a thought out paper and real analysis, the presenter satisfied himself with just talking entirely off-the-cuff and tossing out terms with unfettered abandon, but no connecting tissue. In any event, there is a great deal of work to be done on this subject.
That being said, category theory has (as already been stated) fructified with insights into a variety of other areas of mathematics. One notable spot that I’ll mention here is in computer science, in the study of computation and programming at an abstract level. Looking at structures in terms of relational connections rather than just and only brute-force calculations has opened up a variety of new lines of inquiry for researchers. (For the curious, Barr and Wells excellent introductory book, Category Theory for Computing Science (1998) is free for the download.) These kinds of connections and insights don’t just happen on their own account; they require a standpoint and systematic criteria that invite the relevant kinds of questions to be posed, that novel answers can be discovered. There is a reason why these sorts of connection were never generated by set theory.
An unexpected bit of personal experience presented itself even as I was writing this little piece that offers a superior note upon which to end things than that which I had originally intended. A recent story in WaPo tells about, “A plane fueled by fat and sugar has crossed the Atlantic Ocean.” (Sure, but when I do that, it is ‘bad for me’ …) Among other things, this new form of fuel leads to a very significant reduction in climate-change related pollutants. So it is a big deal. A friend on social media found this quite fascinating, and asked how this was even possible.
Well, I’d played with chemistry a bit when I was a kid (I was one of those kids who never colored outside the lines, so I always followed the instructions and never blew anything up, or unleashed a stink bomb that required evacuating the house) and took a science credit in chemistry in college. So, understanding a little bit about the physical aspects (for example, fossil fuels, fats and sugars, all fall under the broad heading of organic chemistry) I took a stab at explaining it. But as I was typing out my response, I realized I wasn’t talking about chemistry so much as I was talking about functional relationships and transformations that maintain structural features without focusing so much on the individual “things” composing those relations. I was, in a word, applying category theory.
This is why it is not only worthwhile, but important to study such things. The habits of thought that allow such functional and structural relations to leap out are not, in the general course of events, the sorts of things that we are simply born with; gifts that simply float down upon us out of nothing more than the generosity of the gods. (The latter are, as a rule, ill-tempered and stingy bastards.) One does not grow a well cultivated garden by merely staring at the ground. For anyone with pretensions of entering into Whitehead scholarship, the need is even greater, since this is the position from which Whitehead himself is starting.
THE QUANTUM of EXPLANATION 
I enjoyed this quite a bit, Gary. Couldn’t “like” it without logging into a WordPress account. Question (not a math person): why does set theory necessarily presuppose the existence of objects? I guess I’ve always thought about it that way, but why couldn’t sets involve relationships instead? Or maybe the point is just that that’s something that categories are just more adept at treating? At any rate, this short reading actually brought a lot of things together for me. Reminds me of our breakfasts where you would draw some mind-blowing figure on a napkin!
LikeLike
For set theory, the only relationship that “counts” is the membership relationship. (Subset relationships are built out of this.) A member is always an “element” — a solid, enduring, self-identical “thing.” It is almost what is sometimes called a “naked particular,” though it need not actually be “naked”. It is just that any predicable qualities are irrelevant to its status in set theory as an enduring “thing.”
Relationships in set theory are specifically set aside as merely parasitic ways of speaking about things and membership. Any attempt to invert such a foundational move, and you’re no longer doing set theory. In fact, you’d then (in essence) be doing category theory. This is what makes the latter radical. Same with Whitehead’s philosophy. If you start treating relations seriously, and not merely as parasitic forms of speech, then you’ve already abandoned substance based metaphysics; there no way to shoe-horn a relation (and possibilities are forms of relations) into the shape of some manner of self-identical “thing.” (Though Dog knows, Platonists have tried.)
LikeLike
Gary,
I’ve been sort of following along, and you’ve finally come to something upon which I might comment and contribute. I wouldn’t say that “relationships in set theory … as merely parasitic ways of speaking about things and membership.” I’m hesitating at the word “parasitic” and what the thrust of the passage might mean. I cannot tell if this is an issue of diction or disagreement.
