“There was this inescapable sameness, in a way. No matter what I did, I was in the same place doing mostly the same things,” she said. “I was very isolated, and nothing I could do could really change that. I’d wake up on certain days and realize, I’m just older.”
I finally have something in common with a math prodigy.
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, cared about music production or anything interesting! The talk was which boy/girl whatever
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.
I'm 36, but to be fair even in late 90s when we were kids we had
1) Forums like askanexper(can't remember the exact name)
2) great books like sams teach yourself in 24 hours and a habit to spend time in library
3) most importantly not as much competition as we have today. There were like a 2-3 kids in my schools of 2k who were into programming. Today that would be approaching half.
Very difficult to take advantage of if you don't have good control over your brain.
I am learning bit and pieces on how to do this but it's a long and uncertain process. Conflict monitoring is willpower is a new concept to me but it gave me a framework on how to improve my discipline over time.
It's wonderful that Khan Academy played a role in enriching her early education. It's proving to be a solid resource across the spectrum of math ability.
”Cairo applied to 10 graduate programs. Six rejected her because she didn’t have a college degree. Two admitted her, but then higher-ups in those universities’ administrations overrode those decisions."
This is both unsurprising and shocking to me at the same time.
For institutions of allegedly pure higher learning in a field where it's known that youth is where the advancement happens, the fact 80% axle wrap over a piece of paper that, let's face it, in modern times of grade inflation is pretty much worthless of anything beyond money and sitting in a seat for four years.
A lot of programs don't want to have to babysit a teenager no matter how talented they are. Some of the more prestigious programs both have some experience with it and extra staff to (give profs warm fuzzies) handle any issues that come up. I'd expect it depends a lot on her interests and the particular professors that study that at any particular institution.
Even as a student, I'd be more interested in which professors at Johns Hopkins were accepting students, than which school.
I suspect the reason is not about babying her as much as it is a concern for her development, about whether she would be "missing out" on the opportunities for social and intellectual growth in college. This kind of thing is usually why well intentioned higher ed people put roadblocks in front of precocious young people.
i can understand a university administrator not feeling they can give a good educational experience to a 14 year old.
but she'd be the same age as all the first years on campus, any university must be institutionally equipped to deal with a 17/18 year old. seems like an odd choice to personally intervene to cancel an admission.
like presumably if she enrolled in a bachelor's program, and then signed up for grad math classes while doing independent research, this would be fine. but if she takes the same classes and does the same research for PhD credits instead, no good?
> Two admitted her, but then higher-ups in those universities’ administrations overrode those decisions.
I've personally seen universities go both ways, and it comes down both to individuals, and to the culture of the department/university faculty and administration.
(Not to the culture of the student body, which is influenced by the administration culture, but has very little institutional memory, and almost zero power. If you draw an analogy to nations, there might be one with the most awful 'leaders' seizing and abusing power, but that's "way above the pay grade" of the many nice citizens you will meet -- who didn't know what they were being born into, and will do their best to be decent to each other, despite whatever bits they're unfortunate to learn about the upper powers.)
Cairo may have a way out: Commonly the main criteria for a Ph.D. is: (1) Pass qualifying exams in several important topics in the field. (2) Do some research that is an "original contribution to knowledge worthy of publication" where the main criteria for publication is "new, correct, and significant".
No professor or university Dean can keep her from doing well on both (1) and (2). It looks like (1) would be easy enough for her. The work she has already done may satisfy (2). Once she has done well on (1) and (2), tough not to award her a Ph.D.
PhDs in the USA also have curricular requirements, with at least some required courses. These courses are only available after a bunch of paperwork is given to the university. So, it's not so easy.
In my university, when someone is accepted as a graduate student of a different topic (let's say phisics -> biology) it's usual to include a few of the last courses of the major as a mandatory part of the Ph.D.
“After the spring
semester ended, her family moved from Davis
to Berkeley — her brother had decided to
transfer there — and Cairo finally felt able to settle in.”
fascinating that the family follows the kids’ educational steps.
I don’t know about major math achievements- the n is pretty small- but homeschooled kids are overrepresented in geography bees, spelling bees, and other academic competitions at the high school level.
I wonder what would have happened if she said "mom, dad, I want to be a fashion designer/chef/interior decorator." Undoubtedly, they would have supported her ambition and individuality..of course.
I 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
I studied math and knew a girl like that, spending an inordinate amount of times making sure to understand every single details of what we were taught, even going further when I could barely keep up. She would also make those pretty, colorful notes with visual interpretations and drawings in the margin. It takes a special form of love for the discipline. She just finished her PhD and is now starting a postdoctoral contract.
> When I was an undergraduate at MIT I loved it. I thought it was a great place, and I wanted to go to graduate school there too, of course. But when I went to Professor Slater and told him of my intentions, he said, "We won’t let you in here."
> I said, "What?"
> Slater said, "Why do you think you should go to graduate school at MIT?"
> "Because MIT is the best school for science in the country."
> "You think that?"
> "Yeah."
> "That's why you should go to some other school. You should find out how the rest of the world is."
I've been at institutions that explicitly discourage that because of that reason. They want you to get breadth of experience rather than work with the same one person for your early career.
Not only she did grow in Bulgaria during the most turbolent times of regime change from communism to democracy, but later graduates with a PHD from Harvard, and later becomes Director and Founder of the Berkeley Math Circle, and is also organizer of math competitions in Bay Area, and publisher of what seems to be a complete set of Math Books, carefully crafted with her peers from BG and presented here
>Cairo kept reading and thinking. Eventually, she found a way to construct a strange, complicated function out of waves whose frequencies all lay on a curved surface — the type of surface the conjecture required. Usually, when you add these kinds of waves together, they interfere, canceling each other out in some places and reinforcing each other elsewhere.
Rather than speaking about her age or the vague notion of talent, I'd be much more interested in why the rest of the academia was unable to replicate her methods in 40 years or so. From her own admission, it was throwing different ideas and approaches until she saw a disprecancy that ultimately disproved the theorem. There should be many people far more experienced who do have the knowledge to do this, but why did they not? This dosen't look like a intuitive jump, seems more like building a test case that fully stressed the theorem.
Heh, take a look at her paper [0]. It is much more complex that TFA makes it out to be: once you have the intuition you need to prove it actually disproves the theorem.
Also, it's probably a not very well known conjecture that people wouldn't care too much about. Also also, it was a conjecture: people were probably looking to prove it rather that disprove it, failing and then giving up.
As 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.
Offtopic, but I am 17 too just like Hannah Cairo but nothing too groundbreaking till now I suppose and it absolutely brings me delight that I can talk to somebody who was a pupil of Dijkstra,
I have heard a lot about dijkstra's algorithm's and I had forgotten about it and so I searched it right now, but the only thing I knew is that it is pretty popular algorithm.
If I had to ask you kind sir, what would be the biggest life lesson (in coding, or anything general) that you could give me be?
I mean, my interests are just really niche in the sense that I like to glue projects and one of my favourite hobbies is to search if there is an open source replacement to something or if I can use open source things to do stuff
So there is intel tiber cloud and it gave me a free jupyter and I wanted to play minecraft server, so I went to something which is a fork of it and I saw that there was a guy who had created rootless patch of ssh but it wasn't working for me so I kindly asked them and they were really kind enough to help me and then I used something like pinggy or the thousand other cloudflare tunnel-esque solutions which work without any problem whatsoever directly through ssh
Earlier I used to use junest to then get something like ssh working.
Can I be honest? I am so proud of myself because I guess I did all this stuff an year ago but I still remember the day I asked my friend to join the mc server and I joined too, the ping sucked but 8 gigs ram or even more and no storage limits and complete freedom... and I thought of such stuff after almost 2 weeks of trying different things and learned a lot about nat punching etc.
So such stuff always makes really sense to me, just tinkering with software. But I feel like I need to create some professional software too in order to leave a legacy or just, I feel like, if I am being honest, that the things that I am doing aren't that valuable. Anybody can learn such stuff from me from some blog post I write but there is nothing that I can ever monetize.
