Martin Escardo

Professor at the University of Birmingham, UK.
I am interested in constructive mathematics and (constructive and non-constructive) homotopy type theory and univalent foundations, connections of topology with computation, (infinity) topos theory, locale theory, domain theory, combinatorial game theory and much more.

See my meta-blog:
https://cs.bham.ac.uk/~mhe/blog.html

homepage: https://www.cs.bham.ac.uk/~mhe/
Pronouns: He/Him
Inclusivity: Full
Attempting: To get as green as possible and save the planet

Public Key

npub14a5755aglz3euvetuy8z7x25akaz42lgvrgkr8qj2a55ghpwaelqjf9fse

NIP-05 Address

[email protected]

Profile Code

nprofile1qqs276022w503gu7xv47zr30r92wmw32405xp5tpnsf9w62ytshwulspz3mhxue69uhhyetvv9ujuerpd46hxtnfduqs6amnwvaz7tmwdaejumr0dshdr5h7

Publishing to

relay.damus.io

nos.lol

Author Public Key

npub14a5755aglz3euvetuy8z7x25akaz42lgvrgkr8qj2a55ghpwaelqjf9fse

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Last Notes

2026-04-26 00:36:09 UTC

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writes "I thought Gödel's theorem was a shocking negative thing when I first learned about it, but then I grew up."
I was *told* it was a shocking negative result when I grew up.
@npub1lpm…vg96 @npub1x78…xn0m

2026-04-25 17:10:50 UTC

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My view of Gödel's theorem is this.
Consider the following axioms for a given universe of discourse,
We have an element 1, called the neutral element.
We have an operation (x,y) ↦xy called multiplication.
Moreover, we postulate that multiplication is associative and has 1 as a neutral element.
Are we surprised that there are many different universes that satisfy the above axioms (which are called the "group axioms"?
No, we are not.
But then suppose that somebody else said "let's axiomatize sets".
This time people may be more surprised. There isn't a unique notion of set satisfying the axioms of set theory.
Just like there isn't a unique group satisfying the group axioms.
Should you *really* be surprised in the case of sets?
I don't think so. If you give axioms for the gadget you imagine in your mind, it is likely that many other gadgets will satisfy the same axioms.
Why, in the first place, did you originally think that your axioms actually did specify a unique thing?
Gödel is just telling you that, just like the group axioms, which have many models, your axioms for set theory also have many models.
Why would you think that you would be so powerful to come up with axioms that would specify a unique thing?
Gödel's theorem is not a "restriction of what we can do".
It is actually a liberation. We can reason with the same axioms about very many different and interesting, universes of discourse, as topos theory shows.
Gödel's theorem is not a negative result. It is an invitation to explore new, interesting, mathematical worlds.

2026-04-09 20:35:48 UTC

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I humbly don't agree with this approach to communication here.
(And Chris changed her initial post to rectify this.)

2026-04-09 18:49:43 UTC

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Anyway, what I quoted is not necessarily what I think, although there is an element of truth behind it. If the voters don't think enough, they do end up with bad governments. And of course candidates exploit that.

2026-04-09 16:58:22 UTC

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I would prefer that the above were not attributed to me in your post, because I agree with you, and I never said anything in the contrary, or indeed at all about this. Thanks.
@npub1txh…c80s

2026-04-07 21:09:05 UTC

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Maybe I was implicitly alluding to the fact that some people in one particular country elect somebody, and then this person cause havoc in that country and all countries in the world.

2026-04-07 19:51:23 UTC

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“The government you elect is the government you deserve.”
― Thomas Jefferson
The problem is that the rest of the world elect this person.
I don't think the rest of the world deserves that.

2026-04-01 05:22:16 UTC

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It is absolutely fine not wanting to try it.
But then please stop saying things such as "I don't think formalizing mathematics at all makes us think the right way" or other claims about formalization.
If you want to speak with authority about formalization, then learn it first. And, yes, it is not easy or fast.

