Dear Professor Alberto Peruzzi, Thanks so much for sharing what I recognize as a teaching-moment on and about how individual ideation is not individual's own: no mind is an island disconnected from the mainland of one's time et al., and more immediately / importantly: How our thinking is shaped and reshaped by the waves reaching the shores time and time again can't be left to Cognitive Science (a failed enterprise going by their own audit: https://drive.google.com/file/d/1iBIj5yLnnOIjzwRuBy0rlRQXTWBQj5a5/view?usp=d..., and then there's, of course, my curse for refusing to include Mathematics in their logo ;) As Professor F. William Lawvere highlighted: An explicit philosophy, dealing with the mutual transformation of individual thinking and collective thinking, could be very helpful (for example, when reforming the schools) and adequate mathematical models will be an essential basis of clarity ( https://lawverearchives.com/wp-content/uploads/2024/12/1994-tools-for-the-ad... p. 44). If I may, one point that puzzled me is the notion of the primacy of 'set'. Isn't a set a set of things, to begin with; but for elements, what is the Kardinalen a kardinalen of (cf. in speaking of, say, length of a line, is the primacy of line vis-à-vis its length open to debate)? One line of thinking, which seems to leave room for the primacy of sets appears to be that of Professor F. William Lawvere's category of sets, wherein elements of a set are points of the set, i.e., functions defined in terms of sets (there are some extremist set theorists, like my good friend: https://zenodo.org/records/4039859, who after all said-and-written arrived at our beloved category of graphs :). I'll write to you again soon after carefully studying your paper: An Essay on the Notion of Schema. Given that contrast is the basic building block on which rests our consciousness, thinking, sentience, and conscience (cf. Yarbus: stabilized images fade away; discounting the illuminant; as much as I love Cantor, had we all been his lauter Einsen, we wouldn't be ;) isn't it reasonable an approach to deploy arithmos-chaos in analysing given variation and cohesion with their contrasting constancy and discreetness ( https://conceptualmathematics.wordpress.com/wp-content/uploads/2013/02/sets-... pp. 245 - 246)? Last and yet the one that troubles me the most is how much of long-dead philosophy is healthy, especially given the mathematical brilliance of my role model: Professor Stephen H. Schanuel, who wasn't distracted by philosophy ( https://conceptualmathematics.wordpress.com/2023/05/08/mathematicians-mathem... )? My problem is compounded by the fact that I am drawn to "philosophy": beginning with the ancient Indian concept of Dharma, which, in the sense of its etymological roots, means: That which holds it all together, which is none other than Unity-respecting Change (e.g., plants growing into trees, water flowing down the gradient, birds flying high in the sky), or equivalently, Becoming consistent with Being ( https://conceptualmathematics.wordpress.com/wp-content/uploads/2025/02/conce... p. 152), which is all changes inundating us in the sense of: I don't remember turning into a caterpillar on my way to Malabar Cafe for coffee and returing home as a butterfly, all of which can be summed up as miracles-prohibited in our mathematical understanding of our everyday experience populated with varieties of categories of objects, all of which partake in the abstract essence(s) characteristic of their corresponding categories; as such, morphisms between objects of a category are necessarily structure-respecting maps: maps prohibited from tearing apart in mapping, say, a domain graph into a codomain graph (ibid., p. 210). It's this Unity-respecting Change / Becoming consistent with Being, or in a more familiar phrasing: Structure-respecting maps of any category of objects represented as Natural Transformations, appears to be the Zeroth Law of Change, the Dharma that holds it all together (Full Disclosure: Indian-by-birth). Please be kind enough to correct my mistakes (until I found Professor F. William Lawvere and Stephen H Schanuel, Conceptual Mathematics textbook / my Bible :) I used to think: math exams are deliberately designed to trick unsuspecting students into making mistakes, a' la Kahneman; https://philpapers.org/archive/POSFTS.pdf; Fodor provides a more-entertaing denuding of Kahneman's claim to fame: heuristics). Thanking you, Yours respectfully, posina P.S. Knowledge tends to inhibit the flight of imagination. Therefore, a certain naiveté, unburdened by conventional wisdom, can sometimes be a positive asset (Harish-Chandra). P.P.S. The singular Western contributions to the birthing, rearing, and development of modern science seems to have benefitted from not burdening themselves with the ancient intellectual inheritance of India (with its abundance of every concievable concept / thought / theory ;) P.P.P.S. One dangerous delusion that's gained a strong foothold in MathEd is that students' intellectual success / failure depends on that of their ancestors. This is eugenics: inheritance of intellect, or simply put, you need a Blockchain dad to invent blockchain, the rediculous nature of this anti-social paid-career of direspecting the disadvantaged amongst us is all around us since the dawn of sense and reason (http://disq.us/p/2ubeted). Notwithstanding my schooling, Kim, got promoted: from Quanta magazine to The Notices of AMS¿ On Thu, 11 Jun, 2026, 2:36 pm Alberto Peruzzi, <alberto.peruzzi@unifi.it> wrote:
Dear Posina,
perhaps it might be of interest for you to know that when Bill referred to the notion of *arithmos *he relied on many previous conversations in Florence during that period, between Bil and John Mayberry (and also John Bell and me). John stayed in Florence for one year during which he wrote a book which is based on the *Euclidean* concept of “set” (*Foundations of Mathematics in the Theory of Sets, *2000, and at the time Bill liked to spend at least one week each year in Florence to discuss with me about topics concerning issues of philosophy and logic. So, John Mayberry’s pioneering, and deep, investigation on the notion of *arithmos *was a recurrent topic of conversation. There were specific points on which Bill and I didn’agree with John. What you read in the mail you kindly allow us to access is a byproduct of Bill’s disagreement with John, who argued for the primacy of the notion of set over any other, so that - along similar lines to those of Kreisel and Feferman -, Bill’s categorical approach to foundations could not be accepted.As for my objections to John, they were implicit in a section of a paper also published in 2000 "An Essay on the Notion of Schema*"* but I preferred not to say that explicitly. Later, John had a brain ictus and our dialogue had to stop. John was a man of incredibly wide culture and had a “classic” mind. He didn’t accept Bill’s dialectical view and looked at my attempt to free Bill’s intuitions from Hegel’s frame and reframe them into a naturalistic view (while keeping some aspects of Kantian philosophy of knowledge) as an attempt by a "neo-Aristotelean", something I didn’t know to be and didn’t want to look, though mainly for reasons related to Aristotle’s perspective on metaphysics and epistemology rather than on his “naturalistic” reorientation of Plato’s views.
Best, Alberto
Il giorno 5 giu 2026, alle ore 10:58, Posina Venkata Rayudu via Categories <categories-list@categories.org.au> ha scritto:
Dear Professor Graham White,
Reading your email made me happy in more than one respect, one of which is that which you suggest emphasizing, i.e., the conceptual and the geometrical side of mathematics dating back to ancient Greece (in the words of Professor F. William Lawvere):
I just thought you might find it of some interest.
Thanking you, Yours respectfully, posina P.S. Early this morning, I asked the shopkeeper, who gave me a bill of Rs. 6009, for a 1 rupee candy, so that I can pay a round figure of Rs. 7000 (happy to be in your good company ;)
On Mon, Jun 1, 2026 at 2:31 PM Graham White via Categories <categories-list@categories.org.au> wrote:
This is very depressing. I've always been on the geometrical, conceptual side of mathematics (and I've never really been any good at numerical calculation). And the geometrical side of mathematics goes very far back, to ancient Greece at least. Could we not try to emphasize this?
Graham White London
On Mon, 1 Jun 2026, 08:20 peklund via Categories, < categories-list@categories.org.au> wrote:
Why do we have this development?
Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community.
Young students now, unfortunately for CT, have more options to feel self-confident and proud.
Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged.
Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert?
Just wondering.
---
Is there a silverlining? What can we do?
CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching).
Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT.
---
"Real applications", what's that?
I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors.
Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)?
Am I right or am I right?
If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines.
Am I wrong or am I wrong?
If I'm wrong, then there is hope, otherwise not.
And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play").
I rest my case.
---
Best,
Patrik Eklund
PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other.
On 2026-05-31 16:29, Steven Vickers via Categories wrote:
Hi Peter,
I'm sad to hear of the decline of category theory in the Maths Faculty.
All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason?
To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory.
All the best,
Steve.
________________________________ From: P.T. Johnstone via Categories <categories-list@categories.org.au> Sent: Friday, May 29, 2026 10:11 AM To: Marino Gran via Categories <categories-list@categories.org.au> Cc: P.T. Johnstone <ptj1000@cam.ac.uk> Subject: [categories] Category Theory under threat in Cambridge
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As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid!
I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year.
I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so.
Peter Johnstone
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