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John Baez (Sep 08 2025 at 07:48):The title of this article about applied category theory sounds very negative, but the article itself is more thoughtful:
John Baez (Sep 08 2025 at 07:49):Here's a quote:
How Demand Shapes the Supply of Ideas
This external demand for useful tools creates a powerful incentive structure that feeds back into the academic world.
Selective Adoption: The applied world eagerly adopts the parts of category theory that are immediately useful. Computer scientists, for instance, use categorical concepts to bring order to programming languages, which are themselves built on object-like "types." This creates a demand for mathematical work that applies categorical tools to existing set-based structures, reinforcing the idea that set theory is the fundamental layer and category theory is a useful descriptive layer on top.
The Funding Filter: This pragmatic pressure is formalized through funding mechanisms. Government and corporate grants are more likely to flow to research that promises concrete applications and measurable outcomes. A proposal to "use categorical methods to solve a problem in X" is far more legible and fundable than a proposal to "explore the philosophical foundations of X using topos theory." This system doesn't need to be hostile to foundational work; it simply starves it of resources by rewarding incremental, tool-oriented research.
The Career Filter: This incentive structure shapes academic careers. Young mathematicians, facing the pressure to publish and secure tenure, are rationally incentivized to work on well-defined problems within the established paradigm, where success is recognizable and rewarded. Exploring a new philosophical foundation is a high-risk, slow-burn endeavor that the career structure implicitly discourages.
The result is a system where the evolution of mathematics is subtly guided by the needs of its primary consumers. The deep philosophical questions are not suppressed; they are simply sidelined by a pragmatic culture that values mathematics for what it can do.
John Baez (Sep 08 2025 at 07:57):I don't think the rise of applied category theory is reducing the world's interest in its more conceptual or philosophical side. Even if most people only care about what category theory can do, that exposes more people to its new ways of thinking.
And there's certainly nothing new about how "the evolution of mathematics is [not so] subtly guided by the needs of its primary consumers".
I think what's happening is that the market for category theory is dramatically expanding, while the market for the philosophical side of category theory is expanding too, but less.
David Michael Roberts (Sep 08 2025 at 08:02):I was hoping to see more than just philosophical thinking from that article, maybe the "co-opted" part is just lamenting somehow that CT is getting more applied? I confess I only skimmed the article very quickly.
Ultimately ACT will either live up to its promise to deliver actual value for people who don't a priori care about CT, but want to solve their own problems, or it will be an intellectual curiosity. I very much hope the former, even though as a pure mathematician I love intellectual curiosities and would happily work on them if paid to do so.
John Baez (Sep 08 2025 at 08:05):Their attitude:
While the dominant cultural current pulls toward instrumentalism, we at Holon Labs are consciously swimming against it. Holon Labs, a non-profit organization, represents a deliberate attempt to engage with category theory not as a tool, but as a philosophy.
Ralph Sarkis (Sep 08 2025 at 08:30):Another peek at their attitude from here:
We humbly ask the same attitude from our donors. We are asking you to give — without expectation — to a fledgling organization trying to carve a new path. We are asking you to let go of the need for a goal and to rescind control of the outcome this organizational model creates. This isn’t donors providing sustenance for a team of researchers who will, in return, develop a miracle solution. This is emptiness meeting emptiness.
John Baez (Sep 08 2025 at 09:04):David Michael Roberts said:
I was hoping to see more than just philosophical thinking from that article, maybe the "co-opted" part is just lamenting somehow that CT is getting more applied?
They are lamenting that increased use of category theory is not breaking the hold that set theory has on the foundations of mathematics. I'll quote more:
Perhaps the most sophisticated form of co-option is not outright rejection but selective adoption. The mathematical establishment has become adept at incorporating the language and diagrams of category theory in a way that gives the appearance of progress while leaving the foundational philosophy of set theory entirely untouched. It is now common to speak of the category of sets (Set), the category of groups (Grp), or the category of topological spaces (Top).
This practice is a Trojan horse. On the surface, it seems to embrace the categorical perspective. In reality, it does the opposite. By defining categories whose objects are themselves structures built axiomatically within ZFC (a group is a specific kind of set with a binary operation, etc.), it reinforces the idea that set theory is the fundamental, substance-providing layer of mathematics, while category theory is a secondary, descriptive layer that deals with the relationships between these set-based objects.
However, I don't think the key revolutionary goal of category theory should be for it to overcome set theory as the "substance-providing layer" of mathematics. And I think the stakes in applied category theory are much higher than anything one could say about the ontology of mathematics.