2026-03-31 17:22:12 UTC

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writes "But in fact I don't think formalizing mathematics at all makes us think the right way."
You should try it once to see by yourself. You are talking about something you've never done.
I say this because I also said the same thing as you said in the past.
And here is the first time I repent in public:
Computing an integer using a Grothendieck topos
https://math.andrej.com/2021/05/18/computing-an-integer-using-a-sheaf-topos/
@npub18a6…szuj @npub1vlc…mzqq [@JacquesC2](https://types.pl/@JacquesC2 ) [@pigworker](https://types.pl/@pigworker ) @npub1vzx…ncd7

2026-03-30 09:10:46 UTC

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I, on the other hand, like using Agda as a blackboard for thinking, just like I like using paper and pen for thinking.
I do believe that mathematics is inherently informal (and that ZFC, MLTT, ETCS etc. are not the ultimate answer to anything, but just distillation of what various people did informally in various different fronts).
This doesn't prevent me from both liking and finding formal mathematics with the aid of a computer useful (for myself - I am not advocating it would be useful for e.g. you).

2026-03-28 16:17:20 UTC

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No, I didn't ask, because I am pretty sure where this comes from, namely a paper in which I proved a particular case of my question to the AI.

2026-03-28 15:23:33 UTC

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In any case, I love the circularity, and the wrong answer being attributed to me, the very same person asking the question to the AI.

2026-03-28 15:08:44 UTC

Ha ha.
I was playing around with generative AI, mostly for entertainment, regarding an open question I have.
So I asked the question, and the genAI gave a "definitive" answer, crediting Escardo for the solution. 🙂
Which I don't have. 🙂
And nor does the AI.

2026-03-26 06:47:03 UTC

I have a calculator that is correct 80% of the time. But don't worry, every time I use it, I check the results myself.

2026-03-14 17:44:19 UTC

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I like the post but I don't like the picture. She is much more lovely in her Wikipedia page, not only in pictures but in her life.

2026-03-14 08:21:19 UTC

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writes "It's possible they are doing this because they believe Trump will continue his campaign against universities until 2028 at least, and they want to insulate the arXiv from Cornell."
If that's the case, it is crucial that they move away from Amazon servers, or any American corporate server.

2026-03-13 21:40:02 UTC

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Oh, no, I thought the arxiv was going to be for ever, at least for as long as we last in this planet.
First we lost the mirrors in Europe. And now this.
I am not optimistic about this move.
Who gets to decide about this?

2026-03-07 19:36:50 UTC

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This painting and your thread caused an interesting discussion with my 16-year old son and my wife, starting with my son asking "why do you have this painting open in your computer".

2026-03-07 18:11:21 UTC

Dear websites under a paywall,
I get hundreds of links to you from my friends here. I am obviously not going to subscribe to all of you to just read one link to you they send me on an odd day.
I understand that you must make profit to survive. But do you really think that everybody going to you wants to subscribe long term just to see a link a friend sent?
In any case, I don't care. If it is a one-off link sent to me under a paywall, I will just close your page immediately. I don't know you, and so I have no reason to subscribe to your publication, unless you tempt me to. But even if you try to tempt me, it will be difficult, because I won't subscribe to a 1000 things.

2026-02-22 19:06:20 UTC

Regarding my last message, the "v" versus "b" in Portuguese and Spanish is very annoying. There is no consistent rule for mutual departures.

2026-02-20 22:03:28 UTC

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I don't know. I am just learning.

2026-02-20 20:36:58 UTC

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A bonus of this trip is that I got a cousin of mine to explain modern inclusive language in Argentinian Spanish to me.
For example, instead of saying "todos o todas" you can say "todes", or "todxs" if you prefer.
I quite like the "x" version.
3/

2026-02-20 19:55:11 UTC

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Accents should be abolished, as much as possible. In particular, they prevent me from writing fast. Or perhaps, we could conjecture that accents are good, because if you have to write lots of accents, you will write less in the same amount of time.
2/

