iiinova — where your digital self keeps evolving

Fenner Tanswell @fennert.bsky.social · 13/10/2026

The new Wykeham professor of Logic @wiglet1981.bsky.social taking the field in new and unexpected directions. www.theguardian.com/us-news/2026...

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Fenner Tanswell @fennert.bsky.social · 19/08/2026

The internet reacts to an improvement in matrix multiplication efficiency from 2.371339 to 2.371177.

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Fenner Tanswell @fennert.bsky.social · 13/08/2026

Some top work from the Daily Nous comments section today.

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Fenner Tanswell @fennert.bsky.social · 13/07/2026

This includes in checking, where it uses agents to check each other. Reminiscent of the more dialogical approach to proofs, such as proposed by @cdutilhnovaes.bsky.social

Quote from the OpenAI prompt: "Use adversarial agents throughout: every candidate proof must be checked for exact-two
multiplicity, repeated-edge closed trails masquerading as cycles, parallel-edge 2-cycles,
disconnected graphs, cutvertices, bridges introduced by reductions, and circular use of an
equivalent CDC statement."

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Fenner Tanswell @fennert.bsky.social · 13/07/2026

Another week, another LLM-generated proof, this time of the Cycle Double Cover Conjecture. en.wikipedia.org/wiki/Cycle_d...

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Fenner Tanswell @fennert.bsky.social · 07/07/2026

There may be the possibility of us finally getting away from the darkest timeline.

Quote from The Guardian's Politics live blog: "The Lib Dems may not be standing in Clacton, but Count Binface (see 5.19am) says he is up for it.

(Quote Binface) Labour, Tories, Lib Dems and Greens: I demand you stand down in Clacton.

I will be a unity candidate and pledge to build at least one affordable house.

Nigel Farage says he wants The People versus the Establishment.

So be it. Leave him to me.

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Fenner Tanswell @fennert.bsky.social · 15/06/2026

Finally, lots of fine-grained information about the specific uses of interesting and important as applied to different things. e.g. how often activity phrases like "important to note" are used.

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Fenner Tanswell @fennert.bsky.social · 15/06/2026

2) Lots of the uses were about directing attention, something not really discussed much in the literature on adjectives in maths. 3) Clear differences between pure and applied maths (even though the categorisation of these in the study are very coarse). Applied is much closer to physics.

Table showing the frequencies of the term "important".

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Fenner Tanswell @fennert.bsky.social · 15/06/2026

The *ahem* interesting results were: 1) Lots of different things are described by mathematicians as interesting or important. Crucially, the philosophers' favourites (definitions, theorems, proofs) were not really the main ones.

Table from the paper listing the frequency codes for the uses of the term "interesting".

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Fenner Tanswell @fennert.bsky.social · 09/06/2026

Some discussion points to get my students talking about proofs and rigour tomorrow.

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Fenner Tanswell @fennert.bsky.social · 17/04/2026

The interesting thing is that this wasn't "low-hanging fruit", it was something that experts had worked on and gotten partial results in. The solution was achieved by taking a different approach than previously, something that will be useful beyond the particular problem.

Quote from Terence Tao: "In any case, I would indeed say that this is a situation in which the AI-generated paper inadvertently highlighted a tighter connection between two areas of mathematics (in this case, the anatomy of integers and the theory of Markov processes) than had previously been made explicit in the literature (though there were hints and precursors scattered therein which one can see in retrospect). That would be a meaningful contribution to the anatomy of integers that goes well beyond the solution of this particular Erdos problem."

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Fenner Tanswell @fennert.bsky.social · 14/04/2026

Hi Dave! Yeah, nice question. This quote was surprising to me, for example, because I had thought that being a reverse centaur was not something anyone wants to be, but they seem to be delighted.

Quote from the Quanta article reading "When Ryu asked ChatGPT, “it kept giving me incorrect proofs,” he said. “But the lead-up to the inevitable error had interesting steps, correct partial results that seemed potentially useful.” As the LLM made incremental progress, he would check its answers, keep the correct parts, and feed them back into the model with a new prompt. “I had to play the role of the verifier,” Ryu said. “With ChatGPT, I felt like I was covering a lot of ground very rapidly, much more quickly than I could do on my own. That’s what kept me going.”"

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Fenner Tanswell @fennert.bsky.social · 14/04/2026

Seems a bit contrived to make ships spar with the pope before eliminating them.

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Fenner Tanswell @fennert.bsky.social · 10/04/2026

He also mentions the potential use of autoformalisation. The big results in this so far like the Math Inc formalisation of sphere packing, relied substantially on human created blueprints.