John Baez (Sep 08 2025 at 09:10):This passage seems more on the right track to me, except that I would change "the old Set-Theoretic framework" to something much broader, like "the old framework in which science is a tool for humans to stand above, dominate, and exploit nature":
The core philosophy—that an entity's identity is constituted entirely by its relationships, that "you are the sum of your relationships"—is a radical reorientation of the modern conception of the self and the world. It is a direct refutation of the atomistic individualism that underpins so much of our economic, political, and social logic. The challenge is immense: how does one teach this profound, holistic insight without it being immediately co-opted and trivialized by the very paradigm it seeks to replace?
The danger is that the relational philosophy will be instrumentalized, reduced from a transformative worldview to just another tool for optimization within the old Set-Theoretic framework. "Network theory for career advancement," "systems thinking for market dominance," "collaboration tools for increased productivity"—these are the ways the dominant paradigm would seek to absorb and neutralize the Categorical challenge.
John Baez (Sep 08 2025 at 09:30):The whole argument is undercut a bit by also being a funding pitch, and a funding pitch that doesn't really say much about what Holon Labs does. But a lot of the issues it raises weigh heavily on my mind these days.
Nathan Corbyn (Sep 08 2025 at 10:05):I was immediately put off by the following:
We at Holon Labs argue that the prevailing object-centric
worldview, rooted in set theory, became dominant not just because of its
mathematical elegance, but because it perfectly mirrored the needs of
an industrial civilization focused on discrete products, quantifiable
assets, and manageable parts.
I don't think this claim admits evidence-based justification.
Nathan Corbyn (Sep 08 2025 at 10:06):The claim about programming languages and types being 'set-like' also suggests a misunderstanding of categorical semantics.
Chad Nester (Sep 08 2025 at 11:16):I don't really think it makes sense to describe category and set theory as part of a "conflict of foundations". I like Maddy's version: https://bpb-us-e2.wpmucdn.com/faculty.sites.uci.edu/dist/3/399/files/2023/09/What-do-we-want-a-foundation-to-do.pdf
Chad Nester (Sep 08 2025 at 11:16):Full disclosure: I have also only skimmed the article
Cole Comfort (Sep 08 2025 at 12:51):Nathan Corbyn said:
I was immediately put off by the following:
We at Holon Labs argue that the prevailing object-centric
worldview, rooted in set theory, became dominant not just because of its
mathematical elegance, but because it perfectly mirrored the needs of
an industrial civilization focused on discrete products, quantifiable
assets, and manageable parts.I don't think this claim admits evidence-based justification.
One of the motivations for the development of set theory was to establish a foundation with which to formally define the real numbers. Surely this is the opposite of the claimed ``discrete'' teleology of our industrial civilization.
John Baez (Sep 08 2025 at 13:08):I don't think they were talking about discrete versus continuous in the sense of topological spaces. I think they were talking about how we try to chop the world up into separate pieces to make it seem more manageable, when in fact it's an interconnected web where pulling on anything shakes many other things.
I seem to be more sympathetic to a lot of the thrust here than some of you. I actually do think set theory arose as part of the "objectifying" approach to science and technology that dominates our world. Heidegger invented the term present-at-hand for this. But it's extremely hard to break out of this mind-set and category theory does not, by itself, necessarily break out of this. I think that's one of the things the authors are worrying about.
daniel gratzer (Sep 08 2025 at 13:32):One thing I find a bit odd: is really the case that "category theory has been co-opted"? The original work on category theory was quite explicitly motivated by concrete applications to algebra, topology, etc. Perhaps I am simply ignorant of the early history here, but my understanding is that category theory as a foundational alternative was a relatively late idea to the development of category theory (at least a few decades after its inception) and even then, it was motivated by pragmatic concerns of other areas of mathematics/physics. Reading this, I feel as though this is presenting a bit of a longing for a past that did not, broadly speaking, exist.
Fernando Yamauti (Sep 08 2025 at 15:38):I think their claims of discrete/continuous are to be understood in the setting of Lawvere's view of how dialectics (he probably later extrapolated to dialectical materialism and Marxism, though) emerges in category theory. For instance, as in his paper on Cantor's lauter Einsen criticising how Zermelo corrupted set theory by conflating Cantor's set (a potentially structured collection) with the idea of a cardinal (the discrete version). Of course, that’s related to the concept of Vorhandenheit mentioned by @John Baez above and the idea of intensionality/extensionality.
Btw, I don't know how all that thing of "dialectical materialism = cohesion, continuous" started. Perhaps that's because of Marx's "Mathematical Manuscripts", which apparently even made nonstandard analysis the foundation of the basic calculus teaching practice in Maoist China.
Perhaps the connection to dialectical materialism aka Marxism is more explicit in other writings of Lawvere, but I can't remember where I've seen that before...
