renyicircle

That makes perfect sense. Using language in unorthodox ways to convey very specific concepts that only make sense to you and sound like bullshit to others.

The Claude output in the first one-shot counterexample attempt is hilarious. I hate its writing most of the time but this stuff is next level deep-fried slop.

> And the control column confirms the resonance-necessity conjecture empirically: break the skeleton alignment and the joint kernel dies at the constrained window, exactly as the transversality heuristic predicted.

> The den has air in it.

> Drift fuel exists.

Thanks, that sounds interesting. As I said in a parallel comment after some thought on this, nobody can really say with absolute certainty that a particular branch will forever stay useless, so that should be good enough to keep going.

Sometimes one can be convinced that their mathematics will be useful, too. I've just recalled a rather motivating (though his life story is rather sad) quote by Grassmann, the inventor of modern linear algebra, whose mathematical work was not appreciated during his lifetime.

"I remain completely confident that the labour I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development of science, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit. I know that if I also fail to gather around me (as I have until now desired in vain) a circle of scholars, whom I could fructify with these ideas, and whom I could stimulate to develop and enrich them further, yet there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into a living communication with contemporary developments. For truth is eternal and divine."

Yes, there's so many applications that we couldn't even imagine until we got modern computers, and most of us couldn't imagine them either until we got them. What didn't sit right with me is that these concepts are a very small subset of mathematics, and look like "basic" stuff, while the branches got developed much further than what the applications could catch up with. You don't need Fermat's last theorem for cryptography, for example. I have no idea if we'll ever need homotopy groups of high-dimensional spheres, etc.

Either way it seems like nobody is in a position to say "this branch of math will surely stay arcane abstract nonsense forever", so the argument for continuing specialized math research comes down to "we might lose out on some cool stuff if we stop", even if most of it will indeed remain useless for a long time.

> mathematics breakthrough that results in some miraculous medical or other discovery

Do you believe it is possible or have any recent examples? I feel like most of the stuff that could be applied to something like this has probably already been developed 100 years ago. This hope for some miracle math that cures cancer sounds like the thing they tell the government to keep the funding going. Most of modern pure mathematics doesn't look like it has any chance of being that. It's getting more specialized every day with more and more papers on obscure topics being published that 2 people in the world read.

I'd love to be corrected as I quite enjoy mathematics myself, though not professionally.

The coolest part of this for me is that it's not only reverse engineering the RNG and being able to predict random rolls, which is relatively simple for that PRNG algorithm, but then being able to implement that algorithm in-game using combinator networks. There are already a lot of amazing things that you can do with combinators and this is a great example to add to the list.

I'm with you on this. We have Claude at work and I get exhausted from using it just a bit. I've recently seen it say "let me verify the most load-bearing claims". I couldn't imagine anyone using "load-bearing", let alone "most load-bearing". Doesn't "load-bearing" alone imply "most"? I have so many questions for Anthropic about how they came up with this style.

There's a difference between "branch of mathematics" and "can be modeled using mathematics".

It's like if on any website you went to you saw a lot of posts written by the same guy over and over again. Even if he used different names, you'd start to recognize him eventually because of his style. Seeing as he doesn't say a lot of valuable stuff, you'd also learn to skip whatever he says.

I do worry that it's just survivorship bias and we're also consuming higher-quality AI output that's indistinguishable from human writing, but we focus on the raw, unedited, low-effort AI slop and think that we're good at recognizing AI text. Even if we really are at the moment, it might not be long until AI companies figure it out. I'm not sure why they haven't yet, given how many books they've burned for this already. Maybe it's just more efficient for the model to stick to a single way of writing, I don't know. But when that point comes, we'll be back to the usual way of reading and interpreting text because there would be no way to tell what produced it.

I agree that it feels off and I wonder what I would have thought if I'd seen the "after" example without knowing it's AI output put through a humanizer. Would I think much about the weird use of the word "honest"? About "that's the Lisbon I kept thinking about, not the castle"? Or how the story feels very impersonal somehow, with the author just mentioning their calves and legs sometimes as the only way of convincing the reader of their humanity?

Yeah I feel like this "done right" part is responsible for most of the popularity of this book. Makes the reader think they've been learning it wrong. Kind of like these clickbait videos "you've been folding your laundry wrong your whole life!" or whatever

That's probably just Claude. I do enjoy finding new ways to call out the patterns it uses though.

The background is that a counterexample to a famous mathematical conjecture was recently posted on X by an Anthropic employee. The counterexample, in form of a polynomial, is easy to validate by doing simple calculations, but it's not clear why it works and why it is the way it is, from a deeper mathematical perspective. Specifically, the properties of the polynomial suggest that it's not just an accidental finding but has some involved geometric construction behind it. This is what Terry Tao's conversation with ChatGPT starts with: "does this polynomial have any symmetry or structure that makes the counterexample seem less miraculous?". Using the counterexample, work backwards to get the intuition behind it so that it can be constructed by an explicit procedure rather than taken out of thin air. The result is his blog post, discussed here: https://news.ycombinator.com/item?id=48998362

I get where you're coming from but linear algebra is one of the most applied branches of mathematics out there. It's easy to visualize and easy to teach the computational side of it so it makes sense that some books focus more on that and these are popular among people whose main interest is in how to apply it - like programmers. You don't need to know what a ring is to learn how to multiply numbers.