Quote from Mochizuki: "By contrast, the mathematical content of
Stages 4 and 5, as well as (to a lesser extent) Stage 3, are understood by a much larger class of mathematicians and have never
been a source of confusion/debate, hence should be regarded as
long-term goals that seem much more amenable to LeanForm by
research groups not so closely connected to my research group
at RIMS, e.g., (perhaps) via machine-based autoformalization."

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Fenner Tanswell @fennert.bsky.social · 10/04/2026

For instance, he is already talking about a Lean formalisation that isn't in the mathlib:

Another quote from Mochizuki with crazytown bananapants underlinings and italics, reading "In particular, it follows that the LeanForm of species/mutations becomes essentially immediate once one has a LeanForm
of ZFC as a first order theory. In the fall of 2025, I discovered,
on the one hand, that such a LeanForm had in fact already recently been achieved by a young Japanese researcher named Shogo Saito, but, on the other hand, (was quite surprised to learn!)
that such a LeanForm does not yet exist in MathLib (the standard library for refereed Lean code). Here, we observe that the
technical significance of formalizing ZFC as a first order theory
lies in the fact that this is necessary in order to be able to work
with “some indeterminate set-theoretic formula φ” (i.e., as opposed to specific set-theoretic formulas “a ∈ A”, “B ⊆ C ∩ D”,
etc., which are “natively” available in Lean)."

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Fenner Tanswell @fennert.bsky.social · 10/04/2026

The first paragraph is pretty audacious, though. He says that the main benefit of formalisation is normally verification. But that is just one benefit, alongside getting a better understanding of the maths. Most of what is in MathLib was never in doubt anyway.

A Mochizuki quote from the new paper, with wackidoodle underlinings and italics, reading: "The significance of LeanForm: verification vs. communication:
Classically, the significance of LeanForm is typically regarded as
lying in the verification of the logical correctness of a mathematical theory. In the case of IUT, however, although this verification aspect of LeanForm is welcome, in fact the logical structure
of the theory is rather simple, so this verif. aspect of LeanForm
is not a central focal point of interest from the point of view of
researchers in IUT. Rather, in the case of IUT, the significance
of LeanForm lies in producing a precise record of the logical str.
of IUT that is immune to false misinterpretations and hence
can be used, in a pivotal way, to communicate this simplicity
in a maximally efficient/precise way to other mathematicians."

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Fenner Tanswell @fennert.bsky.social · 25/02/2026

Wake up, honey! A new extension to the "maths as journeying" metaphor just dropped.

Tao: These problems are like distant locations that you would hike to. And in the past, you would have to go on a journey. You can lay down trail markers that other people could follow, and you could make maps.

AI tools are like taking a helicopter to drop you off at the site. You miss all the benefits of the journey itself. You just get right to the destination, which actually was only just a part of the value of solving these problems.

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Fenner Tanswell @fennert.bsky.social · 10/02/2026

Human mathematicians do have experience and expertise spotting human errors, but LLMs do not fit the same patterns, so their purported proofs are harder to check. They are not following norms, merely imitating them. They lack common sense. They make weird and unpredictable mistakes.

“[…] in many cases, the solution the chatbots provide is
very long, detailed, and written in a “professional” way,
but it still may be completely wrong, or make no sense
at all when examined more carefully. This may fool a
human to think that such a detailed and long solution
would be correct, so extra caution is needed when we use
such tools for solving similar exercises.” (Emphasis ours.
Plevris et al. 2023, p. 18)

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Fenner Tanswell @fennert.bsky.social · 10/02/2026

In that case, the mathematician becomes the reverse centaur i.e. a horse head making decisions and held aloft by puny human legs (worst of both a human and horse!). Humans are supposed to be keeping vigilant for mistakes and errors, but they are not particularly good at this.

A reverse centaur with a massive horse head and puny human legs. Cannot run fast or think good.

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Fenner Tanswell @fennert.bsky.social · 30/10/2025

So. Never let your guard down.

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Fenner Tanswell @fennert.bsky.social · 30/10/2025

I was stuck on a computer game (Blue Prince, it's really good) that has a lot of puzzles that are quite tricky. I'd been playing with my son, and it was getting a bit frustrating not making progress.

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Fenner Tanswell @fennert.bsky.social · 28/08/2025

The new issue of Synthese emails bizarrely always load the first image from each article. Today, finally, this paid off.

A man with prominent buttocks feels the heat from a campfire using a cable of nerves that snakes up his leg, past said buttocks, along his back and into his brain covered in fuzzy hair. The man's expression is of stoned glee.