I don't know why I am seeking monetization that much but its just that, I see all these kids hustling my age too and I feel like everyone's doing everything for the money... so why should I be any different. I get a lot of ideas but I feel like I can never earn a single cent out of it and I find it a little frustating. I feel less confident myself regarding if I will ever earn decent money or will just be a low level cog of this whole system.
Found it https://github.com/mkj/dropbear
I really love this project so much, it really helped me do something really wonderful.
Not the one you asked, but here's what I would have told my 17 year old self.
* "Slope beats y-intercept." The best computer scientists and engineers I've ever worked with and/or mentored embodied this principle more than anything else.
* It can be tempting to over-optimize for short-term milestones (e.g., an important admissions exam, the next job or promotion), but there is a significant compounding value to knowledge accumulation and truly learning your craft well. Read and learn as much as you can, all the time, even if it isn't immediately necessary or useful.
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.
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.
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.
AFAIK there's no general procedure; it's just that the admissions committee can admit anyone whom they're convinced can do Ph.D.-level work, and that almost always involves getting an undergraduate degree first but in extremely exceptional cases it might not.
The basis would be getting a dean's approval (or lack of a denial). That is the level at which "Exception Handling" is normally done at a university (above professor, below vice president), and it would normally be the dean of the department that housed the graduate program (some grad programs span multiple departments, which makes it more complicated).
Likely, there are also rule systems that would need to be circumvented; for example, while the admissions program for a graduate program would accept the student, the registrar or other entity might then negate that acceptance due to standing policy. At that point, I wouldn't be surprised if it reached the vice president level of the university (VP of the School that housed the department).
Every university handles this slightly differently. Exceptions can often be made.
To qualify you would normally have to do something truly exceptional or show a pattern of continuously excellent (grad student level). From what I can tell (I am not a math expert) she is considered a 'prodigy' and the problem she solved would require considerable skill, in a way that predicts continued successful performance at the university professor. The history of prodigies who skipped various steps of the normal schooling path is mixed- personally, I think people undervalue the importance of spending time with people in your age group during the 15-25 range, but honestly, without knowing more detail, it's hard to say.
For context, Stephen Wolfram dropped out of Eton to start at Oxford, then was admitted to Caltech's physics PhD before finishing his Oxford BA, receiving his doctorate at the age of 21.
Essentially, you have to convince a math professor that you able to complete the Ph.D program. There are a variety of ways to do that. Solving a major open conjuncture is clearly one of them but just getting to know one or another professors can work (I went to school with the semi-famous Paul Lockhart who also did this).
– were all homeschooled. Cairo started learning math using Khan Academy’s online lessons, and she quickly advanced through its standard curriculum. By the time she was 11 years old, she’d finished calculus
No normal childhood and certain kinds of parents, disproportionately more math education, there is certain downsides to all this, that articles like this conveniently avoid to mention.
“There was this inescapable sameness, in a way. No matter what I did, I was in the same place doing mostly the same things,” she said. “I was very isolated, and nothing I could do could really change that. I’d wake up on certain days and realize, I’m just older.”
I don't think they're omitting it, but yeah. A lot of homeschooled kids don't fare nearly as well.
I have yet to move into the woods but, given my current levels of burnout, I think such a proposition is highly underrated. What's wrong with moving into the woods?
Wait, if everyone moves into the woods there will be no more woods that offer seclusion. Yeah moving into the woods is awful! I hope she doesn't burn out!
One option could be to get a hammock and go program in a tree. That’s sort of like moving to the woods, but you don’t have to live there full time and you still get to program.
how would you know for sure? I don't have the skill to evaluate somebody who has disproved one conjecture (maybe some recent techniques made it easier?) versus somebody who had an established pattern of doing excellent work.
no, he's saying Kacyznski is not on the level of Hannah Cairo. If he meant Grothendieck, that's another conversation. Reading the context of this subthread, it should be clear we're discussing Kacyznski vs. Cairo.
I don't have the knowledge to evaluate this, but I still think Cairo has not esta blished enough work to say she's far ahead of Kacyznski.
thanks for the clarification, I also dont have the ability to evaluate who is better kacyznski v cairo. hence why i assumed the original commenter meant kaczynski vs grothendieck because for one kacyznski has a phd in maths, cairo does not have that metal yet
> 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 excuse to make them forego other curriculum, especially if it is required for other students to take. Schools have a responsibility to educate people to a certain standard, and give them some general breadth.
The woman is not being "made" to do anything. She's already world-class at math and wants to maximize the impact of her specialized talents, so she's going straight to PhD.
I don't think we need to nanny people who break the mold with extraordinary talents to conform to some generic correct educational sequence. They've proven they know how to make something of themselves and their own ideas should count for more.
Imagine if every AI researcher out there was allowed to skip even the most fundamental philosophy class(es).
I don’t know about you, but exposing these prodigies to some shared classes is not all that bad. Personally I think every student should be forced to take a class on ethics.
I'm not sure higher ed's educational philosophy is serving students all that well. The breadth of education is a shallow survey at best that is quickly forgotten exactly one semester later.
Thankfully the workforce has common sense and will happily snap her up into employment.
> The breadth of education is a shallow survey at best that is quickly forgotten exactly one semester later.
Your own experience isn't generally applicable. Although it was a couple decades ago, I still use various things I learned from my non-major classes pretty much daily.
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
I've not studied this, but my guess is that the liberal arts education as the foundation is necessary to allow young people a chance to figure out who they are. I certainly found this to be true for me and would guess for my closest college friends.
If a young person is exceptional, do we force them into a liberal arts box? Surely there is value in literature and history. But this one young woman had found her passion. I have to believe that is she found out about something else, she would take that on.
In my teens I worked with the statistics department at UTMB. That had a cast of characters there; many profs in the 70s and 80s, who'd gotten their degrees before WW2. A number of them had schooling of the form:
Start school at 9-10, do 5 years of public school, got to a 1 year prep, do a year or two of college, do a two year PhD. Most of them had their PhD's by 22.
I think the biggest flaw with higher education today is that we're pushing people into doing undergraduate degrees who are already well beyond coming out of self learning from high school or other experiences.
The number of high schoolers who might be ready to go into postgrad programs is very small.
There is no way this could be the “biggest flaw” in higher education today because the number of people possibly impacted is so tiny.
Although I think you’re striking at something that is a real problem with undergraduate degrees today: Many universities have become so watered down and softened that students spend the first 1-2 years doing what they should have been learning in high school.
My friends who still teach at university constantly complain about students arriving for undergrad with very poor writing, communication, and listening skills.
Very few people are too “advanced” to be challenged by a sufficiently difficult undergraduate degree, ridiculous thing to say imho, I went to a UK university this decade and I can give you a laundry list of issues more significant than “exceptional students being slowed down”.
This woman is undoubtedly exceptional. But we don't know how exceptional, because she's an outlier, educated using different methods.
We have no idea how many other people would achieve something similar with a similar background. Personally I'd bet almost anything it's a larger number than most people expect.
I'd also be surprised if she doesn't already have a pretty solid background in undergrad-level math.
The irony is she's actually more typical than not. Universities in the past were open to giving unusual talents special treatment.
Historically, the idea that everyone must follow the same path on the same timetable is unusual.
Australia's debt is in the form of HECS/HELP. It's indexed to inflation, and integrated into the tax system such that it's only repaid on income over a minimum threshold. The first threshold is 1% of income above $56k. The last threshold is 10% of income above $166k.
My understanding is that US student loans accrue interest at rates above inflation. They do have repayment plans based on income, but because the interest rate is higher than wage growth, the debt just keeps growing if you're on these plans. They also don't have progressive thresholds like we do and the repayments tend to be higher.
Australia has a similar but less severe problem. Inflation and wage growth are closer together but they aren't the same. Still, the situation in the US seems worse.
