Tom de Jong
Postdoc at the University of Nottingham working on type theory. PhD from the University of Birmingham. Mathematician, computer scientist and runner.
Postdoc at the University of Nottingham working on type theory. PhD from the University of Birmingham. Mathematician, computer scientist and runner.
@ecavallo Nice! Looking forward to reading it after I've made more progress on my CSL reviews and gone on holiday 🙂
Thierry and I have something in the TYPES 2025 post-proceedings: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2025.10
This is about the "type theoretic axiom of replacement", which you may be familiar with from @egbertrijke's book. We describe a direct cubical semantics for it (not going through Egbert's join construction reduction to pushouts) and suggest how you can think of it as a HIT.
Besides these new things, the first half of the article is dedicated to a survey of applications of this axiom in the literature. I hope you'll find it interesting and/or useful!
I don't have any talks planned about this stuff, so I count on you to pass it around 😇
Call for ESSLLI 2027 Lectures and Workshops
University of Tartu, Estonia, August 2-13, 2027
ESSLLI seeks courses on diverse topics, each course taking
one or two weeks of 90 minutes per day. ESSLLI workshops follow the
same structure (5 days, 90 minutes per day) with content assembled
from external contributors.
Call for proposals: https://easychair.org/cfp/ESSLLI27
Submission link: https://easychair.org/conferences/?conf=esslli2027
ESSLLI 2027 web page (under construction): https://digits.ut.ee/esslli-2027/
Important dates:
* October 1, 2026 (AoE): Deadline for submitting course/workshop titles
* October 15, 2026 (AoE): Deadline for submitting course/workshop proposals
* December 10, 2026 (AoE): Notification sent to course/workshop proposers
@emilyriehl But the fact that there are existing norms, such as frequent flying, that we now recognize to be harmful doesn't give free rein to create new harmful norms......
The "minimal individual impact" argument, as used for topics like meat farming, plastic use and recycling, only holds water when the negative norms are (1) sufficiently established and embedded into society, and (2) perpetuated by large non-individual actors, such that collective individual action is not the majority contributor to the issue (this holds less for meat eating). It seems to me that neither of these holds for the current, *nascent* push of "AI" into all areas of life; the companies are all desperate for validation to build their power plants and datacenters (orders of magnitude more resource-intensive than most currently existing), and by normalizing "AI" use in our area we are collectively giving our acquiescence.
Surely this *is* an area to focus on where mathematicians can collectively have real impact: by keeping the huge social and environmental costs of this tech at the forefront of the debate.
Higher structures (Aberdeen, 8-10 Sep 2026)
The second meeting of the network ‘Higher Structures in Category Theory, Homotopy Theory and Type Theory’ will take place at the University of Aberdeen, with talks on 8 and 9 September 2026 and optional discussion sessions in the morning of 10 September 2026.
Invited speakers will include:
- Paige North, Utrecht University
- Tashi Walde, University of Regensburg
In addition, we invite abstract submissions for contributed talks. Participation is free and everyone is welcome, but please complete the registration form and abstract submission Friday 31 August 2026:
https://docs.google.com/forms/d/e/1FAIpQLSdJUFHlKVHhiwGo-LvGEt1iDeXRTLf3xG6dIbRmrP3W9Jb_yw/viewform?usp=header
The network website is available here:
https://sites.google.com/view/higherstructures/home
Any enquiries about funding should be sent to simona.paoli@abdn.ac.uk.
The network organizers:
Eric Finster, Nicolai Kraus, Nicola Gambino, Simona Paoli.
EDIT: I jumped too fast on the easy story, and overplayed the role of AI here. Had I known how it would blow up, I would have been more careful... Things are still not entirely clear, but I've tried to make the post less wrong.