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Fenner Tanswell @fennert.bsky.social · 26/06/2025

@garymarcus.bsky.social got Imagen 4 to draw and label the parts of a bike. The image gets funnier the longer you look.

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Fenner Tanswell @fennert.bsky.social · 26/06/2025

A poster for our Symposium on Phil. Math. and Epistemology on the 11th July at the TU Berlin.

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Fenner Tanswell @fennert.bsky.social · 05/06/2025

Did you know I'm in a computer game? At least, my name is, but actually named after me! Here seen quizzing an unscrupulous politician. #TechnoBanter

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Fenner Tanswell @fennert.bsky.social · 28/03/2025

Would you let pharmaceutical companies do this to your kids?

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Fenner Tanswell @fennert.bsky.social · 24/03/2025

Found my wee book in a bookshop for the first time ever!

Fenner holding a copy of his book "Mathematical Rigour and Informal Proof" in a bookshop.

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Fenner Tanswell @fennert.bsky.social · 11/03/2025

Thanks Sara! That sounds amazing. In part, I was interested because I wrote a little bit about this before for rigour in maths.

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Fenner Tanswell @fennert.bsky.social · 24/02/2025

Over the weekend, I joined colleagues and students at the protest march against Berlin spending cuts. The TU Berlin buildings are in a horrific state of disrepair, yet the CDU-led government has reneged on vital funding for building and renovations.

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Fenner Tanswell @fennert.bsky.social · 15/01/2025

en.wikipedia.org/wiki/G._K._C...

A quote from wikipedia reading: ""Chesterton's fence" is the principle that reforms should not be made until the reasoning behind the existing state of affairs is understood. The quotation is from Chesterton's 1929 book, The Thing: Why I Am a Catholic, in the chapter, "The Drift from Domesticity":

In the matter of reforming things, as distinct from deforming them, there is one plain and simple principle; a principle which will probably be called a paradox. There exists in such a case a certain institution or law; let us say, for the sake of simplicity, a fence or gate erected across a road. The more modern type of reformer goes gaily up to it and says, "I don't see the use of this; let us clear it away." To which the more intelligent type of reformer will do well to answer: "If you don't see the use of it, I certainly won't let you clear it away. Go away and think. Then, when you can come back and tell me that you do see the use of it, I may allow you to destroy it.""

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Fenner Tanswell @fennert.bsky.social · 06/12/2024

Ultimately, we won't know the answer. The closest to textual correspondence are their comments on language dynamics and statics in maths.

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Fenner Tanswell @fennert.bsky.social · 06/12/2024

Even cooler, the archive contains a copy of Kneebone's papers with Lakatos's underlinings and marginalia. So we know that Lakatos read Kneebone's work.

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Fenner Tanswell @fennert.bsky.social · 06/12/2024

The Lakatos archives contain correspondence between Kneebone and Lakatos between 1962 and 1969, about parallel logic seminars, grading, chairing talks, and comments by Kneebone on Lakatos's 1962 paper "Infinite Regress and the Foundation of Mathematics".

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Fenner Tanswell @fennert.bsky.social · 06/12/2024

Amazingly, this paper outlines a whole "dialectical" philosophy of mathematics, as carried out by Lakatos in Proofs and Refutations, but a decade before Lakatos wrote this as his PhD. This already is an exciting find for the history of the mathematical practice turn.

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Fenner Tanswell @fennert.bsky.social · 25/10/2024

At some point in the recent past, I thought a journal information system was redundant as they already had my info, so didn't need to be filled out seriously.

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Fenner Tanswell @fennert.bsky.social · 08/11/2023

When you make the logic exercise set tricky, but then have to write up the model solutions yourself.

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Fenner Tanswell @fennert.bsky.social · 11/10/2023

If anyone would like to attend, you can ask Daniel Isaacson for the meeting link by email.

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Fenner Tanswell @fennert.bsky.social · 06/10/2023

Broadly we replicated existing findings that it's a bit of both. More interestingly, CJ gives very fine-grained information about people's relative judgements.

A graph displaying the weighted rankings for the three studies.

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Fenner Tanswell @fennert.bsky.social · 06/10/2023

Comparative judgement has participants making binary choices between prompts. With enough judgements from enough judges, you can form a weighted ranking of prompts, as well as measure agreement between participants.

An example of the binary decision prompting for our study. The question reads "A sign says "No Vehicles in the Park". Which example would be the worst violation?". The two options are "Ambulance" and "Helicopter".

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Fenner Tanswell @fennert.bsky.social · 07/09/2023

@impractknow.bsky.social seems to think this is where to philosophy is now, so here I am. To get started, a silly joke. Behold, the Platonic heaven:

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