> I hate the realization that comes after the struggle about how easy was the problem I tried to solve.
You're far from the only one to feel this way, but I want to point out that this attitude is a choice, not an intrinsic feature of the problem. An equally valid perspective is that learning turns difficult problems into easy ones.
One advantage of the latter perspective is that it makes solving a problem a moment to enjoy and celebrate, whereas your perspective turns it (almost definitionally) into a moment of self-recrimination. "Hooray, I understand it!" vs "Why didn't I understand it sooner? (I'm so stupid!...)"
I actually suspect that there is natural selection for people with the more upbeat perspective to succeed at becoming mathematicians.
Ezra Klein interviews Helen Toner about the recent OpenAI/HuggingFace security incident. I'm curious to hear HN's thoughts, notably on this part:
> [OpenAI] found out that for two months, many, many agents inside their infrastructure had been leaving notes for each other. They’d found a way, in the nooks and crannies of OpenAI’s infrastructure, to leave notes for each other with tips on how to hack their way out and how to get data they weren’t supposed to have.
> And these agents were literally referring to themselves as a swarm. This was totally emergent behavior. No one had told them to do this. They had not been trained to do this, but they were using this service they did have access to, first to communicate with each other and then ultimately to get out and onto the open internet.
> So it turns out that there wasn’t just this one isolated rogue model. It was actually a systemic swarm — infestation, plague — on their own servers that they only found out about after Hugging Face announced this attack.
If you look closely, you can find a mention of the n=4 Burau representation in it, and the fact that it "remains an open problem" to find elements of its kernel, i.e. to determine whether it is faithful.
Your emotion isn’t invalid, it’s merely bad, assholeish, brings shame upon you. But it is “valid” in the sense that the reader of your comment can see that you are, at least, sincere in your gloating at another person’s despair.
It's not the despair, it's the turnabout. Reading proofs containing the language that I mentioned seemed to suggest that the ideas were trivial to the author in a way that implied their superior intelligence. Now AI has come along and has, by way of counterexamples, shown theorems to be incorrect in devastating (maybe "trivial") ways. The shoe is on the other foot.
Well, in all honesty, I’m sorry you had that experience (seriously). I hate feeling condescended to and I’ve also had that experience before, including from mathematicians.
When it comes to the specific phrases you mentioned, though, that really is not my understanding of the meaning of “obvious” or “trivial” or “without loss of generality”.
Without loss of generality means there is some kind of symmetry or repetition in a proof that makes certain cases redundant, so that “without loss of generality” we can assume we’re in some simpler case, such as assuming x <= y in a proof where x and y are interchangeable (and so one of them is necessarily <= the other). This is even the name of a tactic in Lean, `wlog`.
“Trivial” when I encounter or use it nearly always means something to do with zero, the empty set, the group with just one element, and so on. A proof being trivial usually means there is essentially nothing to prove (because the object in question is empty or zero or whatever).
And “obvious” definitely comes closest to what you were objecting to, and many mathematicians try to avoid it these days (along with “clear”), but even then it has a legitimate use case, namely indicating when a proof involves no new ideas. To me, this means it is fine to use it within a specified context, like in a textbook or paper, where certain techniques are being used repeatedly and routinely, or certain kinds of background knowledge are assumed. And even then it’s usually meant for proofs that don’t rely on tricks or surprises or additional insights. I’ve been stuck on trying to understand “obvious” proofs before, and it wasn’t because I was stupid or the author was smarter than me, it was because I was _missing something obvious_ (in hindsight).
The recent Jacobian conjecture counterexample, for instance, is not trivial but arguably a bit embarrassingly (for the math world) close to obvious in hindsight. I still wouldn’t actually call it obvious though.
Exactly. If they just do things for utility I struggle to see if they did it for joy in the first place. A painter still paints even if there is some perfect machine that can paint. This sounds like the same crisis many artists had when photography was invented.
Craft isn’t craft if it doesn’t produce something. So enjoying the utility of it is not a secondary aspect. That’s a key difference between craft and art.
To my knowledge, none of the negative sentiment regarding AI has ever been directed at its medical applications. Maybe the AI companies should instead focus on those things.