I think research and higher education does have value for the most part. It's undergrad that's really worthless and something people only do for the experience.
Undergrads who care about learning and research will take the most challenging classes, do research with professors, and surround themselves with other strong students who will push them.
Even at top universities, very very few freshmen are capable of doing high-quality research immediately. They'd be better served learning the foundations inside and out with a cohort of similarly strong students to challenge them.
To agree with you: I've worked with several really brilliant undergrads doing and publishing great research. But all of them were rightfully undergrads. Even if they were actually capable of doing great research, they benefited from the breadth.
If you have bright enough undergrads, you change the curriculum for them within their field of expertise, so that they still get the breadth of things outside it while not wasting time with things they know. You let them not take as many classes, take graduate courses, do more research, take more courses from other departments in related areas but with different perspectives, and so on.
When I was an undergrad, in physics, there was a professor in the department who had done his undergrad there and was legendary, as was quietly mentioned in awe, for not taking any undergraduate physics courses while there; the department had let him skip all of them, and instead take graduate courses and do research.
I'm not sure that's a simple argument and can't imagine many would agree.
Undergrads who do research generally aren't very good at research yet. A major reason is they either lack or don't fully understand the pre-reqs, which they progressively and cumulatively learn during undergrad. A student can be incredibly smart, but acquiring a strong rigorous math background will still take years.
About pre-reqs: third and fourth year PureMath classes at UofWaterloo consisted of math I already took in HighSchool in Romania: group theory, ring theory. Plus some calculus I already read in high school out of curiosity: measure theory and the Lebesgue integral. Another Romanian guy at UofW was auditing 4th year classes while in his first year (he is now a math professor at an American university)
I can see a committed and gifted student being able to get most of the pre-reqs for doctoral studies in America or Canada while in high school.
If you don't know the foundations well, you don't belong in a postgrad program. That's the reality and how it currently works. Undergrad teaches you those foundations.
Anyone can try doing research, even undergrads who half-know the foundations. However, trying research doesn't mean you have the background to do great research or to succeed in a postgrad program.
Indeed. I've long felt that most undergrad students would be better served by a typical college minus the formal classes. Basically dorms and all the other amenities found in a typical college campus, where you mainly gain life skills and mingle with other people your age. Because most people I met at an average 4-year school were there because it's a societal expectation among certain classes, it's less scary than just getting a job and figuring out life completely on your own, and it is 10x-100x easier to make friends at college than just "out in the world." Not on the list: to learn from college classes, which at an average school teach you less than you'd get from a $200 a year subscription to Great Courses Plus or Brilliant. Or free from Khan Academy.
I know a few very special schools give undergrads access to brilliant minds in their field, but I also have been told that undergrads at those schools are mostly taught by grad students, so I'm not sure that Ivies provide a lot either, beyond the opportunity to hobnob with the legacies that will be running Goldman Sachs in 20 years.
Perhaps the term "canalization" fits? Coined by C.H. Waddington, describing the process of forming a chreode in a homeorhetic system. But it doesn't really encompass the "everything becomes overly generalized" concept you mentioned. Rather, it's more about robustness against perturbation of a trajectory in a conceptual space. Christopher Alexander called them 'paths in configuration space'.
Of course she was, but that's not what this is about. Letting her in proves that the bureaucratic credentials offered by schools are meaningless. The university system in its present form in the US is entirely predicated on the fiction that those credentials actually mean something and are worth paying six figures for.
> Jeez... what a damning indictment of today's Universities.
> She could just use her publication as a dissertation and be done with it.
I’m not suggesting this person is doing anything fraudulent as she seems quite impressive.
However, educational institutions get constant requests from parents who want their children to skip far ahead before they’re ready. It’s a competitive world and they know that being able to claim a child skipped several grades or even skipped undergrad entirely is a unique and very impressive achievement for the resume. It also theoretically provides a few additional years of earning potential by giving a career head start.
The first problem is that many of these parents (again, no accusations for this specific case) see this and want to make it happen for their child at any cost. There are some wild stories about parents trying to cheat their kids forward or falsifying their accomplishments to try to skip grades.
The secondary problem is that it can be hard on kids to be thrust forward so far past their peers. I had several friends who skipped a grade in middle school and most of them didn’t have a great experience for social reasons. Skipping undergrad altogether would thrust someone into a foreign world with a lot of baseline expectations and norms that they hadn’t yet learned, combined with no peers their age to discuss it with.
It creates a high chance for burnout or failure, which could leave them worse off than when they started.
That’s why the recommendation is generally to do undergrad at a challenging institution that allows students some upward mobility in specific areas where they’re ahead. No reasonable undergrad program is going to have this person taking Algebra 101, but there are a lot of opportunities for them to jump right into advanced programs and go deep and broad.
IIRC Erik Demaine was also homeschooled, skipped college, got a very early PhD, and joined the MIT math faculty as a teenager (he is a full professor there now). Johnny von Neumann OTOH went through a "normal" secondary education partly because his parents wanted him to have traditional social exposure to kids his age while he was growing up. His math training was very accelerated though, and he had professional level research publications at 17.
> She could just use her publication as a dissertation and be done with it!
The purpose of a PhD is not writing a dissertation. It is a research school, and I'm sure she could still learn a thing or two about research (and teaching).
Conversely: if that's all it takes, there is no point in going to University just to get a piece of paper that says "you did the thing you already did".
University (for folks serious about continuing in academia after) is (obviously) about making sure you have the same base knowledge as everyone else, but also for you to come to terms with how academia actually works, who the bad players are, who the good players are, and who you need to know to get shit to happen for you. So in that sense, most Universities going "no" is literally the most accurate reflection of what life's going to be like on a continuous basis on the inside.
I don't see why it would matter. she could quit math now and still be ahead of the majority of mathematician in terms of contributions. The rest is just formalities. I can see her breezing through the undergrad, quals, etc . It would just be a small delay.
To add slightly more detail:
Most research math programs in the US are 5(ish) year 'combined' masters+phd programs. The first couple years are basically course work, seminars, finding your area and advisor, and then the rest is the actual research work. It's not uncommon to leave after the first couple years with a Master's degree.
> Normally you have a master of science as well. And for that you require a bachelors.
Many Master’s programs require Bachelor’s as a prerequisite for admission, and some of the others include a Bachelor’s at some point, but not all do. (The same is true of the Ph.D. and the stereotypically preceding degrees, and also of some professional degrees and Bachelor’s degrees.)
I have a Master’s and never got a Bachelor’s. Universities can admit people more or less as they please, subject only to the very forgiving scrutiny of accreditation agencies or whatever government department officially supervises them.
There is a story about when Stephen Wolfram applied for a job at AT&T. The HR person asked him where had gone to college. Wolfram confessed that he had never gone to college. Well ok, did you graduate high school? Um no, I'm afraid not. GED? Nope. What about primary school? Wolfram looks more and more more embarrassed, and admits that he didn't even finish primary school. HR person by now is quite uncomfortable when Wolfram manages to think of something. He says "I do happen to have a Ph.D. Does that help?"
Thinking about my GE requirements at undergrad, I think it would be a waste of this girl's time to be forced to learn about and write about random subjects that don't interest her. She has but one lifetime, and can contribute much to her field.
The subjects such as English composition, she should be allowed to test out of if she is already a good writer.
> I think it would be a waste of this girl's time to be forced to learn about and write about random subjects that don't interest her.
This is a common STEM view, but it is inherently wrong. IMHO it dates back to the unfortunate divide between science and the liberal arts, whereas both were once considered a single field, now days there is disdain and mistrust between the two sides.
The point of history classes isn't to memorize the dates of wars, is it to understand the motivations of humans, it is to understand how the world we lived in has been shaped throughout time, and it is to learn how to do, and understand, research about the history of a place.
The point of English classes isn't just to get good at writing, it is to get good at various types of writing, it is to learn how to read different forms of literature, and it is to have a guided tour through a chosen selection of literature to hopefully develop one's character and thoughtfulness.