Kevin Carlson (Sep 08 2025 at 17:07):John Baez said:
I don't think they were talking about discrete versus continuous in the sense of topological spaces. I think they were talking about how we try to chop the world up into separate pieces to make it seem more manageable, when in fact it's an interconnected web where pulling on anything shakes many other things.
...and certainly, the Weierstrassian foundation of analysis is the apogee in all intellectual history of this tendency to understand a thing by chopping it up into nice well-defined little pieces!
Kevin Carlson (Sep 08 2025 at 17:08):I also more or less like the vibe here but I don't really know that anyone has a proposal for an alternative. We know that trying to re-found mathematics on category theory in the same sense as it is currently founded on set theory (thinking of ETCC) doesn't really work. Perhaps the true categorically-minded approach to mathematics wouldn't even need a foundation, in some sense. But again I don't think anybody really knows what that would mean. Or perhaps homotopy type theory is already the best answer to these kinds of concerns.
Mike Shulman (Sep 08 2025 at 18:51):Yes, this sounds to me like just yet more people misunderstanding category theory. Hardly anyone has ever claimed seriously that "category theory" can be a foundation for mathematics in any sense resembling the way set theory is. ETCS can, but it's just as reductionist in principle as ZFC. Same with HoTT/UF. Category theory is more of a "co-foundation" that gives us ways to organize mathematics from the top down, rather than from the bottom up, and as such is complementary rather than in competition with whatever foundation one may choose.
John Baez (Sep 08 2025 at 18:57):daniel gratzer said:
One thing I find a bit odd: is really the case that "category theory has been co-opted"?
I think some idealistic visions of applied category theory - like in my manifesto, and the remarks of David Spivak quoted in the essay - have run into the reality of how the world works. I feel very much that category theory is getting "co-opted" by big business and the military, right now. I wouldn't normally use the word "co-opted", but I don't think it's such a terrible word for the process whereby it may become just another tool for making rich and powerful people more rich and powerful.
But the dreams David and I had came quite a bit later than Lawvere's rather different idealistic visions for category theory as a revolutionary form of mathematics.
And those came quite a bit later than the original inception of category theory.
I say a tiny bit more about my views here:
I don't think the business about "set theory versus category theory" is very interesting here, not compared to other issues.
Valeria de Paiva (Sep 08 2025 at 23:39):@John Baez I'm intrigued by this sentence in your blog post "And some of us are using category theory to develop new software for epidemiology, and soon labor economics, trying to model aspects of the green transition." Is there anything written about the labor economics and green transition yet?
David Michael Roberts (Sep 09 2025 at 01:55):I'm not sure what it means to "co-opt" a whole research field. Can one say "algebraic geometry" is being co-opted?
Mike Shulman (Sep 09 2025 at 02:18):At first I thought it didn't make any sense, but apparently there's another meaning of "co-opt" that I wasn't really familiar with: "appropriate" in the sense not of "take over control of" but rather in the sense of "cultural appropriation", i.e. incorporate someone else's ideas into your own work.
Mike Shulman (Sep 09 2025 at 02:19):If that's the meaning, I would tend to celebrate it. Let's get as many people as possible to incorporate the ideas of category theory into their work!
David Michael Roberts (Sep 09 2025 at 04:52):Yes, I think it's more people adopting things for their own purposes and using it in more-or-less in the way they are accustomed to rather than reading MacLane's book, or Spivak–Fong, and then taking over the intellectual culture of CT.
David Corfield (Sep 09 2025 at 06:05):John Baez said:
I actually do think set theory arose as part of the "objectifying" approach to science and technology that dominates our world. Heidegger invented the term present-at-hand for this.
So what would a mathematics look like of the ready-to-hand, the usual contrasting term? Recent thoughts I've had along these lines would depart from later phenomenology than Heidegger's. Merleau-Ponty brings in the body interestingly, but I would start from Eugene Gendlin's A Process Model. The word 'space' appears over 400 times in the book.
"We have and use our space, of course, but let us permit new concepts of space to arise from our interactional concepts." (p. 7)
There are indications of intricate structure there, yet he finds mathematical framings to be limiting:
"Mathematics is a very human process, but we have not chosen it to begin our model. I have shown that such a model must drop out much of what is most important to living things." (p. 58)
But I wonder if this isn't just down to lacking a full sense of what mathematics could be.
Morgan Rogers (he/him) (Sep 09 2025 at 08:06):I agree with most of the criticisms of the essay that people have expressed here, but I agree with John that it is an important topic to be writing about, so I hope a better attempt comes along soon.
I've said before that science is political (not that I can lay any claim to that observation!) but the ways in which it is political are often subtle enough that theoreticians can avoid taking responsibility for the political ramifications of their research. Here we see an articulation of how the mathematical formalisms at our disposal shape what we are able to talk about and account for in science more broadly, and dually which aspects become "externalities" (to borrow an economics term) -- things we are unable to properly reason about within our formalisms.