2026-02-20 19:54:50 UTC

My mother tongue is Spanish, and my native tongue is Portuguese.
And I have lived in an English speaking country since 1993, so 36 years, more than half of my life.
At home we speak mostly Portuguese - my wife is Brazilian, like myself, but our sons, born here, oscillate between the two when we meet.
I hardly speak Spanish in practice, other than with my parents from time to time, irregularly by video call, who are Argentinian, but moved to Brazil when they got married.
I was surprised to be able to speak fluently in Spanish in the last 7 days to lots of relatives who came from Argentina to Brazil to celebrate my mother's 80th birthday.
I was even more surprised I could actually also write in Spanish in the various WhatsApp chats people set for that event.
I did live for three years in Argentina and went to school there, from 11 to 13 years of age. But that was all.
One thing I find very annoying is that, mathematically speaking, the rules for writing accents in Spanish are suboptimal.
Almost every word gets an accent in Spanish, at least in the way it is spoken and written in Argentina. Portuguese is much more efficient in this respect. The rules minimize the need for accents. Still, I think that accents in both languages are mostly unnecessary.
One rather important exception, in Portuguese, is "e" and "é". Not only are they pronounced differently, but also the former means "and" and the latter means "is", which can indeed cause unwanted ambiguities in practice. So we definitely don't want to drop that.
1/

2026-02-16 02:47:51 UTC

I'm about to be back.
Back home, and back online.
Fun fact. Lufthansa redirects me to LATAM for check-in, and LATAM redirects me back to Lufthansa, in an infinite loop.
Less fun fact. I was supposed to arrive back home at 12:00, but they changed the arrival time to 22:45, and I have to teach early the day after. To add insult to injury, they gave me a 15 Euro voucher as "compensation".
Anyway, I have the luxury of going to the beach 7-9am before going to the airport.
So all is good. Sort of. See you all soon here.

2026-02-04 23:11:19 UTC

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writes "When disable quote-replies it means I can't publicize their quotes with an introductory explanation, and for more technical posts this often means I don't publicize the post at all."
Fair.
Even better, it seems that the bad quote, in my case, has been removed in the mean time.
As I said, I don't blame whoever did this. It was probably without thinking enough, and they backtracked, which makes me happy.
I can't see the post I was indirectly complaining about any more, which is good.

2026-02-04 20:40:47 UTC

Oh. I will disable quote replies.
Somebody quoted me out of context by saying "cyclists of any kind are horrid left-wing people".
But actually *I* was quoting somebody else (British newspapers).
[@FediTips](https://social.growyourown.services/@FediTips ) I invoke your knowledge to help me disable quote replies like this - Thanks!

2026-02-01 22:31:22 UTC

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I haven't. Do you recommend it?
@npub1nf4…nqe4

2026-02-01 22:03:51 UTC

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But the real question is not whether she is reading it. Did she ask you about the Banach-Tarski Paradox?

2026-02-01 21:58:37 UTC

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Small world. 🙂

2026-02-01 20:48:29 UTC

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I should read this book!

2026-02-01 20:48:14 UTC

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Now she is querying me about the prisoner's dilemma, in the next pages of the book.

2026-02-01 20:44:16 UTC

My wife is reading Katabasis next to me. Then she came to the Banach-Tarski paradox. She asked me whether this is a real thing.

2026-01-24 16:38:50 UTC

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Or lots more plants to fill the void. (And give you physical work watering them.)
[@jer_gib](https://functional.cafe/@jer_gib ) @npub1j8e…ncgu

2026-01-21 09:03:38 UTC

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writes "though he may be tired of talking about it."
🙂
Lean is better for classical mathematics, both because of its large mathlib and its community.
Agda is better for other things, in my case (constructive) homotopy type theory.
You can define tactics in Agda, but generally people prefer to write terms.
Then there is Rocq, which is good for both, and does have tactics.
Agda is based on Martin-Löf type theory, Rocq on the Calculus of Inductive Constructions, and Lean on something similar to Rocq (you would have to check the literature for precise details).
I hope this helps. Other people can say more.
@npub1z33…rwxg

2026-01-15 18:14:09 UTC

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(Minor remark: "their" is ambiguous for people reading your post casually.)