One of the most valuable classes I ever took at University was the Art Of Listening To Music. We started off around 500ad or so and went forward through time up until about 1920. We learned the vocabulary of music, how to sit down and listen to a piece of music and describe what we were hearing. After I was done with the class I went from appreciating a handfuls of genres of music to appreciating music itself no matter the genre. It was a 3 credit guaranteed A class that had enriched my life by an enormous amount.
If you really love your major, then the point of going to university was NOT your major, odds are you would've studied that field with or without the school. (Barring fields that require large capital investments, chemistry, physics, playing with an entire orchestra, building airplanes, etc) The point is everything else.
But that isn't how it works in reality, at least in the US. In reality, outside of their major (and sometimes inside it too), students usually pick the absolute easiest classes that satisfy the requirements. The ones that are known that the teacher doesn't take attendance are heavily desired. And the university is happy to oblige. Departments are funded based on a formula of how many students are in their classes, and they know that if they gain a reputation for being hard, students won't take their classes for GE requirements. It's a race to the bottom. So most departments offer enormous 1 level classes with 200 students taught with minimal rigor, and where you really only have to study a few days before the final to collect your A. And on top of that the frats all keep collections of graded tests from every class for years past, so basically anyone who wants to cheat is able to do so easily. This isn't education. This isn't worth six figures.
> In reality, outside of their major (and sometimes inside it too), students usually pick the absolute easiest classes that satisfy the requirements.
In reality, no one eats healthy food, everyone eats fast food hamburgers all day. Just look at the sales numbers of fast food / junk food VS organic lunch salad bars!
Except, that isn't true. Some people eat junk food all day, and some people choose to eat healthy. Obviously in America we have a bit of a bias, but just looking at averages doesn't give a complete picture.
> The ones that are known that the teacher doesn't take attendance are heavily desired.
Almost none of my university classes took attendance. Why would they? We were paying to be there, if we wanted to waste our money, it wasn't the university's problem.
> So most departments offer enormous 1 level classes with 200 students taught with minimal rigor
Reading books and writing essay's doesn't require rigor, the learning is in the doing. I put in honest work to learn and I got honest feedback from my 100 and 200 level professors, which was all I expected.
> And on top of that the frats all keep collections of graded tests from every class for years past
Almost none of my GE classes used multiple choice tests. They were typically essay tests, written in class.
I should note I did my GE requirements at a local community college, where class sizes averaged ~20-30 students, professors had office hours, and I think I only saw a TA once.
> This isn't education. This isn't worth six figures.
The price is too high yes, but a university is place to go where you can dedicate yourself fully to learning, hopefully w/o other outside worries (sky high tuition ruins that...). What a person chooses to do with that time is up to them.
Now one can argue that the worth of a degree is lessened by students who didn't actually learn all that much also being in possession of one. That is a closely related, but separate topic.
That said, the poster I was originally replying to was indirectly advocating for not caring about one's GE classes, and I was replying that one should indeed care, because those classes are incredibly important!
>One of the most valuable classes I ever took at University was the Art Of Listening To Music.
I also benefited tremendously from such a class. I would recommend it to anyone interested in music, or not interested in music and trying to figure out why others are interested in music.
Insurance is a pretty common one. Lots of insurance companies have offices there - and if you're a top-notch dev, you can make a crap-ton of money as well.
But I don't know that from the proposed re-titling of the article. The importance of the conjecture is more salient for the general reader than the (uninformative) names of the people who conjectured in the first place.
There's so many photos of just her staring off into the distance, and only one photo with her presenting the actual thing that she's supposedly famous for. I don't get the point of just all these random photos of this girl.
I never got excited for math, thus didn't care much to start with. But then add the issues of high school math and that solidified it for me. Math word problems were the worst. With a dyslexic brain I would consistently read the important words as something different, thus correctly solving the wrong problem and being derided for it.
Geometry required rote memory exceptionalism which also works against my brain design (adhd as well), but let me have the formulas to choose from and I'll get it done. Algebra II? I kept getting in trouble because I would do homework assignments in class instead of paying attention to the teacher trying to teach concepts that were easier for me to learn by reading the examples and following the book. After that, who would ever want to continue beyond the required credits and not think of further math as a masochist hellscape
Thank God she found math instead of Factorio.
the factory must grow ;-)
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, cared about music production or anything interesting! The talk was which boy/girl whatever
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.
I'm 36, but to be fair even in late 90s when we were kids we had
1) Forums like askanexper(can't remember the exact name) 2) great books like sams teach yourself in 24 hours and a habit to spend time in library 3) most importantly not as much competition as we have today. There were like a 2-3 kids in my schools of 2k who were into programming. Today that would be approaching half.
Very difficult to take advantage of if you don't have good control over your brain.
I am learning bit and pieces on how to do this but it's a long and uncertain process. Conflict monitoring is willpower is a new concept to me but it gave me a framework on how to improve my discipline over time.
So many options but also a corresponding amount of competition and pressure
And with social media, you aren't just comparing yourself to your classmates, but with basically the entire world
For those who speak strong English, yes
khan academy is available in many language
Loneliness is maththematician best friend.
Straight to jail, right away
>A few more months in prison and I would have proved all the great theorems of the time.
André Weil
Who is he ?
Boubarki group founding member, medal Fields recipient, brother of Simone Weil
It's wonderful that Khan Academy played a role in enriching her early education. It's proving to be a solid resource across the spectrum of math ability.
”Cairo applied to 10 graduate programs. Six rejected her because she didn’t have a college degree. Two admitted her, but then higher-ups in those universities’ administrations overrode those decisions."
This is both unsurprising and shocking to me at the same time.
For institutions of allegedly pure higher learning in a field where it's known that youth is where the advancement happens, the fact 80% axle wrap over a piece of paper that, let's face it, in modern times of grade inflation is pretty much worthless of anything beyond money and sitting in a seat for four years.
A lot of programs don't want to have to babysit a teenager no matter how talented they are. Some of the more prestigious programs both have some experience with it and extra staff to (give profs warm fuzzies) handle any issues that come up. I'd expect it depends a lot on her interests and the particular professors that study that at any particular institution.
Even as a student, I'd be more interested in which professors at Johns Hopkins were accepting students, than which school.
As if that was the reason.
Also, I've seen a great deal of adult babies in academia, so let's not be ageist here.
I suspect the reason is not about babying her as much as it is a concern for her development, about whether she would be "missing out" on the opportunities for social and intellectual growth in college. This kind of thing is usually why well intentioned higher ed people put roadblocks in front of precocious young people.
i can understand a university administrator not feeling they can give a good educational experience to a 14 year old.
but she'd be the same age as all the first years on campus, any university must be institutionally equipped to deal with a 17/18 year old. seems like an odd choice to personally intervene to cancel an admission.
like presumably if she enrolled in a bachelor's program, and then signed up for grad math classes while doing independent research, this would be fine. but if she takes the same classes and does the same research for PhD credits instead, no good?
> Two admitted her, but then higher-ups in those universities’ administrations overrode those decisions.
I've personally seen universities go both ways, and it comes down both to individuals, and to the culture of the department/university faculty and administration.
(Not to the culture of the student body, which is influenced by the administration culture, but has very little institutional memory, and almost zero power. If you draw an analogy to nations, there might be one with the most awful 'leaders' seizing and abusing power, but that's "way above the pay grade" of the many nice citizens you will meet -- who didn't know what they were being born into, and will do their best to be decent to each other, despite whatever bits they're unfortunate to learn about the upper powers.)
Cairo may have a way out: Commonly the main criteria for a Ph.D. is: (1) Pass qualifying exams in several important topics in the field. (2) Do some research that is an "original contribution to knowledge worthy of publication" where the main criteria for publication is "new, correct, and significant".
No professor or university Dean can keep her from doing well on both (1) and (2). It looks like (1) would be easy enough for her. The work she has already done may satisfy (2). Once she has done well on (1) and (2), tough not to award her a Ph.D.