Someone said to me recently, "Tell me how constructive mathematics is used to make bombs," as a way of dismissing the idea that war in general (and the genocide in Gaza sepcifically) has anything to do with their work, and so shutting down discussion of it. There may be an actual direct answer to their demand; I haven't looked into it because even I would rather not know(!), but a now-banned participant in this Zulip had a habit of rubbing people's faces in the military applications of their work. It is happening!
For me the specifics of the connection and how direct it is are secondary: even if I manage to convince myself that I have minimized my own responsibility for something, that leaves plenty of room for discussing how we may be able to act collectively to actively reduce the negative impacts of the thing that do concern us, even if the only answer we come up with is to continue our theoretical work in a conscientious way, or redirect our efforts in that work to try to make neglected concepts expressible.
John Baez (Sep 09 2025 at 08:52):David Michael Roberts said:
I'm not sure what it means to "co-opt" a whole research field.
The authors explain pretty clearly what they mean:
Selective Adoption: The applied world eagerly adopts the parts of category theory that are immediately useful. Computer scientists, for instance, use categorical concepts to bring order to programming languages, which are themselves built on object-like "types." This creates a demand for mathematical work that applies categorical tools to existing set-based structures, reinforcing the idea that set theory is the fundamental layer and category theory is a useful descriptive layer on top.
The Funding Filter: This pragmatic pressure is formalized through funding mechanisms. Government and corporate grants are more likely to flow to research that promises concrete applications and measurable outcomes. A proposal to "use categorical methods to solve a problem in X" is far more legible and fundable than a proposal to "explore the philosophical foundations of X using topos theory." This system doesn't need to be hostile to foundational work; it simply starves it of resources by rewarding incremental, tool-oriented research.
The Career Filter: This incentive structure shapes academic careers. Young mathematicians, facing the pressure to publish and secure tenure, are rationally incentivized to work on well-defined problems within the established paradigm, where success is recognizable and rewarded. Exploring a new philosophical foundation is a high-risk, slow-burn endeavor that the career structure implicitly discourages.
The result is a system where the evolution of mathematics is subtly guided by the needs of its primary consumers. The deep philosophical questions are not suppressed; they are simply sidelined by a pragmatic culture that values mathematics for what it can do.
All this is of course perfectly normal for fields of mathematics: they are unsubtly guided by the needs of their consumers. Some fields like mathematical physics wear this proudly on their sleeve. So what do the authors think is "bad" about all this? Clearly they consider the philosophical significance of category theory to be its key virtue that's subtly being suppressed through its adoption as a mere means to an end.
It will be interesting to see if the appearance of books like Categories for the Working Philosopher declines, or even increases, as category theory continues to go mainstream.
My own concern about "co-opting" is different: it's about the application of category theory for bad ends. This too is perfectly normal for fields of mathematics. But I still think we should still fight it.
John Baez (Sep 09 2025 at 09:03):David Corfield said:
John Baez said:
I actually do think set theory arose as part of the "objectifying" approach to science and technology that dominates our world. Heidegger invented the term present-at-hand for this.
So what would a mathematics look like of the ready-to-hand, the usual contrasting term?
I think of Dasein as the real opposite. I can't imagine what a mathematics of Dasein would be.
David Michael Roberts (Sep 09 2025 at 20:37):This makes it sound like somehow everyone in category theory will pivot to only work on these applied problems, when in reality it's more like ACT is developing/has developed as it's own subfield inside CT.
Valeria de Paiva (Sep 09 2025 at 21:13):hmmm, @David Michael Roberts , I'm sure ACT has developed and is developing its own subfield, but not inside CT, more like, parallel to it. At least I'd like to think that's parallel and collaborative. and it means that we're doubling the numbers...
David Michael Roberts (Sep 09 2025 at 21:25):@Valeria de Paiva well, that's an interesting question of when one subfield of maths is "inside" another, but I agree that there is nontrivial overlap (witness the interest in using double categories more even in non-ACT settings, I believe inspired by the ACT work), but that you are right that there isn't proper containment.
Mike Shulman (Sep 09 2025 at 23:39):But pure mathematics and applied mathematics are both part of mathematics, right? So shouldn't pure category theory and applied category theory both be part of category theory?
Valeria de Paiva (Sep 09 2025 at 23:42):well, the way the term was being used CT seemed to stand for pure category theory, not the union of pure and applied category theory. Or so it was my impression.
Mike Shulman (Sep 10 2025 at 00:00):used by who?
Valeria de Paiva (Sep 10 2025 at 01:30):In the thread above?
Mike Shulman (Sep 10 2025 at 01:34):Okay. I didn't have that sense, but even if it was, I would argue that it shouldn't be used that way.