2026-01-11 22:20:13 UTC

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Amazing!

2026-01-09 23:05:00 UTC

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Isn't this an instance of the fact that ideas tend to be discovered simultaneously by many people at about the same time, but some particular people push them further?
(For example, this happened regarding Darwin with evolution. But even the piano was invented twice at the same time independently, as an improvement or modification of previously existing keyboard instruments.)

2025-12-30 23:59:54 UTC

A short post about constructive mathematics in the way I see it.
I am interested in constructive mathematics, which I see as a generalization, rather than as a restriction, of classical mathematics.
In constructive mathematics, in the way I conceive it, computation is a side-effect, rather than its foundation.
What distinguishes classical and constructive mathematics is that the latter is better equipped to explicitly indicate the amount of information given by its definitions, theorems and proofs, which is related to topology and domain theory.
This is particularly the case for Martin-Löf type theory, and even more so for its univalent extensions, such as Book HoTT (homotopy type theory) and Cubical Type Theory.
Constructivity is not a binary notion, but a matter of degree. The correct question is not whether a mathematical theorem is constructive, but instead how much information its formulation (type) and proof (inhabitant of the type) give.
A good foundation has a rich supply of types and ways to build their inhabitants so that the available amount of information can be precisely expressed.

2025-12-29 15:43:44 UTC

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writes "I just installed @npub1rk6…hv4q "
Nice.
A couple of months ago my 15-year old son installed nextcloud for me in our Linux server at home.
This was mostly for being able to use Saber [1] to write notes by hand on both Android and iPad tablets, and having the notes visible in all computers, at home and at university.
It works like a charm. I can write notes at university in the Android tablet I keep there, and then continue writing at home in my iPad. And I can read the notes from my laptop.
It just works, thanks to both nextcloud and Saber. Thanks both!
@npub1rk6…hv4q
[1] https://saber.adil.hanney.org/
And both are free, and also rather professional. More professional than e.g. what Apple provides on an iPad or a Mac.

2025-12-24 18:46:33 UTC

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writes "We give the notes letter names. [...] A,B,C,D,E,F,G"
Thanks for the nice thread.
I think I've already said this here a few years ago. I find it very difficult to get used to use "A,B,C,D,E,F,G" automatically without thinking. I grew up playing musical instruments with "do, re, mi, fa, sol, la, si" corresponding to "C,D,E,F,G,A,B".
Question. I know the origins of the two notational systems. But why does one start with "A/la" and the other with "do/C"?
You kind of mention this in (5/n).
But when did this actually occur?

2025-12-13 20:10:10 UTC

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quoted Barenboim saying "I've never met any child prodigies, but I did meet many of their parents."
Oh, this happens so often in my experience. But I don't fault their parents. They are not experts. They are just trying. If non-experts parents don't try, a lot of talent will be missed (which it is).
In any case, although I love Barenboim, I love Martha Argerich even better (also Argentinian).

2025-12-07 16:12:37 UTC

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writes "But I've never been interested in solving precisely stated hard problems. To me, the interesting part of math is finding beautiful structures that help us understand reality better - and even more importantly, clearly explaining these structures"
+1.
Yes, understanding is what entices me to do mathematics. Most theorems I prove (and definitions I come up with) are to further my own understanding and to try to explain it to other people.

2025-12-07 13:54:04 UTC

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asks "do you have to supervise students who work on AI even if you don't really enjoy it?".
No, I don't.

2025-12-06 19:09:53 UTC

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And lastly
“What I tell students is that, if what you’re trying to optimize publishing papers, you know, it’s actually honestly not that hard to do. Just do really crappy low-quality work and bomb conferences with it. But if you want to do really thoughtful, careful work, you’re at a disadvantage because you’re effectively unilaterally disarmed”
It is worth reading the whole article (linked in the leading post of this thread).