PhDs in the USA also have curricular requirements, with at least some required courses. These courses are only available after a bunch of paperwork is given to the university. So, it's not so easy.
My Ph.D., applied math, Johns Hopkins, supports my post.
In my university, when someone is accepted as a graduate student of a different topic (let's say phisics -> biology) it's usual to include a few of the last courses of the major as a mandatory part of the Ph.D.
- moved between countries or first/second gen immigrant? check
- home schooled? check
This on top of her extraordinary talent and hard work. Institutional education truly is a great leveler, at both the top and bottom.
> moved between countries or first/second gen immigrant?
“Cairo grew up in Nassau, the Bahamas, where her parents had moved so that her dad could take a job as a software developer”
“Travel restrictions stranded her family at her grandparents’ house in Chicago. While they were there, she joined the Math Circles of Chicago”
This doesn’t read like an immigrant. It kind of reads like her dad is a fully American finance dev.
“After the spring semester ended, her family moved from Davis to Berkeley — her brother had decided to transfer there — and Cairo finally felt able to settle in.”
fascinating that the family follows the kids’ educational steps.
Are home schooled over-represented among mathematicians that solved major math mysteries?
Wait that would not even prove anything.
Are there no mathematicians who solved major math mysteries that went to school?
I don’t know about major math achievements- the n is pretty small- but homeschooled kids are overrepresented in geography bees, spelling bees, and other academic competitions at the high school level.
https://readlion.com/homeschoolers-dominate-inland-northwest...
It appears that homeschooling enables students to pursue singular passions further than students in schools. It’s not for everyone though.
My point was just to note some other interesting characteristics beyond the age and gender that show up in coverage.
I wonder what would have happened if she said "mom, dad, I want to be a fashion designer/chef/interior decorator." Undoubtedly, they would have supported her ambition and individuality..of course.
Earlier discussion: https://news.ycombinator.com/item?id=44481441
I wish her the best in her coming career.
Thanks! Macroexpanded:
A 17-year-old teen refutes a mathematical conjecture proposed 40 years ago - https://news.ycombinator.com/item?id=44481441 - July 2025 (105 comments)
I 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
Her 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...
I studied math and knew a girl like that, spending an inordinate amount of times making sure to understand every single details of what we were taught, even going further when I could barely keep up. She would also make those pretty, colorful notes with visual interpretations and drawings in the margin. It takes a special form of love for the discipline. She just finished her PhD and is now starting a postdoctoral contract.
Looks like a prepared presentation rather than notes.
If all presentations were prepared like this I’d probably attend them
Indeed. All presenters can take a lesson from this talented person.
Wonder why Berkeley didn’t offer her PhD admission given that she was already working with the Prof there.
> When I was an undergraduate at MIT I loved it. I thought it was a great place, and I wanted to go to graduate school there too, of course. But when I went to Professor Slater and told him of my intentions, he said, "We won’t let you in here."
> I said, "What?"
> Slater said, "Why do you think you should go to graduate school at MIT?"
> "Because MIT is the best school for science in the country."
> "You think that?"
> "Yeah."
> "That's why you should go to some other school. You should find out how the rest of the world is."
-- Surely You're Joking Mr. Feynman
I've been at institutions that explicitly discourage that because of that reason. They want you to get breadth of experience rather than work with the same one person for your early career.
Here's a link to the paper on the arxiv: https://arxiv.org/abs/2502.06137
Zvezdalina Stankova who comments on miss Cairo is on her own super out of the ordinary.
https://math.berkeley.edu/~stankova/
Not only she did grow in Bulgaria during the most turbolent times of regime change from communism to democracy, but later graduates with a PHD from Harvard, and later becomes Director and Founder of the Berkeley Math Circle, and is also organizer of math competitions in Bay Area, and publisher of what seems to be a complete set of Math Books, carefully crafted with her peers from BG and presented here
https://archi-math.com/
Curious whether miss Cairo was a student of hers or is to be.
Alexander Givental is also involved with that, I think.
https://sumizdat.com/homepage/index.html scroll down.
This math stuff is corrupting our youth.
Besides how can she just skip college and go right into a PhD? She hasn’t even taken intro to Sociology or poetry yet.
>Cairo kept reading and thinking. Eventually, she found a way to construct a strange, complicated function out of waves whose frequencies all lay on a curved surface — the type of surface the conjecture required. Usually, when you add these kinds of waves together, they interfere, canceling each other out in some places and reinforcing each other elsewhere.
Rather than speaking about her age or the vague notion of talent, I'd be much more interested in why the rest of the academia was unable to replicate her methods in 40 years or so. From her own admission, it was throwing different ideas and approaches until she saw a disprecancy that ultimately disproved the theorem. There should be many people far more experienced who do have the knowledge to do this, but why did they not? This dosen't look like a intuitive jump, seems more like building a test case that fully stressed the theorem.
Heh, take a look at her paper [0]. It is much more complex that TFA makes it out to be: once you have the intuition you need to prove it actually disproves the theorem.
Also, it's probably a not very well known conjecture that people wouldn't care too much about. Also also, it was a conjecture: people were probably looking to prove it rather that disprove it, failing and then giving up.
[0] https://arxiv.org/abs/2502.06137
As 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.
Offtopic, but I am 17 too just like Hannah Cairo but nothing too groundbreaking till now I suppose and it absolutely brings me delight that I can talk to somebody who was a pupil of Dijkstra, I have heard a lot about dijkstra's algorithm's and I had forgotten about it and so I searched it right now, but the only thing I knew is that it is pretty popular algorithm.
If I had to ask you kind sir, what would be the biggest life lesson (in coding, or anything general) that you could give me be?
> what would be the biggest life lesson (in coding, or anything general)
Pursue what interests you and what fires up your passion, not what grown-ups tell you is “lucrative”, “prestigious”, “profitable” or somesuch.
I mean, my interests are just really niche in the sense that I like to glue projects and one of my favourite hobbies is to search if there is an open source replacement to something or if I can use open source things to do stuff
So there is intel tiber cloud and it gave me a free jupyter and I wanted to play minecraft server, so I went to something which is a fork of it and I saw that there was a guy who had created rootless patch of ssh but it wasn't working for me so I kindly asked them and they were really kind enough to help me and then I used something like pinggy or the thousand other cloudflare tunnel-esque solutions which work without any problem whatsoever directly through ssh
Earlier I used to use junest to then get something like ssh working.
Can I be honest? I am so proud of myself because I guess I did all this stuff an year ago but I still remember the day I asked my friend to join the mc server and I joined too, the ping sucked but 8 gigs ram or even more and no storage limits and complete freedom... and I thought of such stuff after almost 2 weeks of trying different things and learned a lot about nat punching etc.
So such stuff always makes really sense to me, just tinkering with software. But I feel like I need to create some professional software too in order to leave a legacy or just, I feel like, if I am being honest, that the things that I am doing aren't that valuable. Anybody can learn such stuff from me from some blog post I write but there is nothing that I can ever monetize.
I don't know why I am seeking monetization that much but its just that, I see all these kids hustling my age too and I feel like everyone's doing everything for the money... so why should I be any different. I get a lot of ideas but I feel like I can never earn a single cent out of it and I find it a little frustating. I feel less confident myself regarding if I will ever earn decent money or will just be a low level cog of this whole system.
Found it https://github.com/mkj/dropbear I really love this project so much, it really helped me do something really wonderful.
I have nothing to add to that.
Not the one you asked, but here's what I would have told my 17 year old self.
* "Slope beats y-intercept." The best computer scientists and engineers I've ever worked with and/or mentored embodied this principle more than anything else.
* It can be tempting to over-optimize for short-term milestones (e.g., an important admissions exam, the next job or promotion), but there is a significant compounding value to knowledge accumulation and truly learning your craft well. Read and learn as much as you can, all the time, even if it isn't immediately necessary or useful.