Nathanael Arkor (Sep 10 2025 at 04:55):David Michael Roberts said:
witness the interest in using double categories more even in non-ACT settings, I believe inspired by the ACT work
I'm curious what prompts this impression?
John Baez (Sep 10 2025 at 10:13):Valeria de Paiva said:
John Baez I'm intrigued by this sentence in your blog post "And some of us are using category theory to develop new software for epidemiology, and soon labor economics, trying to model aspects of the green transition." Is there anything written about the labor economics and green transition yet?
There's a ton written about these subjects, of course, but not using category theory! But someday there will be.
You may remember Owen Haaga, who was a summer intern at Topos in Berkeley. Now he's a grad student at Oxford, at the Institute for New Economic Thinking, working on models of "labor shocks" caused by the green energy transition. He's been working with my gang on these models using first AlgebraicJulia and now CatColab. He's now spending a month and a half in Edinburgh, talking to me.
He's doing his thesis on this stuff, and it will take a while before he comes out with anything.
Valeria de Paiva (Sep 10 2025 at 17:09):Thanks @John Baez ! I remember Owen, but didn't know he was working in Edinburgh. I'm interested in work formalizing economic arguments in category theory.
John Baez (Sep 10 2025 at 20:11):Owen just decided to visit Edinburgh for a while to talk to me. He's still doing his PhD down in Oxford, though he had to change advisors.
I don't know work formalizing economic arguments using category theory. Owen is trying to address some of the oversimplifications of economic modeling. In labor economics it's common to assume everything is in equilibrium at any moment even when external conditions are rapidly changing. The most common attempt to go beyond this is very limited, based on wage-stickiness. But Owen is trying to lay the groundwork for more realistic models, like agent-based models.
Joe Moeller (Sep 10 2025 at 20:40):Morgan Rogers (he/him) said:
I agree with most of the criticisms of the essay that people have expressed here, but I agree with John that it is an important topic to be writing about, so I hope a better attempt comes along soon.
I've said before that science is political (not that I can lay any claim to that observation!) but the ways in which it is political are often subtle enough that theoreticians can avoid taking responsibility for the political ramifications of their research. Here we see an articulation of how the mathematical formalisms at our disposal shape what we are able to talk about and account for in science more broadly, and dually which aspects become "externalities" (to borrow an economics term) -- things we are unable to properly reason about within our formalisms.
Someone said to me recently, "Tell me how constructive mathematics is used to make bombs," as a way of dismissing the idea that war in general (and the genocide in Gaza sepcifically) has anything to do with their work, and so shutting down discussion of it. There may be an actual direct answer to their demand; I haven't looked into it because even I would rather not know(!), but a now-banned participant in this Zulip had a habit of rubbing people's faces in the military applications of their work. It is happening!
For me the specifics of the connection and how direct it is are secondary: even if I manage to convince myself that I have minimized my own responsibility for something, that leaves plenty of room for discussing how we may be able to act collectively to actively reduce the negative impacts of the thing that do concern us, even if the only answer we come up with is to continue our theoretical work in a conscientious way, or redirect our efforts in that work to try to make neglected concepts expressible.
There is a mythos within pure mathematics that in part I believe helps keep people in this mindset that they are exempt from real-world consequences of their research. Every math undergrad is told that pure mathematics research is at minimum 50 years off from applications. The attitudes embedded in the culture of pure math enable and encourage people to believe this. The point is well-made by that banned guy that however fancy your math is, there's someone in the US DOD (I guess DOW now) who has a phd in your topic and is reading your papers between meetings on increasing efficiency in their search-and-kill systems. I think this sort of touches on Mike's point of two definitions of co-opt. Nobody is taking control of category theory, this is nonsensical. But people are incorporating into their work, and some of them might be doing things you don't like with your work. Of course I don't conclude from this that people shouldn't engage in pure math research. I do think it's irresponsible to do so without making yourself aware of these avenues. I don't believe there is a route towards preventing people from using your research to do things you don't want within the realms of science or academia. People doing bad things is a social problem, and requires a social solution.
Morgan Rogers (he/him) (Sep 11 2025 at 17:41):Joe Moeller said:
I do think it's irresponsible to [engage in math research] without making yourself aware of these avenues. I don't believe there is a route towards preventing people from using your research to do things you don't want within the realms of science or academia. People doing bad things is a social problem, and requires a social solution.
If a person decides in their free time and with their own resources to apply maths to immoral ends, I might agree that this is something we cannot do much to prevent. However, when it comes to allocation of resources to research on a societal level, there is not nothing we can do.