2025-12-06 19:07:55 UTC

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Continuing:
"Meanwhile, says Farid, major tech companies and small AI safety organisations now dump their work on arXiv, a site once reserved for little-viewed preprints of math and physics papers, flooding the internet with work that is presented as science – but is not subject to review standards.
The cost of this, says Farid, is that it is almost impossible to know what’s actually going on in AI – for journalists, the public, and even experts in the field: “You have no chance, no chance as an average reader to try to understand what is going on in the scientific literature. Your signal-to-noise ratio is basically one. I can barely go to these conferences and figure out what the hell is going on."

2025-12-06 18:58:55 UTC

This year, a lot of my final-year project students decided to work on AI. Of course I am disappointed.
Then I quote
"Farid says that he now counsels students to not go into AI research, because of the “frenzy” in the field and the large volume of low-quality work being put out by people hoping to better their career prospects.
It’s just a mess. You can’t keep up, you can’t publish, you can’t do good work, you can’t be thoughtful"
https://www.theguardian.com/technology/2025/dec/06/ai-research-papers

2025-11-30 16:02:00 UTC

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It seems to me that this is something about Anglo-Saxon culture.
Not liking maths, not getting it, being embarrassed if you like it or if you are good at it.
Is this a cultural or an educational phenomenon?
Even worse, I find, is when people make choices, about any subject, including maths, on the basis of whether they "are good at it".
I hate this expression, which is repeated over and over again. You are not innately good at anything. You get good by doing it with love (and hence persistency).
The right question is not whether I am good at it, but instead whether I love it.
But I sense that both the education system and culture are playing against this, although I don't know for sure.

2025-11-26 22:00:45 UTC

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I will try this coming weekend.

2025-11-26 21:39:52 UTC

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I will try, then! (And may report back.)

2025-11-26 21:14:59 UTC

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Is that suitable for interested 15-year olds? I am asking just in case I have the option of planning to watch all this with my son.

2025-11-26 18:52:26 UTC

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writes "you may or may not have seen that the latest episode of my Standard Model videos"
I haven't yet, but it is in my queue.
You also write "[...] for a reason! I don't want to give it away, but the Standard Model matter particles seem to be described by 5 qubits."
So are you saying that the world is a big quantum computer simulating our classical computers? 🙂
Anyway, looking forward to watching this.

2025-11-26 17:55:59 UTC

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Why are there some many powers of two arising in this discussion? I thought powers of two were reserved for computer science. 🙂

2025-11-24 10:47:55 UTC

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I had read both the ZX81 manual and Topology via Logic, but I hadn't realized they were written by the same person, until 1993 when I started as a PhD student at Imperial (Steve was there at that time) and saw the two books next to each other in a stand showcasing the books written by Imperial academics.
@npub1x78…xn0m

2025-11-24 10:36:14 UTC

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Yes, the same one!
@npub1x78…xn0m

2025-10-21 21:08:37 UTC

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writes " we are actually surprisingly unafraid of other humans, able to crowd into subway cars and trains without panicking."
Yes, in certain environments, like the ones you mention, but certainly not all, and certainly very little, and mostly for privileged people only.

2025-10-21 20:42:56 UTC

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I am sure that if you walk in certain parts of the UK or the US, and particularly at certain times of the day, you will be very afraid of humans.

2025-10-21 20:34:07 UTC

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Even humans are afraid of humans. And they should be.

2025-10-12 13:15:41 UTC

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Previously, I supported the idea of you explaining the Standard Model to a general academic audience, which I still do.
But, now, given what you say, I would additionally like an explanation of what Dark Matter is. Or, better, what the dark-matter question is for (mathematical) physicists.

2025-10-07 21:19:24 UTC

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Standard model.

2025-09-26 18:00:28 UTC

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I thought you were primarily a physicist? 🙂
@npub1sq8…xvpd

2025-09-26 14:37:46 UTC

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Edinburgh musings.