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.
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.
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.
What is the general basis for skipping college and getting admission directly in a Ph.D. program? What does one have to generally do to qualify?
AFAIK there's no general procedure; it's just that the admissions committee can admit anyone whom they're convinced can do Ph.D.-level work, and that almost always involves getting an undergraduate degree first but in extremely exceptional cases it might not.
The basis would be getting a dean's approval (or lack of a denial). That is the level at which "Exception Handling" is normally done at a university (above professor, below vice president), and it would normally be the dean of the department that housed the graduate program (some grad programs span multiple departments, which makes it more complicated).
Likely, there are also rule systems that would need to be circumvented; for example, while the admissions program for a graduate program would accept the student, the registrar or other entity might then negate that acceptance due to standing policy. At that point, I wouldn't be surprised if it reached the vice president level of the university (VP of the School that housed the department).
Every university handles this slightly differently. Exceptions can often be made.
To qualify you would normally have to do something truly exceptional or show a pattern of continuously excellent (grad student level). From what I can tell (I am not a math expert) she is considered a 'prodigy' and the problem she solved would require considerable skill, in a way that predicts continued successful performance at the university professor. The history of prodigies who skipped various steps of the normal schooling path is mixed- personally, I think people undervalue the importance of spending time with people in your age group during the 15-25 range, but honestly, without knowing more detail, it's hard to say.
For context, Stephen Wolfram dropped out of Eton to start at Oxford, then was admitted to Caltech's physics PhD before finishing his Oxford BA, receiving his doctorate at the age of 21.
(He was publishing papers in high school.)
I knew someone who was admitted to a Masters program at Stanford sans undergraduate degree. His employer at the time was a large donor.
Essentially, you have to convince a math professor that you able to complete the Ph.D program. There are a variety of ways to do that. Solving a major open conjuncture is clearly one of them but just getting to know one or another professors can work (I went to school with the semi-famous Paul Lockhart who also did this).
be actually gifted
this is what gifted looks like, not legions of students who are told they are "gifted" but are just pretty smart
Yes, but it's not enough.
Be actually gifted and apply to a university that isn't run by dickheads.
This is the most impressive thing I've seen in years.
Amazing story on a no-less amazing teenager.
Also, I love the handwritten slide on one of the photos. Very nice.
Wow, this is remarkable. So inspiring to read, even though I'm terrible at mathematics.
Previously: Hannah Cairo: 17-year-old teen refutes a math conjecture proposed 40 years ago (https://news.ycombinator.com/item?id=44481441)
I hope she doesn't burn out and move into the woods like Grothendieck, Kaczynski,
I have yet to move into the woods but, given my current levels of burnout, I think such a proposition is highly underrated. What's wrong with moving into the woods?
Wait, if everyone moves into the woods there will be no more woods that offer seclusion. Yeah moving into the woods is awful! I hope she doesn't burn out!
One option could be to get a hammock and go program in a tree. That’s sort of like moving to the woods, but you don’t have to live there full time and you still get to program.
Kaczynski was really not on the same level. By several orders of magnitude.
It's generally well documented that Kacynski was a highly talented mathematician.
Yes, but like the gp said, not even remotely close in skill.
how would you know for sure? I don't have the skill to evaluate somebody who has disproved one conjecture (maybe some recent techniques made it easier?) versus somebody who had an established pattern of doing excellent work.
I am familiar with both of their work and can judge the relative quality.
just to clarify, do you mean Kaczynski is not on the level of Grothendieck? In which I can agree
no, he's saying Kacyznski is not on the level of Hannah Cairo. If he meant Grothendieck, that's another conversation. Reading the context of this subthread, it should be clear we're discussing Kacyznski vs. Cairo.
I don't have the knowledge to evaluate this, but I still think Cairo has not esta blished enough work to say she's far ahead of Kacyznski.
thanks for the clarification, I also dont have the ability to evaluate who is better kacyznski v cairo. hence why i assumed the original commenter meant kaczynski vs grothendieck because for one kacyznski has a phd in maths, cairo does not have that metal yet
... and one who was younger than the other students when he was at university.
I mean, force feed her massive doses of LSD and use elite psyops to incessantly harass her as a ̶ ̶p̶r̶a̶n̶k̶ science experiment and maybe
> 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 excuse to make them forego other curriculum, especially if it is required for other students to take. Schools have a responsibility to educate people to a certain standard, and give them some general breadth.
The woman is not being "made" to do anything. She's already world-class at math and wants to maximize the impact of her specialized talents, so she's going straight to PhD.
I don't think we need to nanny people who break the mold with extraordinary talents to conform to some generic correct educational sequence. They've proven they know how to make something of themselves and their own ideas should count for more.
Imagine if every AI researcher out there was allowed to skip even the most fundamental philosophy class(es).
I don’t know about you, but exposing these prodigies to some shared classes is not all that bad. Personally I think every student should be forced to take a class on ethics.
I'm not sure higher ed's educational philosophy is serving students all that well. The breadth of education is a shallow survey at best that is quickly forgotten exactly one semester later.
Thankfully the workforce has common sense and will happily snap her up into employment.
> The breadth of education is a shallow survey at best that is quickly forgotten exactly one semester later.
Your own experience isn't generally applicable. Although it was a couple decades ago, I still use various things I learned from my non-major classes pretty much daily.
Most of that you can get with diligence in high school and a smattering of College classes as HS junior/senior IMHO.
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
This is the entire premise of a liberal arts education. Not everyone pursues one, but it's certainly well-represented at the tertiary education level.
I've not studied this, but my guess is that the liberal arts education as the foundation is necessary to allow young people a chance to figure out who they are. I certainly found this to be true for me and would guess for my closest college friends.
If a young person is exceptional, do we force them into a liberal arts box? Surely there is value in literature and history. But this one young woman had found her passion. I have to believe that is she found out about something else, she would take that on.
Maybe. What interested me at 17 certainly does not interest me now.
You say that like it’s a universal thing. It’s much more of a Liberal Arts thing, which is much more prevalent in the US then in Europe.
> especially if it is required for other students to take
How many of the other students disproved 2 conjectures in advanced calculus that were open for decades?
> It is entirely possible for people to intensely focus on a very, very narrow thing - and ignore everything else.
sounds like Phd
In my teens I worked with the statistics department at UTMB. That had a cast of characters there; many profs in the 70s and 80s, who'd gotten their degrees before WW2. A number of them had schooling of the form: Start school at 9-10, do 5 years of public school, got to a 1 year prep, do a year or two of college, do a two year PhD. Most of them had their PhD's by 22.
I think the biggest flaw with higher education today is that we're pushing people into doing undergraduate degrees who are already well beyond coming out of self learning from high school or other experiences.
The number of high schoolers who might be ready to go into postgrad programs is very small.
There is no way this could be the “biggest flaw” in higher education today because the number of people possibly impacted is so tiny.
Although I think you’re striking at something that is a real problem with undergraduate degrees today: Many universities have become so watered down and softened that students spend the first 1-2 years doing what they should have been learning in high school.
My friends who still teach at university constantly complain about students arriving for undergrad with very poor writing, communication, and listening skills.
Very few people are too “advanced” to be challenged by a sufficiently difficult undergraduate degree, ridiculous thing to say imho, I went to a UK university this decade and I can give you a laundry list of issues more significant than “exceptional students being slowed down”.
This woman is undoubtedly exceptional. But we don't know how exceptional, because she's an outlier, educated using different methods.
We have no idea how many other people would achieve something similar with a similar background. Personally I'd bet almost anything it's a larger number than most people expect.
I'd also be surprised if she doesn't already have a pretty solid background in undergrad-level math.
The irony is she's actually more typical than not. Universities in the past were open to giving unusual talents special treatment.
Historically, the idea that everyone must follow the same path on the same timetable is unusual.
> everyone must follow the same path on the same timetable
We are livestock for corporations, that's why.
To the right people a university education can be an asset rather than a barrier to entry.