I reckon that the times when fundamental research has received the biggest expansion of support in it's own right has been on occasions when it has been championed by people through these and other routes. In a time where there is huge and growing popular opposition to the expansion of military funding (and hence the military control over research priorities), we need to collectively be working to take advantage of any opportunity that arises to wrest back some of that control.
David Corfield (Sep 13 2025 at 06:09):John Baez said:
David Corfield said:
John Baez said:
I actually do think set theory arose as part of the "objectifying" approach to science and technology that dominates our world. Heidegger invented the term present-at-hand for this.
So what would a mathematics look like of the ready-to-hand, the usual contrasting term?
I think of Dasein as the real opposite. I can't imagine what a mathematics of Dasein would be.
For Heidegger, Dasein is the way of being of people. 'Present-at-hand' and 'ready-to-hand' are contrasting ways objects can be for us. Both attitudes towards objects are adopted by Dasein.
Here's one commentator:
For Heidegger, the objects, or the non-human entities, with which Dasein comes face to face in its world do not have the same way of Being. The Being of such objects and things is either ready-to-hand or present-at-hand, according to Heidegger.
For Heidegger, ready-to-hand and present-at-hand, in their characterizing of and referring to objects, things, or entities in Dasein’s world, do not speak about or signify different classes of things. Ready-to-hand and present-at-hand refer to two different ways in which Dasein encounters, experiences, and relates to non-human entities in its world.
According to Heidegger, Dasein’s own relating to a certain object, either practically or theoretically and detachedly, is what determines if this specific object is encountered as ready-to-hand or present-at-hand.
John Baez (Sep 13 2025 at 11:41):By the way, I doubt Heidegger would ever say that Dasein is the nature of being of people, though that's certainly a reasonable summary. Dasein is "being there" - and not being there for something else (like a cup is there for us), but being there for itself (as we are for ourselves).
Thus he steps aside from any argument about whether Dasein is limited to Homo sapiens.
More precisely, he says:
"[Dasein is] that entity which in its Being has this very Being as an issue..."
As usual, by trying to say things just right, he becomes a bit hard to understand.
Matteo Capucci (he/him) (Sep 16 2025 at 09:23):Nathan Corbyn said:
I was immediately put off by the following:
We at Holon Labs argue that the prevailing object-centric
worldview, rooted in set theory, became dominant not just because of its
mathematical elegance, but because it perfectly mirrored the needs of
an industrial civilization focused on discrete products, quantifiable
assets, and manageable parts.I don't think this claim admits evidence-based justification.
lmao I also stopped reading exactly after that paragraph. In hindsight I might have been too hasty in drawing a conclusion, but surely it made my eyes roll.
John Baez (Sep 16 2025 at 09:32):So you folks don't think the prevailing object-centric world-view became dominant because it perfectly mirrors the needs of an industrial civilization? Or you don't think the prevailing world-view places a lot of emphasis on describing everything in terms of objects? Or you don't think this world-view is rooted in set theory? I can't tell what seems ridiculous to you. Maybe just the whole idea of relating mathematics to culture?
To me the biggest mistake seems to be the idea that the prevailing world-view is rooted in set theory: I'd be more likely to argue that set theory is rooted in this world-view! I think the foundations of mathematics tend to formalize ideas that are already around. Set theory seems like a good idea because we like to describe the world in terms of collections of things.
I might also say the mechanisms of our industrial civilization, e.g. inventories of products, are an outgrowth of the prevailing world-view. But they also reinforce it in a powerful way: we all live within this framework, as we walk through supermarket aisles and choose among carefully catalogued products, or flick on our cell phone through sets of potential partners.
(I met my wife the old-fashioned way, not choosing her from such a set.)
Mike Shulman (Sep 16 2025 at 16:04):If I understand you right, John, I agree: the intrinsic human tendency to describe the world in terms of collections of things is likely the root both of set theory and of the characteristics of our civilization. What sounds faintly ridiculous to me is the attempt to attribute the ascendancy of set theory to the characteristics of our civilization (and without even trying to imagine what some other hypothetical civilization might be like) rather than recognizing both of them as stemming from the nature of our minds.
Morgan Rogers (he/him) (Sep 16 2025 at 16:24):So to summarise, we've settled that we were set up to set up set theory, but the set of authors of said paper set forth that we should set about setting it down?
Mike Shulman (Sep 16 2025 at 16:27):The set-up for that set of "set"s is only set down by the fact that "set down" isn't usually the opposite of "set up".
Kevin Carlson (Sep 16 2025 at 17:54):Mike Shulman said:
If I understand you right, John, I agree: the intrinsic human tendency to describe the world in terms of collections of things is likely the root both of set theory and of the characteristics of our civilization. What sounds faintly ridiculous to me is the attempt to attribute the ascendancy of set theory to the characteristics of our civilization (and without even trying to imagine what some other hypothetical civilization might be like) rather than recognizing both of them as stemming from the nature of our minds.