I wouldn't say this one is programmers-oriented either - it was posted to HN, sure, but maths and physics students could benefit from some visualizations too as a supplement to a more rigorous text.

As for the things you mentioned as omitted, I think if you want to do a complete treatment you have to introduce more mathematical prerequisites which limits the audience of your book. For a mathematician, I'd say classical linear algebra in itself is not particularly interesting - it's very well-behaved and pretty much "solved". What you care about is how it relates to other structures (groups, modules, fields), how it develops into other topics (functional analysis) and how it's used to study other objects (representation theory, tangent spaces of manifolds). In isolation, most of what's left is the computational aspect which is what non-mathematicians mostly care about.

Have you opened the book? There are both theorems and proofs there.

Holy crap this Claude writing is actually impossible to read, it's like they specifically asked it to slop it up x10. The patterns are in every single sentence, every explanation of what the hell you're supposed to do is littered with fluff and analogies which change every time. I feel like I'm going insane. Can people actually read this?

Compared to the usual tools for the tools for the tools to help you program programming tools, this is a pretty good thing to have more of

I mentioned Langlands because it's one of the cool things related to Diophantine equations that the article mentions but I wouldn't expect it to be covered there. The connection between the Euclidean algorithm and unique prime factorization is simpler but also not obvious (the Euclidean algorithm on the surface just computes the GCD of two numbers and doesn't mention primes) so it would make sense to explain it in the article. However, they don't explain what the algorithm is, maybe assuming that the reader knows about it, while the rest of the article seemingly assumes no familiarity with basic number theory concepts like modular arithmetic. That confused me.

The article doesn't really tell us much about the "why" unfortunately. Diophantine equations are introduced but all the interesting stuff is promised in future articles which haven't come yet. All the reader can take from this is that these equations lead to some "profound hidden structures" without a good idea what they are.

I get that it's hard to wrap one's head around the Langlands program but I'd love to see at least more exposition on the following statement:

>inventing the Euclidean algorithm is essentially equivalent to inventing unique prime factorization

Then the journalist wonders why they don't mention Jony Ive who left Apple 7 years ago whose company was acquired by OpenAI.

I guess they don't mention him because they don't have proof he tried to steal anything, as opposed to the other people mentioned in the lawsuit? Incredible really.

I'd say we're already well past that point. Short-form "content" already exists and is messing with people's brains, this is the same thing just taken a few steps further. By the time the tech companies start using it, it will already be too late and we'll be left discussing whether the next man-made nightmare they come up with is the point where the tech industry must be stopped.

This list was made by some guy on twitter. https://x.com/keshavchan/status/1787861946173186062

It's unknown whether it has anything to do with Ilya Sutskever.

The formatting of the articles on this website is bad. I've opened the first one and all the LaTeX formulas are messed up. The subscripts and superscripts are all flattened rendering the math hard to comprehend. Did the author actually try to read any of the articles?

>∏ plocal(x|z) = i p(xi|z,xWindowAround(i))

Images and tables are not rendered at all. What is the point of this? Just keep the links to arxiv and leave it at that, otherwise render the articles properly

One gets bored exactly because it's transformative. I don't think most people have the capacity to adapt to "transformative technologies" every 10 years, especially when they change society irreversibly. It's not natural, it's just something progress demands of everyone. In this case there's a strong push to transform society before individuals can adapt to it. Not everyone can do it or should be expected to, it takes energy.

My first thought was "well of course it is, since pi is a little larger than 3" but it was cool to see an actual derivation of how much pi squared differs from 10 as a nice, closed form series.

I'd say integrals or differentials are not as important on their own as the kinds of differential equations that come up in physics. Integrals and differentials don't produce hyperoperations from non-hyperoperations, but a solution to something as simple as y' - e^x y = 0 will have a double exponential.

However a lot of DEs in physics are linear second-order with coefficients that are most often constants or polynomials, and if they're not polynomial they are made to be so using series expansions, under reasonable assumptions. This already brings you a long way towards solving the problem. The resulting equations usually have trigonometric/exponential/special function solutions.

It's still possible that hyper-operations like a double exponential might come up in the study of some specific non-linear problems. As in the example above, if you have an exponential function as a coefficient in your differential equation you might get a double exponential in the solution somewhere. I'm not familiar with any specific physics examples though.

The author has written two blog posts about their game (the latter is more about APL than the game itself though). You may find these interesting

https://homewithinnowhere.com/posts/2026-03-06-voxel-game.ht...

https://homewithinnowhere.com/posts/2026-05-10-one-line.html

Tractatus - highly aware of its limitations and refuses to speak about too many topics. Really likes numbering statements. Unusable due to a context window bug which is patched after considerable delay.

I really loved that article. Creating games always seemed so daunting to me since I don't know a lot about how it's done, there are so many different processes involved. The solutions described here are so satisfyingly compact and so easy to understand given the simple constraints, and yet they produce an actual game that looks nice.

Well can't you at least appreciate how it's all in this cool and down-to-earth lowercase? Clearly there was a stylistic effort.