That’s one of the biggest problems? Not pushing people into higher educational programs with little societal or economic value but huge loads of debt?
The debt part is just a US thing, but the rest of us still have the other problems.
Not remotely true, and the US isn't even the worst. Students in the UK graduate with more debt.
australia copied the US in that regard. they used to say we were the 51st state but i dont even think we qualify as that... a territory perhaps?
Australia's debt is in the form of HECS/HELP. It's indexed to inflation, and integrated into the tax system such that it's only repaid on income over a minimum threshold. The first threshold is 1% of income above $56k. The last threshold is 10% of income above $166k.
My understanding is that US student loans accrue interest at rates above inflation. They do have repayment plans based on income, but because the interest rate is higher than wage growth, the debt just keeps growing if you're on these plans. They also don't have progressive thresholds like we do and the repayments tend to be higher.
Australia has a similar but less severe problem. Inflation and wage growth are closer together but they aren't the same. Still, the situation in the US seems worse.
I think research and higher education does have value for the most part. It's undergrad that's really worthless and something people only do for the experience.
Undergrads who care about learning and research will take the most challenging classes, do research with professors, and surround themselves with other strong students who will push them.
Even at top universities, very very few freshmen are capable of doing high-quality research immediately. They'd be better served learning the foundations inside and out with a cohort of similarly strong students to challenge them.
To agree with you: I've worked with several really brilliant undergrads doing and publishing great research. But all of them were rightfully undergrads. Even if they were actually capable of doing great research, they benefited from the breadth.
If you have bright enough undergrads, you change the curriculum for them within their field of expertise, so that they still get the breadth of things outside it while not wasting time with things they know. You let them not take as many classes, take graduate courses, do more research, take more courses from other departments in related areas but with different perspectives, and so on.
When I was an undergrad, in physics, there was a professor in the department who had done his undergrad there and was legendary, as was quietly mentioned in awe, for not taking any undergraduate physics courses while there; the department had let him skip all of them, and instead take graduate courses and do research.
If you do research during your 4 year undergrad. You shouldn't have been undergrad. It's really that simple.
I'm not sure that's a simple argument and can't imagine many would agree.
Undergrads who do research generally aren't very good at research yet. A major reason is they either lack or don't fully understand the pre-reqs, which they progressively and cumulatively learn during undergrad. A student can be incredibly smart, but acquiring a strong rigorous math background will still take years.
About pre-reqs: third and fourth year PureMath classes at UofWaterloo consisted of math I already took in HighSchool in Romania: group theory, ring theory. Plus some calculus I already read in high school out of curiosity: measure theory and the Lebesgue integral. Another Romanian guy at UofW was auditing 4th year classes while in his first year (he is now a math professor at an American university)
I can see a committed and gifted student being able to get most of the pre-reqs for doctoral studies in America or Canada while in high school.
Working on that skill and ability is the entire point of postgrad. If those are the skills you're working on then you should be in a postgrad program.
If you don't know the foundations well, you don't belong in a postgrad program. That's the reality and how it currently works. Undergrad teaches you those foundations.
Anyone can try doing research, even undergrads who half-know the foundations. However, trying research doesn't mean you have the background to do great research or to succeed in a postgrad program.
Let me ammend my statement. *"Anyone who succeeds at publishing research deserves to be in a postgrad program."
Plenty of people in postgrad programs don't know the foundations. It's ok. You are there to learn.
Completely unfair to expect someone already doing research to slog out 4yrs of classes not furthering their career.
You can definitely do research in an area without having a good background on other topics.
People sometimes accidentally do research. I'm not joking.
Indeed. I've long felt that most undergrad students would be better served by a typical college minus the formal classes. Basically dorms and all the other amenities found in a typical college campus, where you mainly gain life skills and mingle with other people your age. Because most people I met at an average 4-year school were there because it's a societal expectation among certain classes, it's less scary than just getting a job and figuring out life completely on your own, and it is 10x-100x easier to make friends at college than just "out in the world." Not on the list: to learn from college classes, which at an average school teach you less than you'd get from a $200 a year subscription to Great Courses Plus or Brilliant. Or free from Khan Academy.
I know a few very special schools give undergrads access to brilliant minds in their field, but I also have been told that undergrads at those schools are mostly taught by grad students, so I'm not sure that Ivies provide a lot either, beyond the opportunity to hobnob with the legacies that will be running Goldman Sachs in 20 years.
> Jeez... what a damning indictment of today's Universities.
There’s a modern phenomenon I’ve been thinking about but have struggled to put a name to.
Everything just becomes so generalized, streamlined that it becomes impossible to operate outside of the pre defined “happy path”.
AI will make this increase 100x as taking humans out of the loop seems to accelerate this process.
Perhaps the term "canalization" fits? Coined by C.H. Waddington, describing the process of forming a chreode in a homeorhetic system. But it doesn't really encompass the "everything becomes overly generalized" concept you mentioned. Rather, it's more about robustness against perturbation of a trajectory in a conceptual space. Christopher Alexander called them 'paths in configuration space'.
Object orientation applied to life.
The interesting part is that two others admitted her but retracted after "higher-ups in those universities’ administrations overrode those decisions".
Was she not considered properly conditioned?
Of course she was, but that's not what this is about. Letting her in proves that the bureaucratic credentials offered by schools are meaningless. The university system in its present form in the US is entirely predicated on the fiction that those credentials actually mean something and are worth paying six figures for.
"I'm in charge, I enforce the rules, I'm a big deal."
> Jeez... what a damning indictment of today's Universities.
> She could just use her publication as a dissertation and be done with it.
I’m not suggesting this person is doing anything fraudulent as she seems quite impressive.
However, educational institutions get constant requests from parents who want their children to skip far ahead before they’re ready. It’s a competitive world and they know that being able to claim a child skipped several grades or even skipped undergrad entirely is a unique and very impressive achievement for the resume. It also theoretically provides a few additional years of earning potential by giving a career head start.
The first problem is that many of these parents (again, no accusations for this specific case) see this and want to make it happen for their child at any cost. There are some wild stories about parents trying to cheat their kids forward or falsifying their accomplishments to try to skip grades.
The secondary problem is that it can be hard on kids to be thrust forward so far past their peers. I had several friends who skipped a grade in middle school and most of them didn’t have a great experience for social reasons. Skipping undergrad altogether would thrust someone into a foreign world with a lot of baseline expectations and norms that they hadn’t yet learned, combined with no peers their age to discuss it with.
It creates a high chance for burnout or failure, which could leave them worse off than when they started.
That’s why the recommendation is generally to do undergrad at a challenging institution that allows students some upward mobility in specific areas where they’re ahead. No reasonable undergrad program is going to have this person taking Algebra 101, but there are a lot of opportunities for them to jump right into advanced programs and go deep and broad.
IIRC Erik Demaine was also homeschooled, skipped college, got a very early PhD, and joined the MIT math faculty as a teenager (he is a full professor there now). Johnny von Neumann OTOH went through a "normal" secondary education partly because his parents wanted him to have traditional social exposure to kids his age while he was growing up. His math training was very accelerated though, and he had professional level research publications at 17.
Clearly the correct method is: publish your “dissertation” first, then apply for admittance to the PhD program.
Hey, you are not getting your degree without paying for it.
> She could just use her publication as a dissertation and be done with it!
The purpose of a PhD is not writing a dissertation. It is a research school, and I'm sure she could still learn a thing or two about research (and teaching).
Two out of ten is pretty good for anything involving individual decisions, just ask any salesperson.
Conversely: if that's all it takes, there is no point in going to University just to get a piece of paper that says "you did the thing you already did".
University (for folks serious about continuing in academia after) is (obviously) about making sure you have the same base knowledge as everyone else, but also for you to come to terms with how academia actually works, who the bad players are, who the good players are, and who you need to know to get shit to happen for you. So in that sense, most Universities going "no" is literally the most accurate reflection of what life's going to be like on a continuous basis on the inside.