Well, the early modern mathematicians didn't think of the real numbers as a collection of points, for instance, right? It seems at least faintly plausible to try to draw some connection between the defense of, say, the inherent cohesion of the continuum in the 17th and into the 18th centuries as reflective of a lingering Medieval mindset focused more on organic wholes than on atomic parts, only finally overcome in the high modernity of the 19th century.
John Baez (Sep 16 2025 at 19:36):It's fun to try to imagine math never focusing on "atomic parts".
I can imagine an alternate history where continued clarification of Euclidean geometry led to multiple axiom systems for rigorously doing parts of mathematics "synthetically", without trying to unify all of mathematics under the umbrella of set theory. We could have axioms for Euclidean geometry and the first-order theory of a real closed field (both of which are complete and decidable), and maybe the introduction of absolute geometry, where the parallel postulate is dropped. (People had been worrying about that postulate as far back as Proclus (410–485), who complained about Ptolemy's false proof of this postulate, before giving his own false proof.) It seems that only a more second-order approach to the real numbers, featuring statements like the least upper bound property, led analysis down the road which took us to Cantor and then axiomatic set theory. What if math had never gone down that road?
Alex Kreitzberg (Sep 16 2025 at 23:11):I'm mixed on the article, the few definite claims it makes feel confused imo. And I'm not sure I understand the Buddishm well enough to see its relevance. But, the general themes and ideas it's touching on seem genuinely interesting and important.
I don't see Set theory, as a foundation, as wildly different from say, x86 assembly. If your theory is based on sets you're at least as correct as sets; if your program is based on x86 you can at least run on x86 computers. In both cases, it seems obvious to me theories built on these foundations are being unduly influenced by those foundations.
The most popular languages were inspired by designs that were intended to keep the translation to assembly languages as simple as possible.
Similarly, mathematicians keep their translations into sets as painless as possible.
One really nice educational experience of category theory for me, was learning different categories were not the same as Set. The author even played up how much of a perspective shift "explore the philosophical foundations of X using topos theory." Would be - but that example seems to fall short of their own point.
Why can't you take any particular category as a starting point? Forgive my glibness - Why not treat the Yoneda lemma as an axiom, the arrows of a category as atomic, and build your theory from there? What goes wrong when you do that?
I'm assuming the article is causing offense because it's carelessly dismissing the problems tools like homotopy type theory are trying to solve. What are those problems? And assuming Type theories or their analogs are very necessary, do folks have ideas on how to let theories built out of them not look like them?
Kevin Carlson (Sep 16 2025 at 23:59):"Why can't you take any particular category as a starting point?"
Well, what's a category? Isn't there a set, or at least a class, of arrows in between any two objects, and aren't the objects also a class? How are you going to state what it means to take the Yoneda lemma as an axiom if you don't already have a category of sets? There are some answers to these kinds of questions, but most aren't very satisfying and none are completely satisfying, I think it's fair to say; the farther you get from starting with a well-pointed $$(\infty)-$$topos with lots of coproducts, the less plausible it is that you can rebuild math this way.
I don't really think type theory gets you that far away from set theory, as far as the aesthetics of a foundation go. A type is still a collection of things--it's just an intensional collection, rather than an extensional one. So I think it's actually just entirely non-obvious what it would mean to really respond to the kinds of concerns in the post.
David Michael Roberts (Sep 17 2025 at 06:06):The first-order definition of category doesn't presuppose axioms on sets or classes. But I agree that you have a harder time trying to say what the Yoneda lemma is without talking about hom-"sets"
David Michael Roberts (Sep 17 2025 at 06:09):Thinking that one needs sets to define categories is a fallacy that people entrenched in set theory bring up when trying to belittle using eg ETCS as a foundation.
Peva Blanchard (Sep 27 2025 at 05:44):Mike Shulman said:
If I understand you right, John, I agree: the intrinsic human tendency to describe the world in terms of collections of things is likely the root both of set theory and of the characteristics of our civilization.
This reminds me of the work of the anthropologist Philippe Descola. He is reknowned partly because of his views on how various cultures approach the separation of nature and culture.
Broadly speaking, he classifies civilisations according to how they relate the various beings (human and non-human). He posed two criteria: similarity/dissimilarity and interiority/physicality; which give four classes (translated from the french wikipedia article):
- animism: similar interiorities, dissimilar physicalities. Key concept: metamorphosis. E.g., Amazonia, subarctic america, southeastern asia, melanesia.
- totemism: similar interiorities, similar physicalities. Key concept: hybridation. E.g., Australia.
- naturalism: dissimilar interiorities, similar physicalities. Key concept: objectivation. E.g. Occident (our western culture).