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This whole Lean kernel bug is almost too on point to be true, it fits perfectly in the discussions we've had here and elsewhere over the last months/years…
To summarize:
- formal methods researcher (@ramana) sets up a repo with a sorry-free proof of the Collatz conjecture
- the proof is reporting (by @kirancodes) as being a kernel bug, the repo was a tongue-in-cheek way of exposing the bug
- Ramana mentions llms were
involved (but how is unclear atm)
- the bug is related to (nested) inductive types, for which there is no clear theoretical specification: the kernel's code is the reference
- external checkers (lean4lean and nanoda from a week ago) are affected too, either because they copy the reference kernel implementation, or because (nanoda) of a missing check in this subtle part of the code
- the bugs compound, and the invalid proof is accepted by comparator, Lean's gold standard for proof checking
And so
- adversarial AI writing proofs raises the bar for kernel correctness
- without a clear type-theoretic understanding of *what is actually implemented*, we're toast
- external checkers help to catch implementation bugs, but without a clear specification they can't really catch logic bugs
I believe the whole story also showcase the solidity of human measures: the bug was fixed as soon as found, with healthy discussions between experts, in the open, as to what to do. At no point was there any real doubt as to the solidity of a mechanised result.
@ToucanIan Obvious but maybe good to remark: for the least fixed point on a poset with small infima you can take Ord^op as a counterexample. For the least fixed point on a poset with small suprema, I don't know.
@ToucanIan The thing is that versions of Tarski's fixed point theorem for large (locally small) posets with small suprema/infima are just *false*. For the greatest fixed point version of Tarski, you can consider the large poset Ord of small ordinals which does not have a greatest element so that the identity on Ord does not have a greatest fixed point. This was pointed out to us by one of the referees of, and subsequently explained in, the journal version (https://doi.org/10.46298/lmcs-19(2:8)2023) of the paper you linked.
So even though the proof/reduction of [Tarski-fixed-point → resizing] may be of interest, the statement itself isn't really because Tarski-fixed-point is simply false in this generality.
One of the joys of using #Typesetter (a minimal editor for #typst) is that I find it improves after each update which is not usually the case for many other pieces of software 🥲️
The following said above in this thread is indeed true:
* Supervisors are only human, too.
* They are unable to (1) read your mind, (2) manage their time, (2) remember much.
Despite all the above shortcomings, that apply to myself too, I love supervising.
And, remember, then, if you are a supervisor, students are only human, too. And (1)-(3) apply to them too. 🙂
So we are all in the same boat, and we should rather sail together enjoying our mutual company.
I am trying to verbalize to myself why I don't like autoformalization (say to get Claude to write Agda code for me).
As I have said repeatedly before, I use TypeTopology/Agda as a blackboard, to think about mathematical objects and their properties.
When I had Claude to do the experiment I reported earlier, I did get partial results, but the process was painful because Claude was thinking, not me.
The whole point, in my particular case, is to use TypeTopology with Agda to think about things, more than to get things done.
I sometimes feel that my style at odds with the way that Mathematics is traditionally written. A theorem like https://mathstodon.xyz/@jonmsterling/116934135318482820 this would, in a proper mathematics paper, be stated without proof because the proof is easy.
In my opinion, that practice (though defensible on the grounds that professional mathematicians generally make very few fatal mistakes) is a bit unfortunate. First of all, it makes it harder for a newcomer to see just *how* people prove things in an area (unless they are personally taught by an insider). Second of all, much of the beauty of mathematics is in the proof. Seeing that a proof works swimmingly without any difficult parts can provide confidence that the definitions are good.
I do sometimes read books and papers where a lot of proofs are left out, and then when I try to reconstruct them myself, I see that the results are true but the proofs are really really complicated in ways that could have been fixed with better definitions.
Another day, another theorem proved à la “equivalences without tears“ (https://arxiv.org/abs/2408.11501 by @de_Jong_Tom). It took me about four hours to figure out how which type should be the ‘fulcrum’ of my nest of equivalences, but once I did that, it was easy because the diagram commuted definitionally. It never ceases to amaze me how well this works.
@jonmsterling It's funny. I mostly wrote this up in summer two years ago as a welcome change to other less fun things that had to be written up. I'm so glad folks find it helpful 🙂
(And naturally I use this technique in the plan for my final ESSLLI lecture.)
I think that if people have written formal proofs using LLMs and wish to claim them as contributions in their papers, they need to include some account of the methodology they used to ensure that the proof actually formalises the intended results. That would necessarily involve an audit chain of all *definitions* needed in the statements of the main results...