I don't see why it would matter. she could quit math now and still be ahead of the majority of mathematician in terms of contributions. The rest is just formalities. I can see her breezing through the undergrad, quals, etc . It would just be a small delay.
Normally you have a master of science as well. And for that you require a bachelors. So to do it directly you need to do it all at once?
In the US a doctorate usually doesn't require a master's, most people go straight from undergrad to phd.
To add slightly more detail: Most research math programs in the US are 5(ish) year 'combined' masters+phd programs. The first couple years are basically course work, seminars, finding your area and advisor, and then the rest is the actual research work. It's not uncommon to leave after the first couple years with a Master's degree.
> Normally you have a master of science as well. And for that you require a bachelors.
Many Master’s programs require Bachelor’s as a prerequisite for admission, and some of the others include a Bachelor’s at some point, but not all do. (The same is true of the Ph.D. and the stereotypically preceding degrees, and also of some professional degrees and Bachelor’s degrees.)
I have a Master’s and never got a Bachelor’s. Universities can admit people more or less as they please, subject only to the very forgiving scrutiny of accreditation agencies or whatever government department officially supervises them.
There is a story about when Stephen Wolfram applied for a job at AT&T. The HR person asked him where had gone to college. Wolfram confessed that he had never gone to college. Well ok, did you graduate high school? Um no, I'm afraid not. GED? Nope. What about primary school? Wolfram looks more and more more embarrassed, and admits that he didn't even finish primary school. HR person by now is quite uncomfortable when Wolfram manages to think of something. He says "I do happen to have a Ph.D. Does that help?"
That's a nice quip, but aren't degrees meant to offer breadth of knowledge? (I'm sure she has lots, but perhaps is weak in other areas.)
This is ignoring the value of the whole education process, that you go through the years at a university.
I disagree that she should skip the general education.
Thinking about my GE requirements at undergrad, I think it would be a waste of this girl's time to be forced to learn about and write about random subjects that don't interest her. She has but one lifetime, and can contribute much to her field.
The subjects such as English composition, she should be allowed to test out of if she is already a good writer.
> I think it would be a waste of this girl's time to be forced to learn about and write about random subjects that don't interest her.
This is a common STEM view, but it is inherently wrong. IMHO it dates back to the unfortunate divide between science and the liberal arts, whereas both were once considered a single field, now days there is disdain and mistrust between the two sides.
The point of history classes isn't to memorize the dates of wars, is it to understand the motivations of humans, it is to understand how the world we lived in has been shaped throughout time, and it is to learn how to do, and understand, research about the history of a place.
The point of English classes isn't just to get good at writing, it is to get good at various types of writing, it is to learn how to read different forms of literature, and it is to have a guided tour through a chosen selection of literature to hopefully develop one's character and thoughtfulness.
One of the most valuable classes I ever took at University was the Art Of Listening To Music. We started off around 500ad or so and went forward through time up until about 1920. We learned the vocabulary of music, how to sit down and listen to a piece of music and describe what we were hearing. After I was done with the class I went from appreciating a handfuls of genres of music to appreciating music itself no matter the genre. It was a 3 credit guaranteed A class that had enriched my life by an enormous amount.
If you really love your major, then the point of going to university was NOT your major, odds are you would've studied that field with or without the school. (Barring fields that require large capital investments, chemistry, physics, playing with an entire orchestra, building airplanes, etc) The point is everything else.
But that isn't how it works in reality, at least in the US. In reality, outside of their major (and sometimes inside it too), students usually pick the absolute easiest classes that satisfy the requirements. The ones that are known that the teacher doesn't take attendance are heavily desired. And the university is happy to oblige. Departments are funded based on a formula of how many students are in their classes, and they know that if they gain a reputation for being hard, students won't take their classes for GE requirements. It's a race to the bottom. So most departments offer enormous 1 level classes with 200 students taught with minimal rigor, and where you really only have to study a few days before the final to collect your A. And on top of that the frats all keep collections of graded tests from every class for years past, so basically anyone who wants to cheat is able to do so easily. This isn't education. This isn't worth six figures.
> In reality, outside of their major (and sometimes inside it too), students usually pick the absolute easiest classes that satisfy the requirements.
In reality, no one eats healthy food, everyone eats fast food hamburgers all day. Just look at the sales numbers of fast food / junk food VS organic lunch salad bars!
Except, that isn't true. Some people eat junk food all day, and some people choose to eat healthy. Obviously in America we have a bit of a bias, but just looking at averages doesn't give a complete picture.
> The ones that are known that the teacher doesn't take attendance are heavily desired.
Almost none of my university classes took attendance. Why would they? We were paying to be there, if we wanted to waste our money, it wasn't the university's problem.
> So most departments offer enormous 1 level classes with 200 students taught with minimal rigor
Reading books and writing essay's doesn't require rigor, the learning is in the doing. I put in honest work to learn and I got honest feedback from my 100 and 200 level professors, which was all I expected.
> And on top of that the frats all keep collections of graded tests from every class for years past
Almost none of my GE classes used multiple choice tests. They were typically essay tests, written in class.
I should note I did my GE requirements at a local community college, where class sizes averaged ~20-30 students, professors had office hours, and I think I only saw a TA once.
> This isn't education. This isn't worth six figures.
The price is too high yes, but a university is place to go where you can dedicate yourself fully to learning, hopefully w/o other outside worries (sky high tuition ruins that...). What a person chooses to do with that time is up to them.
Now one can argue that the worth of a degree is lessened by students who didn't actually learn all that much also being in possession of one. That is a closely related, but separate topic.
That said, the poster I was originally replying to was indirectly advocating for not caring about one's GE classes, and I was replying that one should indeed care, because those classes are incredibly important!
>One of the most valuable classes I ever took at University was the Art Of Listening To Music.
I also benefited tremendously from such a class. I would recommend it to anyone interested in music, or not interested in music and trying to figure out why others are interested in music.
> what a damning indictment of today's Universities.
Prodigy skips undergrad -> Universities are doomed? What?
50% of my college education was general education though.
And frankly I find it just as valuable.
Wait, what software engineering jobs require you to move to the Bahamas?
Insurance is a pretty common one. Lots of insurance companies have offices there - and if you're a top-notch dev, you can make a crap-ton of money as well.
In my experience, online gambling or banking.
Rich people offshore money there
A less click-baity headline might be "17 year old Hannah Cairo has solved the Mizohata-Takeuchi conjecture"
probably add "in the negative", to that.
"17 year old Hannah Cairo has disproved the Mizohata-Takeuchi conjecture"
Is it an important conjecture, or just something someone came up with last week?
This is answered in the article’s second paragraph.
But I don't know that from the proposed re-titling of the article. The importance of the conjecture is more salient for the general reader than the (uninformative) names of the people who conjectured in the first place.
It's loosely explained deep down in the article.
...which is not the HN title, so parent's point still stands
There's so many photos of just her staring off into the distance, and only one photo with her presenting the actual thing that she's supposedly famous for. I don't get the point of just all these random photos of this girl.
It's not very easy to take photos of mathematics, and the girl is the central subject of the article anyway. What images would you expect to see?
What I'd like to know is how many 17 year olds failed to solve their math mystery, and chose a career in programming instead.
Many people who succeed in solving their math mystery still end up choosing a career in programming
Almost all of them?
I never got excited for math, thus didn't care much to start with. But then add the issues of high school math and that solidified it for me. Math word problems were the worst. With a dyslexic brain I would consistently read the important words as something different, thus correctly solving the wrong problem and being derided for it.
Geometry required rote memory exceptionalism which also works against my brain design (adhd as well), but let me have the formulas to choose from and I'll get it done. Algebra II? I kept getting in trouble because I would do homework assignments in class instead of paying attention to the teacher trying to teach concepts that were easier for me to learn by reading the examples and following the book. After that, who would ever want to continue beyond the required credits and not think of further math as a masochist hellscape