Boarders

Interested in mathematics (homotopy theory, category theory, topos theory), programming languages, and philosophy.

Site: https://boarders.github.io
Locations: NYC
Path: What hope is there in a book about a book

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Last Notes

2026-09-09 17:32:46 GMT

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overall I think people who work towards problems for human understanding should similarly be feeling good

2026-06-16 22:02:30 GMT

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The purpose of a system is what it does

2026-06-16 20:15:07 GMT

mathematicians largely refuse to teach their students inference rules of logic and then complain that none of the students know what it means to disprove a universally quantified statement, nor what it means to prove a negation. At some point complaining about young people might involve asking who raised them

2026-05-29 20:17:33 GMT

“Even within the instrumental logic of capitalism, it is at least sometimes recognized that giving people adequate time and resources, and a measure of freedom from the obligation constantly to justify their existence – leaving them alone, basically – can be beneficial to their performance or ‘productivity’”
https://newleftreview.org/sidecar/posts/irreversible?pc=1767

2026-05-09 04:12:40 GMT

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could you say what it is about the latter that makes it so?

2026-01-09 17:00:25 GMT

The corporate politics of procurement seems to me one of the most interesting under-discussed questions today, especially in the context of universities or schools. There are so many pieces one reads berating academic labour to get serious about future finances; about the future viability of the liberal university, but the follow up is never to ask who is in charge of procurement of e-learning software, never how much it is costing, never whether the department(s) responsible for this collection of choices are ‘living in the real world’. In my experience, many of the same trends re-occur through a glass darkly in other corporate environments where a product manager will push to pay millions of dollars per year for a certain service, but won’t hire a single extra engineer to work on an internally developed custom version of a similar service.

2025-12-22 17:31:03 GMT

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this is from Granström’s book Treatise On Intuitionistic Type Theory which situates Lebnitz’ project in a broader movement towards a universal language ‘for logic’ (in the ancient sense of the word). The book makes the case (I can’t say whether successfully or otherwise yet) that ML type theory should be considered the modern successor to this project

2025-12-21 17:07:58 GMT

“A lot of present day work in computer science has some connection with the lingua characteristica and it is a curious fact that, in the past 50 years, computer science, as a subject, has had more impact on the lingua characteristica than logic, mathematics, and philosophy together. The reason seems to be that a certain amount of realism is, so to say, forced upon the computer scientist and this realism is, as it seems, an important ingredient in the lingua characteristica. This is why few implementations of mathematics on computer are based on Zermelo-Frankel set theory, despite it presently being the most popular foundation of mathematics.”

2025-09-26 14:02:41 GMT

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not totally serious but “Maxwell’s demons”

2025-05-28 19:01:01 GMT

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Merchants of Doubt, for example, is just a straightforward history of widespread corporate conspiracies that we know to be 100% true

2025-05-08 17:01:02 GMT

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GHC notes are a good example too of the kinds of many-to-many source relationships large projects eventually want to be able to communicate

2025-03-28 00:55:59 GMT

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Quine was also against higher order logic due to his very esoteric ideas about what grounds the meaning of the logical quantifiers (still held as serious for reasons I can’t grasp) - a view which I believe will come to be seen as more and more incorrect over time

2025-03-19 02:08:26 GMT

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I’d be interested to hear why? I feel like lenses (+ other optics) are really conceptually amazing as a way of thinking, but probably encourage a certain kind of Java mindset of needless ceremony and needless busywork

2025-03-11 04:15:34 GMT

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I would love to know if this is somehow caused by the relationship between n-type and h-level in HoTT where there is a mismatch of size 2

2025-03-07 20:03:36 GMT

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I hope you write a book (or several) one day :))

2025-03-05 17:35:59 GMT

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that should perhaps be a warning about how we shouldn’t so easily go along with provocateurs simply because we happen to like their target - perhaps especially so

2025-03-05 00:59:46 GMT

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she also just constantly slanders her colleagues for clicks and encourages conspiracy thinking that scientists are mostly taxpayer leaches. She mostly seems to possess all the nuance of a boulder when it comes to almost all controversies she decides to weigh in on.

2025-02-26 02:49:48 GMT

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off topic: did you come up with any interesting ways to approach calculus or that level or is it not worth swimming upstream in such a setting?