2025-09-22 15:17:45 UTC

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Yeah, I found this shocking even in London when I moved there to do my PhD. Then I got a job in Edinburgh, and this was even more shocking.
And you forgot to mention that, in Edinburgh, the birds start to make loud noises at 2am, to try to wake you up as early as possible. 🙂

2025-09-22 14:50:17 UTC

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I agree 100%.
In particular, I love teaching, at all levels, beginner, medium, advanced. It is just the unnecessary bureaucracy that bothers me and requires additional time.
And the excessively large number of students that universities in the UK need to take in order to survive financially.
But I wouldn't be doing this unless I loved it.

2025-09-21 18:11:43 UTC

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writes "It's Sunday, time for idle speculations:"
It would be nice if every day we had time for idle speculations, and then developing them or discarding them.
The current system uses mass teaching to fund research.
I do love teaching. But I hate the bureaucracy that comes with it (and so do the students).

2025-09-14 13:40:11 UTC

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I wrote a thread here and then somebody wrote webpage somewhere, with no citation to my thread.

2025-09-12 15:30:45 UTC

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Or I could tell the students that the mistake was to commit no mistake.

2025-09-12 15:21:45 UTC

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Yes.

2025-09-12 15:19:50 UTC

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Also, I often prepare a mistake for the lecture, and ask the students to identify it. Sometimes they actually catch an actual, unprepared mistake. :-)

2025-09-12 15:18:18 UTC

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We already do this in Functional Programming. The lecture notes are distributed via an internal gitlab. The student who does the most pull requests fixing the notes gets a prize. It is usually a Haskell mug, but it varies.

2025-09-10 16:18:08 UTC

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That would be a good symbol (instead of equality) to denote the identity type in homotopy type theory.

2025-09-08 19:53:51 UTC

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writes "Historians argue about history all the time."
Don't physicists argue about physics all the time, and mathematicians argue about mathematics all the time, etc.?
🙂

2025-08-28 23:18:55 UTC

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Also, can't we just stare at the beauty of this result, and many other results in mathematics, without guilt?

2025-08-26 21:50:31 UTC

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The problem is that
(1) packages depend on thousands of packages,
(2) people haven't learned, in theory or in practice, how to make "improvements" to a particular package without breaking all the packages that depend on it,
(3) people don't seem to care about this problem, in theory or in practice.

2025-08-26 21:31:55 UTC

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This is true, but I don't think this is the problem.
The problem is that, even if there was an infinite amount money available, people are interested in particular, exciting things, and so after cleverref (which is clever) is implemented, nobody is interested in the bureaucracy of keeping it working after rogue modifications in the ecosystem, particularly if the thing that breaks cleveref happens a decade after it was implemented.

2025-08-26 21:15:01 UTC

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So I guess cleverref was turned into dumbref after your remarks.
So here is the protocol:
(1) Implement something I find useful for myself.
(2) Make it publicly available to everybody.
(3) Lots of people adopt it.
(4) It is unmaintained because it just works, for years and years.
(5) But then suddenly the platform that supports it makes a change, and then it breaks it.
(6) The person who created it is not around any more. And nobody was appointed to maintain it. So there is no pathway to fix it.
(7) Everybody using it is on their own.
As you all know, this is not particular to cleveref. It happens all the time.
It happens even with packages used by industry (Google, Apple, etc.). Many such packages.

2025-08-26 18:14:54 UTC

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What Tom and I found was https://tex.stackexchange.com/questions/730148/cref-refers-to-lemmas-as-theorems
(I saved it as a comment in the latex file!)
I don't know about your fix, but our fix doesn't break files for people who use older versions of texlive.
This is important in our case, because I have the latest version and Tom has an older one, and we are working on joint projects.
You need to add
```
\usepackage{thmtools}
\renewcommand{\cref}{\Cref}
```
I think that's all, according to the comments we wrote in our latex file.

2025-08-26 16:43:06 UTC

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I confirm it broke for me as well when I updated to the latest version of texlive.
But [@de_Jong_Tom](https://mathstodon.xyz/@de_Jong_Tom ) found a fix by changing something in our latex file.

2025-08-21 20:57:07 UTC

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(But I like your (well-known) joke, too.)

