I am rereading rereading Robert Goldblatt’s book, Topoi, though in many respects it seems like I’m reading it for the first time. When there is enough time and space between myself and some volume or other, that experience of ‘(re)reading it for the first time’ is not all that uncommon. It occurred not too long ago with E.P. Thompson’s The Making of The English Working Class, and Durrell’s Alexandria Quartet. Those works both had in the neighborhood of forty years between the first and the second readings, so I feel less guilty at the sense of surprise and pleasure. With Topoi, my excuses are somewhat more thin, though I can still assert with considerable truth and honesty that there’s been considerable intellectual development on my part since the first time I tackled the book. I mention this not just to make what amounts to little more than a peculiar Facebook post (some people share pictures of their meal, after all), but to set up a discussion of why a Whiteheadian should pay special attention to that area of abstract algebraic thinking known as Category Theory. I’ll first spend a few words talking about the book itself.
The word “topoi” is the plural form of “topos,” which seems rather more elegant than saying “toposes.” A topos is a category theoretic structure that is rich in a variety of “nice” formal characteristics, the details of which I’ll spare you (as that would require an entire book on category theory to explain.) Now, a category (such as might take on the structural features that would further specify it to be a topos) is a mathematical constructions that turns away from “objects” so-called to devote particular attention to functions, transformations, and operations without any special concern for the supposed “what” that is being transformed or operated on. As such, category theory is arguably the purest form of algebraic thinking around. It is scarcely an accident that Leo Corry’s magnificent history of the development of abstract algebra, Modern Algebra and the Rise of Mathematical Structures, ends with the emergence of category theory.
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!
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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.)
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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.
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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.
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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.”
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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.
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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.
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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/
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