2025-01-10 04:41:15 GMT

“At this historic moment—when the ascendancy of Europe is so rapidly coming to an end, when Asia is swelling with resurrected life, and the theme of the twentieth century seems destined to be an all-embracing conflict between the East and the West—the provincialism of our traditional histories, which began with Greece and summed up Asia in a line, has become no merely academic error, but a possibly fatal failure of perspective and intelligence. The future faces into the Pacific, and understanding must follow it there.”

2024-12-17 19:49:26 GMT

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mathematicians mostly “program” against abstract interfaces, they just need a kind of first order local set theory with sorts for the natural numbers, real numbers, set theoretic operations etc. Hardly any use the ZFC implementations.

2024-11-30 18:35:13 GMT

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saw this via @npub1ste…j70v
https://media.mathstodon.xyz/media_attachments/files/113/573/344/605/101/804/original/ae188aa7636c27f7.jpeg

2024-11-15 01:19:54 GMT

The comparison of applied category theory with cybernetics would have been more accurate if the cat theorists had managed to sell their ideas to at least management consultants, there is still time

2024-11-09 03:04:14 GMT

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I think you will find looking at the exit polling data about how registered republicans voted very sobering

2024-11-08 20:47:53 GMT

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I do hope, in the short term, that the Democrats realize they cannot win in a battle of oligarchs (with their own oligarchs) and citizens united was a total disaster that they need to actually talk about again. Nadar-ist consumer protectionism is also something they never think about even as they had Lina Khan and the best FTC in years, their leadership seems incapable of finding winnng messages - they stopped calling JD Vance weird as some strategist apparently said it was “too negative”.
I’m not delusional enough to think this stuff would necessarily work in our current propaganda environment, but I do think it should be tried as this Cheney family stuff has totally failed on its own terms - registered republicans just do not vote democrat

2024-11-06 23:03:10 GMT

I don’t think these results are explained by policies, that the democrats should move left or right or not run as republican-lite or that the campaign made a strategic error or etc etc.
This essay is a better starting point for me: https://www.weareplanc.org/2014/04/we-are-all-very-anxious/
Having a child now means having to set up a 529 college savings account from day 0 and immediately thinking about getting your kid into a top college. Immediately thinking of the costs of playing sports, the cost of summer camp etc. This might be the elite version of this psychosis, but it is impossible to miss.
Being a teen now means facing intense, unbearable competition, the overwhelming sense that you need to sprint to stand still, that you need AP classes, extra curricula’s, 4.0 GPA - I’ve even seen teens trying to become coauthors on academic publications…
Having a job now means facing the constant sense that you are near obsolete, that you need to change company every few years, that you need to expand your outdated skill set - the old fantasy of a job for life is just that.
Being a non-college educated male right now (I have witnessed first hand in my family) means constant FOMO that you’re not getting in on some stock market scam, crypto scam, supplements scam, properties scam etc etc.
We have a culture of unbearable, overwhelming anxiety about every aspect of your life, whether you will have a place to live, a stable job, a worthwhile degree, be able to afford groceries, whether you will get to have friends, endless rat races, and it is causing all different kinds of people to have all sorts of fight or flight responses. Algorithmic, video-based social media is parasitic on all this anxiety and is eating us alive.

2024-10-29 02:09:38 GMT

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the stuff I had in mind is what Mike Stay was interested in at me point (bicategories constructed out of lambda calculus) just in case it isn’t wasn’t very clear

2024-10-29 00:33:18 GMT

If one forms the bicategory with objects given by the types of a simple type theory, morphisms given by (appropriate) terms, and 2-morphisms given by rewrite rules (I’m not sure if that is even meaningful on its face) or perhaps, more sensibly, given by equations in the associated equational theory (so it has beta, eta, congruence rules etc.) then does this satisfy a nice universal property still?
The associated category which we get when we take connected components of each morphism category should be the free Cartesian closed category. I seem to recall this was discussed years ago on the n-category café but I am unsure if they arrived at a satisfactory answer (cc. @npub1nf4…nqe4)

2024-10-24 20:25:54 GMT

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I never understood in this line of reasoning* - how someone would possibly go from any fact about our best physical theories perhaps making a god of Einstein/Spinoza more or less plausible to Christianity - if anything the “divinely inspired” books not mentioning these facets of reality should decrease my belief in that particular creed (as should many other things about it)
*obviously I actually do understand that it is just motivated reasoning and wishing to have a certain kind of (unearned) intellectual credibility