2025-08-21 20:54:06 UTC

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(I wasn't suggesting to name it as the discoverer, but instead to like the name of the discoverer better than the law, as a joke, if you read my post. 👨‍🦲 )

2025-08-21 20:14:21 UTC

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One can be tempted to like the name of the discoverer, in unabbreviated form, better than the law discovered by them 🙂

2025-08-12 18:41:27 UTC

Mathematicians: What are the important *definitions* in your field?
For example, in topology the notions of compactness and of Hausdorff space are crucial (among many others).
Or in complexity theory the notion of NP-completeness.
And so on.
Can you list the ones in your mathematical field?
Non-practicing-mathematicians: this time just listen what the practicing mathematicians say.
This is an exercise, for me, in learning which concepts working mathematicians of fields different from mine consider to be important for their own field.
PS. I will consider Theoretical Computer Scientists and Theoretical Physicists for this question. And I am also willing to be more general, without any specific example in mind.

2025-08-06 14:16:08 UTC

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The analogy with automatic transmission is not very good, as it generally works safely, whereas "vibe coding" with genAI doesn't generally work safely, and often fails badly (with many examples reported by the press and some in court for releasing private information as a result of not implementing any security measure, for example).

2025-08-05 18:33:18 UTC

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Yes, I agree with explaining things intuitively first. Intuition does come first.
In fact, today I spent the whole afternoon doing that with somebody.
But at some point, if you want to answer a tricky question, you have to get rigorous.
And this did happen to me this afternoon too, when the person in question asked "why" to an assertion I made and didn't feel intuitive to them.
I love this take on what we are discussing here, by Terry Tao:
https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/
(And I have used it in lectures about formal reasoning.)

2025-08-05 18:00:15 UTC

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I think the essence of the problem is that students (and sometimes teachers) are not told what real numbers are, explicitly by a construction or axiomatically by a list of properties they are expected to obey.
If the thing you are discussing is not well defined, then one can expect any answer to all sort of questions, including the one under discussion.

2025-08-05 17:01:16 UTC

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writes " Yes, they say the difference is 0.000...001, where the string of zeroes is infinitely long."
How do they explain an infinitely long string of zeros that *ends* (with zeros as well)? Have they thought about this, or have they written "..." without thinking what this could mean in this case?

2025-08-05 16:41:17 UTC

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Have you tried to ask them to compute the subtraction 1.000... - 0.999... and tell you what the result is?
And can they write this in decimal, if they think numbers are decimal expansions?
Am I curious to know what their answers to these two questions would be.

2025-08-05 16:33:36 UTC

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Start with a cake, and in the first day eat 9/10 of it.
Then in the second day, eat 9/10 of what is left.
Keep doing this forever.
How much cake is left after all "natural" days 1,2,3,4,5,...?
The same amount as if you had eaten the whole cake on day 1.

2025-07-07 19:40:02 UTC

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The behaviour I currently get is that I get lots of little notification bundles, not a single notification bundle, for favourites and boosts of my posts.

2025-07-07 19:31:50 UTC

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I fail to find this option on mathstodon (web interface). Where is it? Thanks.

2025-07-06 16:59:06 UTC

Academics, including myself, should create an equivalent of the arxiv for git.
I am not sure how long it will take, but gihub will certainly go the google+ and jamboard way, by discontinuing things that are no longer interesting or profitable to Microsoft (or, even worse, acceptable by whatever US administration decides to pressure Microsoft).
While there are already alternatives, I don't (at least not yet) trust any of them.
We need something made by academics for academics, just like the arxiv, where there is some hope of long-term persistence.
For example, I care about the future of TypeTopology, even after I die, but there is no hope for this in the hands of Microsoft, or any alternative I have seen.

2025-06-27 16:57:50 UTC

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This is many more than the ones we have to take here in Birmingham.
But we have some you are not listing:
* Information security.
* Data protection.
* Exporting technology and know how that has potential military applications.