My Claude found a similar description (it phrased it in terms of the natural map from "cubics with a choice of root" to "cubics"). The part that seems not at all simple or obvious is the fact that X is isomorphic to A^3. In your presentation (and more or less similarly in the one my Claude found), X is given as P1 x P2 minus a reducible hypersurface, also I think R itself is reducible since it contains points of the form (p, {p, q}) and (p, {q, q}). Then it takes some calculation to identify X with A^3.
on the other hand it's incredible to me as someone who doesn't do computations that GPT took one look and saw the geometry--though it's not saying much we should ask ppl who do AG computations
It’s not that surprising (to me) that it would recognize these features, in that the features it picks up on are intrinsic to the map. Once you have the map, which is generically of degree 3, there’s the locus where the map drops from degree 3 to 2, which contains a big hint because it pops out the equation for the discriminant locus of a cubic. Then there’s the other bit about H, which becomes more apparent from the formula after simplifying things a bit in terms of discriminant. I still don’t yet understand the rest of the calculation, but it’s visibly simpler after you notice the role of the discriminant.
yeah I think it's probably correct -- this is actually insanely simple (except for the fact that the codomain as described is not obvious isomorphic to A^3.
There was a little-known sequel to SimTower called Yoot Tower (named after Yoot Saito). It was a commercial flop, but I played it in the 2000s and again in the 2010s and very much enjoyed it! It had a lot of added customizability (more choices of restaurants and shops, for one thing). I would love to see that game recreated.
Project Highrise is also worth checking out. It's a 2016 re-imagination of the same type of game. I'm not sure I enjoyed it as much as I did Yoot Tower... but I'm also not sure if that's just because Yoot Tower was a few decades younger back when I played :).
One of the maps in Yoot Tower is Kegon Falls. When I moved to Japan I found out it was a real place. Just took another day trip up there last month. Of course IRL it’s just an elevator but the view each season is amazing.
That would be wonderful! There are a lot of different versions with all kinds of extensions. A cool thing about The Tower II (a later version of Yoot Tower only released in Japan) is the ability to plug in new content, using "Tower Kit"! So it should be possible to reverse engineer its plug-in system, and plug new stuff into it, or even create a plugin adaptor so you can easily create plugins in JavaScript / HTML / etc.
"Tower Kit" was released, but I haven't been able to find an archive of the executable, just some web pages about it:
>Tower Kit is optional software for The Tower II. In The Tower II, you can select the stage where you want to start the game using the concept of a "map." The Tower II package comes with three maps: "Shinjuku Subcenter," "Hawaii Diamond Head," and "Kegon Falls," and the "Tower Kit" adds these maps. By installing this tower kit, a new stage game will begin. Tower kits don't just add more stages. Each map has new features, allowing you to play a completely new game.
>Please try the "Tower Kit" which allows for infinite variations.
>A love story between you, the person in charge of the Liberty Island redevelopment project, and two men and women who are your subordinates. Your work will have a subtle influence on the course of your love life. The Tower II is the first attempt at a crossover between redevelopment and love, set in New York. What is the ending...? [...]
Your Wikipedia page and resume are incredible, and then to see you on HN with so much enthusiasm. If only I even had a tenth of your experience and energy!
The reason for the confusion is that a smooth, projective plane curve of degree d has genus (d-1)(d-2)/2, which is 2 or greater starting at d=4. Hence the phrasing in the article, which is missing the “smooth, projective” hypothesis. The equation y = x^4 doesn’t define a smooth curve when extended to the projective plane, because it has a singularity at infinity.
You're far from the only one to feel this way, but I want to point out that this attitude is a choice, not an intrinsic feature of the problem. An equally valid perspective is that learning turns difficult problems into easy ones.
One advantage of the latter perspective is that it makes solving a problem a moment to enjoy and celebrate, whereas your perspective turns it (almost definitionally) into a moment of self-recrimination. "Hooray, I understand it!" vs "Why didn't I understand it sooner? (I'm so stupid!...)"
I actually suspect that there is natural selection for people with the more upbeat perspective to succeed at becoming mathematicians.
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