- analogism: dissimilar interiorities, dissimilar physicalities. Key concept: chain of beings. E.g. Brahmanic India, West Africa [...]
It seems that the human tendency to describe the world in terms of collection of things is mostly a modern western tendency (naturalism). So set theory would be "naturalistic".
I am wondering now how various mathematical theories confront to this "civilization classification".
ps: Descola's book is Beyond nature and culture
Kevin Carlson (Sep 27 2025 at 18:01):I tried for a few minutes, including reading the article, but I don’t think it’s possible to get a real sense of these categories just from that micro-summary or even the mini-summary at Wiki. “Animism” is a fraught word with moving target semantics to begin with, and my impression is that Descola is setting local jargon with the opposition to totemism.
Kevin Carlson (Sep 27 2025 at 18:02):In particular, in what sense does an animist think non-humans have different physicalities and the same interiorities? And what does this have to do with collections of objects?
Peva Blanchard (Sep 28 2025 at 15:58):True, my summary is a bit too hasty. I've skimmed through Descola's book, and will try to answer your questions.
To be more precise, Descola classifies modes of identifications. Here identification means establishing differences and resemblances between myself and other entities, by inferring analogies or contrasts between appearance, behavior or properties I ascribe to myself, and those I ascribe to them.
Two kinds of such properties are "interiority" and "physicality". Interiority encompasses features such as having a mind, soul or consciousness, or a subjectivity, reflexivity, feelings, or immaterial things such as breath or vital energy. In contrast, physicality includes the external form, substance, the physiological and sensorimotor processes, the morphology or anatomical characteristics.
Once these identifications are posed, I can develop many relations wrt these entities (e.g. predation, friendship, etc.), which can be translated to concrete social norms.
Animists ascribe to non-humans an interiority similar to their own. Hence, a bird has a mind, a subjectivity, an intentionality which does not differ from mine much than from another human being. However, their bodies are "obviously" different: birds have feathers, fish scales, and so on.
Naturalists, on the other hand, don't recognize any interiority to non-humans. Humans are special as they are the only entities to have received a soul, an intentionality, an ability to signify using language, etc. However, our bodies, those of animals, and rocks, and so on, are made of the same stuff (we used to think this stuff was springs and pneumatic devices, now we think it is atoms, molecules, etc.). I think Descartes' animal machine provides a good example of naturalist thinking.
It seemed to me that the "object-centric worldview" that has been mentioned earlier fit in the naturalistic mind: there is a nature out there, a collection of objects, which our bodies are part of, that we can observe, and infer laws about. But the non-human objects don't have interiorities, and thus don't receive any civic or moral statutes.
Then, I was too hasty when I said that set theory would then be "naturalistic": I don't have a definite argument to support it.
Mike Shulman (Sep 28 2025 at 17:14):That sounds to me as though all "modes of identification" are subsequent to the inborn human tendency to describe the world in terms of collections of things. Before I can establish differences and resemblances between myself and other entities, I have to first have described the world as a collection of entities including myself.
Peva Blanchard (Sep 28 2025 at 21:02):I don't know if this human tendency is inborn, I guess this is a question for psychologists.
It seems plausible to me that the act of taking a series of encounters with various entities as a whole, i.e. thinking about all of them as forming a collection, is secondary. My naive understanding of "collection of things" is a collection of things of the same type. So I need to acknowledge their sameness in order to acknowledge their belonging to a collection.
Peva Blanchard (Sep 28 2025 at 21:21):Mike Shulman said:
What sounds faintly ridiculous to me is the attempt to attribute the ascendancy of set theory to the characteristics of our civilization (and without even trying to imagine what some other hypothetical civilization might be like) rather than recognizing both of them as stemming from the nature of our minds.
Connecting the ascendancy of set theory with the characteristics of our civilization is a difficult task, and Holon should provide more substantial arguments (as otherwise, I agree, they sound faintly ridiculous). But I'm not sure either if the ascendancy of set theory stems from the nature of our mind, or namely our ability to form abstract collection of things. We have many other abilities, so why this one in particular has been so useful to lay down a solid ground to serve as foundations of mathematics?
ps: I'm assuming that "ascendancy of set theory" refers to its prominent role in the debates around the foundations of mathematics.
Mike Shulman (Sep 28 2025 at 21:29):Peva Blanchard said:
We have many other abilities, so why this one in particular has been so useful to lay down a solid ground to serve as foundations of mathematics?
Perhaps because mathematics is also a creation of our mind, and therefore consists of collections of things?
Peva Blanchard (Sep 29 2025 at 06:49):I just noticed the other thread set theoretic and categorical foundations. Reading the first message there, I think I was about to reboot the same discussion here. I'll probably continue over there.