2025-06-27 14:53:57 UTC

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writes 'they're trying to redefine "equals"'
Correction. They are not.
Equality is not redefined in HoTT/UF.
In type theory, the notion of equality is given by the identity type, and this existed way before before HoTT did.
It turns out that the univalence principle for this identity type is *independent*. You can't prove or disprove it.
HoTT/UF resolves this independence by postulating it. Voevodsky's motivation for this is that it is true in the simplicial-set model of type theory, which was his intended model of type theory.
The nice thing is that there is only one equality which captures all the equalities that people use
Equality of groups is equivalent to group isomorphism. But also, equality of two subgroups of an ambient group is *not* isomorphism. It is "having the same elements", which is what we want.
Equality of topological spaces is equivalent to homeomorphism.
Equality of metric spaces is equivalent to isometry.
Equality of 1-categories is equivalent to the usual notion of equivalence of categories.
Equality of whatever mathematical object you build in type theory is typically what a mathematician would expect.
And these are not definitions. They are theorems that follow from the univalence axiom.

2025-06-24 18:47:29 UTC

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says "A lot of people I talk to think they're intelligent. "
I don't know. I don't like intelligence contests or beauty contents.
What do I like is the excitement of thinking about things and finding new things or understanding what other people found in their own excitement.
I don't think you can measure (or even define) intelligence as you can measure money, and, in particular, "twice as much intelligent" is meaningless.
I think that, for example, the great results in mathematics are driven more by curiosity and persistency than by intelligence.
Of course intelligence (whatever it is) helps a lot. But without curiosity and persistency, it is useless.
Also, there is another rather important factor. Some people just can't stop thinking. This also helps.

2025-06-15 17:21:43 UTC

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There is no contradiction of this with what I said in other posts in this thread.

2025-06-15 17:00:02 UTC

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writes "okay, that's a fine solution: then you just need to change the math curriculum so that students learn enough type theory to know that there are different types of 0."
I don't think it will harm students if they are told that a function comes with a domain and a codomain, as you well know as a category theorist.
There are several exponential functions, including at least the following three:
ℕ × ℕ → ℕ (arises in combinatorics)
ℝ × ℕ → ℝ (arises in Taylor series and the binomial theorem)
ℝ × ℝ → ℝ (wants to arise in analysis, but doesn't really make sense)
As I said before, it is kind of a coincidence that we have an embedding ℕ → ℝ of counting numbers (where the count can be zero) into the real line, mapping "counts" to "points" or "positions" in the line.
It is a *useful* coincidence, but we should avoid being misled by it.
When we write x^n in a Taylor series or in the binomial theorem, clearly x and n have different roles: (1) x is real number, (2) and n counts how many times we differentiated a function in the case of Taylor series (figure out yourself what n is counting in the case of the binomial theorem).
But also I want to say this. A lot of undergrad and PhD students in both mathematics and computer science are well ahead of their academic supervisors, and taking up computer formalization without their permission or blessing. Such people are very likely to be aware of the distinctions I have made above, by necessity, rather than philosophy.
The proof assistant doesn't allow you to pretend that there is only one zero (the natural number, the real number, the ordinal, the cardinal, the initial object of a category, and so forth).
1/2

2025-06-15 06:27:45 UTC

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This is again conflating 0 of different types. When n is a natural number and an exponent and zero, then the exponential should be one. It is x^y which doesn't make sense, with x and y *real* numbers.

2025-06-14 16:09:36 UTC

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Exactly. If you want m^n to count the number of functions from a set of cardinality n to a set of cardinality m, then there is no other option but to have 0^0=1.

2025-06-14 16:03:45 UTC

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"My least favourite version is when they give the definition for 0 and say that it's 'by convention'"
Yes. I remember when I was a university student, I was upset a lecturer said that 0!=1 by convention.
Of course it isn't by convention. There is exactly one permutation of the empty list, namely the identity.
If you want n! to count the number of permutations of a list of length n, then there is no other option but to have 0!=1.

2025-06-02 18:36:04 UTC

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The thing with crackpots is that they often give arguments that "are not even wrong".
They don't even make sense.
(And, in particular, there is no self-consistency as you claim there would be.)
I don't know, @npub1nf4…nqe4 , if this criterion is in your crackpot index.