2024-10-23 00:50:27 GMT

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there is a lot of people including Hossenfelder, Brian Keating, Horgan, sometimes Woit.
I have no problem with people holding that empiricism is the highest and final standard in physics. I also don’t have a problem with them heavily criticising absurd hype. I have a problem with them being anti-intellectuals, with them doing bad amateur philosophy. For example, hossenfelder has repeatedly said that there is no reason to expect the universe to conform to any mathematical principles and taking inspiration from mathematics is ridiculous, has said that there is no reason for physical constants to have “special values”, and seems to be a kind of nihilist on us ever going beyond the standard model. This is just ahistorical Gnosticism - almost all theoretical physics goes against the view and it is not sensible to bring up just because you want to kick string theory for not paying off.

2024-10-23 00:32:17 GMT

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personally I am frustrated by the entire crew of online people who have made a shtick out of calling modern physics (particularly people doing anything adjacent to string theory) completely unreasonable and anti-scientific etc.

2024-10-09 16:40:57 GMT

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what a truly gibberish opinion that is. I am no great fan of Chomsky, but this is a totally embarrassing opinion - LLMs only give more fuel to Chomsky’s primary criticism of behaviourism about the poverty of stimulus. Also, why would interest in the deep structure of language be a somehow suspect - our innate ability to recognize the grammaticality of sentences remains something that should give awe. It confirms to me that machine learning people are vapid, anti-intellectual, and overly impressed by their own techniques.

2024-10-06 00:14:28 GMT

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I’m sadly not sure, I think I recall hearing something like that and that it is implemented in maple, mathematica etc. but also that it is wildly complicated

2024-10-05 23:13:20 GMT

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differential Galois theory aims to gives an answer to the question of which functions have an anti-derivative in terms of elementary functions i.e. which programs have normal forms

2024-10-05 23:10:26 GMT

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high school integration: axioms are the antiderivatives of elementary functions plus theory of equality, arithmetic etc., There are rules of integration by parts, linearity and substitution, the theorems are all equalities of integrals in terms of other terms in the language. One is particularly interested in normal forms which are terms which no longer involve the integral sign

2024-10-03 17:30:50 GMT

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I think old scientific theories, such as the pre-Socratic theories of the world as made up of fire/water/earth, or early enlightenment theories of the corpuscular nature of light, or the phlogiston theory of fire, or the theory of humours etc. give us certain ontologies of the world which now seem naive and laughable (as ours might to future people).
Still, I think studying these historically as early forms of systemic theorizing is worthwhile and I am sure even the theory of humours is in some ways deeply connected to how we think about the modern world.
Theories of the self are not so easily graspable by current empirical methods and so we should also expect to be better served by studying historical conceptions since the question of what it is to be a person remains evergreen.

2024-10-03 16:17:01 GMT

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that’s fascinating and you’re completely correct that I should be less dismissive (that wording was inspired by reading a Russell essay on it which whilst beautifully written seemed to go nowhere)

2024-10-03 15:03:48 GMT

Platonism vs nominalism: an old fashioned debate in philosophy is that of universals - talking of triangle as a property commits oneself to the existence of the abstract concept of triangle or perhaps a “generic triangle”.
In this case, we can, to some extent, resolve the question mathematically in a precise way - there is a mathematical object called the moduli stack of triangles, the generic triangle is the generic point of this moduli stack. We can answer various precise questions about this generic triangle such as what automorphism group it has etc. It also specializes to various more particular triangles with a precise notion of specialize due to algebraic geometers. It seems to me this is a better formal apparatus for thinking about mathematical universals than any kind of vague waffle available, but the idea has not managed to escape the purview of mathematicians.

2024-09-19 01:23:45 GMT

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my point was just that generic elements of the categorical product are given by pairs of generic elements and this fact means “Cartesian” is extremely appropriate (though I am not generally a fan of eponymous concepts, but neither do I see sense in top-down naming schemes since from where to do we look downwards from to take such a perspective)

2024-09-19 01:19:52 GMT

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maybe I’m just confused about what “given by pairs” means, unless it is just means that it is given by pairs of elements in some underlying set or something like that?

2024-09-19 01:11:28 GMT

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I re-read a couple of times, but no wiser as to what I have misinterpreted unfortunately

2024-09-18 15:08:47 GMT

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Cartesian product is given by pairs of generic elements, thinking it should be given by pairs of global elements is just a misintuition