I feel like a constructive proof is the best scenario for a young talent like this, they really can use their vivid imagination and manipulate what they are after.
At 17, Hannah Cairo solved a major math mystery
131–140 of 186 posts
Re: At 17, Hannah Cairo solved a major math mystery
#132As a pupil of Dijkstra and seeing at least some rise in formal verification because of the modern tooling and as a follower of Lean (and Agda, Coq, Idris* etc), I hope it will be at least a strive to deliver parts of proofs in code verifiable form. More machine verifiable building blocks will lead to a bettering of everything.
I'm on the practical side of things and have dabbled quite a bit with languages like Idris, I think we are mostly far from using them because of ergonomics.
Even in Scala (which is a very advanced language, but still far behind Idris) I deliberately don't use certain type-level features because it increases the compilation times too much (even with incremental compile!). It's very sad, but this is reality.
So, the problem isn't really that we need to "invent airplanes" - we already have them! What we need to do is make them usable and affordable by everyone.
I see more and more languages trying to add type-level features or try to embed other languages like Prolog-like ones into them. I hope that gets traction and becomes ergonomic, otherwise no one will use it in practice.
Re: At 17, Hannah Cairo solved a major math mystery
#133Wonder why Berkeley didn’t offer her PhD admission given that she was already working with the Prof there.
Re: At 17, Hannah Cairo solved a major math mystery
#134Wait, what software engineering jobs require you to move to the Bahamas?
Rich people offshore money there
Re: At 17, Hannah Cairo solved a major math mystery
#135Earlier quoted context omitted.
Maybe hot take… I can see the point of sameness in homeschooling, but compared to traditional education? I’m not sure how much flexibility one would have to teach oneself calculus by 11 or the equivalent of an undergrad in math by 14! That flexibility must be found in something non-traditional! I’m no prodigy at all whatsoever but school was mostly dull and filled with teenager drama! Nobody knew what Linux was, care…
its so much better nowadays. I"m 40 now and I'm low key jelous of kids today. Today if you want to learn to code, you have freecode camp and chatgpt to ask questions. Math? there's mathacademy and khanacademy. There are so many options now for learning stuff that we didn't have
Learning opportunities are indeed better. But time sinks such as TikTok and YouTube have gotten exponentially better (read: addictive). I think there's a higher overall likelihood that a kid gets trapped in doomscrolling than in Khan Academy.
Re: At 17, Hannah Cairo solved a major math mystery
#136Her notes are so clear and so artfully wrought! I wonder if learning from online resources makes one naturally focus more on presentation. From the article: https://www.quantamagazine.org/wp-content/uploads/2025/08/Ha...
Looks like a prepared presentation rather than notes.
If all presentations were prepared like this I’d probably attend them
Re: At 17, Hannah Cairo solved a major math mystery
#137As a pupil of Dijkstra and seeing at least some rise in formal verification because of the modern tooling and as a follower of Lean (and Agda, Coq, Idris* etc), I hope it will be at least a strive to deliver parts of proofs in code verifiable form. More machine verifiable building blocks will lead to a bettering of everything.
I'm on the practical side of things and have dabbled quite a bit with languages like Idris, I think we are mostly far from using them because of ergonomics. Even in Scala (which is a very advanced language, but still far behind Idris) I deliberately don't use certain type-level features because it increases the compilation times too much (even with incremental compile!). It's very sad, but this is reality. So, the pr…
I agree; I would say that most can be taught to people without advanced math/logic degrees/backgrounds, but because the tools are created by (and usually for) people with formal verification majors), they are just quite bad ergonomically as you say. I think we are in a good place of making it these types of proofs easier and faster more mainstream effort, not just one professor and their students, is put behind it. I believe AI will play a part here.
Re: At 17, Hannah Cairo solved a major math mystery
#138Earlier quoted context omitted.
No, that was the purpose of high school. As not practiced in public schools, as not practiced in the US
No, it's both. The purpose of high school is to give you a wide foundation on everything. The purpose of an undergraduate degree (in math) is to give you a wide foundation (in math). In a (math) PhD, you are generally hyper-specialized in a very, very narrow area (of math).
Although the end goal of a PhD is a specialized thesis, the first couple of years generally involves courses with a wide coverage of analysis and algebra at the graduate level.
Given her achievements, I'd be very surprised if Cairo hasn't already covered the material in an undergrad degree
Re: At 17, Hannah Cairo solved a major math mystery
#139I find the Soviet idea of Math Circles so interesting and important. I bought books on the subject, but it's difficult to implement for your own children only. Nothing beats it like having an actual one, run by math teachers and in your city.
Absolutely. Key for me was to invite children's friends (and family) along, host it in our house and make it a recurring weekly thing. The books (presumably you also have these from the MSRI's 'Mathematical Circles Library') are great, but week-to-week I've found the free online NRICH resources much more directly useful: https://nrich.maths.org/about-nrich
Re: At 17, Hannah Cairo solved a major math mystery
#140> Only the University of Maryland and Johns Hopkins University were willing to welcome her straight into a doctoral program. She’ll start at Maryland in the fall. When she finishes, it will be her first degree. Jeez... what a damning indictment of today's Universities. She could just use her publication as a dissertation and be done with it!
I see your point, but undergraduate degrees should provide a wide foundation, with little specialization. As you progress to a masters degree, you become more specialized. A doctorate is as specialized as it gets. It is entirely possible for people to intensely focus on a very, very narrow thing - and ignore everything else. Even to such a degree that they can write a doctorate on it. But I don't think that's a good…
> It is entirely possible for people to intensely focus on a very, very narrow thing - and ignore everything else.
sounds like Phd