Depending upon what we’re doing in set theory–and here I’m thinking more groups than sets–the relationship is the identity. That is, anything that has a certain mathematical relationship as described by the set just is that element. That’s how a set is defined. To put it another way, there is no essential set membership. Much of group theory, in particular–and for those reading along group there is both a more particular and much more advanced form of set theory–is about how seemingly different “elements” reduce to the same thing, as identity is about relationships or maybe the better word is “functions” since too many people think that the related items are what matter … but they don’t in group theory. The relationship / function / transform is what matters.
To give an example and application, I studied much of this and then applied it to cryptography as an undergrad. In crypto, you don’t know whether you’re seeing a legit communication, what languge it is in, what the encryption is, etc. So, in short you’re checking for some basic patterns that communication must have, especially when you can guess a language and its representation. But my point is that the calculations work regardless of the language, representation, message, etc. because the only thing that matters is the pattern of relationships. It’s a functional analysis.
So, again, I’m not sure we’re disagreeing, but … I did want to geek out a bit.
LikeLiked by 1 person
In set theory, the elements that can be members are enduring “things.” The only relation that holds even a partial status of its own is the membership relation. All other are merely façon de parler, short hand forms that aren’t genuinely real on their own account, and hence are parasitic upon the “real” language of set theory. The CompSci example is appropriate, and one I mention in the post. A number of researchers are finding Category Theory a useful instrument for understanding issues in CS that set theory is unable to offer. Because in CT (unlike in sets), “the only thing that matters is the pattern of relationships. It’s a functional analysis.”
By the bye, in category theory a group is a structure with a single “object” (the group itself) and the members of the group are all invertible transformations (“arrows”) operating on that one object. Bringing to bear any form of additional algebraic structure is already a very significant move away from set theory. Category theory is “simply” the ultimate codification of such moves. What happens when you press that algebrization far enough is that you no longer are no longer saddled with set theoretic metaphysics.
LikeLike
Gary,
Apparently, I’m getting notifications about comments but not new posts … I thought this was still the Whitehead and God thread.
Regardless, I looked up “category theory” and it’s functionally equivalent to the group and field theory that I know. Seems to be more a disciplinary naming difference than anything else, especially since all my professors were algebraists/number theorists and not topologists.
So, your talk of objects and groups is maybe not equivalent to what I’m thinking, but it doesn’t really matter. I was conceiving / recollecting a “group” as a set of distinct homomorphisms, and contrasting that with simple sets where the elements are substantially rather than functionally defined. E.g., “one” is the element that includes all possibilities and forms of singularity as opposed to understanding it as simple counting.
Still geeking. I was not much into topology, just algebraic structures. The language might be very similar, but of course the latter is not inherently about “spatial mapping” or advanced “geometry.”
LikeLike
Category theory is far more generalized than just and only groups and fields, so it is well beyond a mere renaming. Also, I’m not getting any notifications at all from WP, so the only time I see a reply is when I go up and look.
LikeLike
Awesome explanation. So set theory just by definition has to do with these naked particulars, things, or elements. I like how you equate inverting the foundation of set theory with moving away from substance metaphysics. In my initial reading of your blog post, my reaction to what you were saying about set theory was to question why it couldn’t be done relationally. But the move toward categories seems to mark a transition to the use of a set of tools more appropriate to describe fields of relations. Like Dewey said, we just “get over” some problems or ways of doing things that have become outmoded. Trying to describe a world of relations with set theory just doesn’t work very well.
LikeLike
This has become something of a “war path” with me and a great deal of what passes as Whitehead scholarship these days, especially from the process theology camp. I gave a talk a few months back on the subject of “Learning the ‘Language’,” with regard to internalizing some of ways of thinking associated with the algebraic approach. https://garyherstein.com/2023/10/23/what-a-math/
LikeLike