Math is still catching up to the genius of Ramanujan
quantamagazine.org
quantamagazine.org
What really stuck out to me was how R failed in a bunch of other subjects except math because he wasn’t interested in them.
I know society and norms expect students to learn all these other subjects.
But what if those just aren’t interesting to someone?
I wonder how many geniuses we skip on because doing the chores of homework and getting through boring classes is busywork and memorization for the sake of getting an A.
Meanwhile, hardly anyone actually remembers anything about those topics and even the best students mostly go on to achieve only above average things.
My class valedictorian went on to become a doctor and while that is certainly impressive to me, there are many doctors and he practices (like almost every other doctor) and isn’t pushing the boundaries of medical science. I feel terrible writing that because I’m certainly not as smart as him, but R is just so impressive and I’m glad he got his lucky break.
People like R would be lost in the sea of averages because their genius would be kept shut by norms.
Almost every extraordinary person I read about seems like they were 1 step away from being forgotten, and got some huge universal break that boosted them.
It's good that public school exposes children to many subjects - hopefully most of them. So that they can discover if they click with one of them. The real danger is that someone never gets exposed to a subject at all. College is the place to specialize in a subject.
Why is that useful besides for the employer trying to impose Framework v3.0 onto their subjects?
To me at least, learning things one is not excited about is only useful to capitalist society that views human beings as replaceable resources.
Let's use another example. Communicating with other people is a very important part of my job as a software engineer. I'd say that at this point in my career, writing good proposals, documentation, project charters, task orders, change requests, etc. is vital to my individual and collective success, so I'm glad I was taught lots of rhetoric and literature and writing and all that artsy stuff my younger self spent too much time disdaining.
Or another example. I have written precisely zero lines of Lisp or ML for work, and I flunked the first computer science class I took that used Lisp (good old _Essentials of Programming Languages_). And yet I use the concepts I learned in those classes _all the time_. Heck, I was just talking about side effects the other day in the context of 21 CFR Part 11 computer system validation.
My last metaphor. Brushing one's teeth or taking a shower isn't particularly exciting, but that kind of routine maintenance work is a vital part of your physical health. So it goes for mental exercise. Reading something challenging, learning the work's historical context, and writing a critical response well all helps keep you mentally healthy. That's good no matter what socioeconomic system you live in.
My university didn't allow any classes above the most introductory ones to be considered as fulfilling the general education requirement. I signed up for a history class that would involve doing research and having weekly discussions with a small group. I was stoked. Then the professor made a note that it didn't fulfill the general education requirements. I had to drop it and switch to a huge-ass mindless lecture of hundreds of people. I would've liked to still take the more in-depth history class even if it didn't fit the gen ed requirement, but so many of those BS classes are required that my schedule was completely packed all 4 years with zero leeway.
I still find it funny and informative that multivariable calculus, linear algebra, and high level math classes did not automatically fulfill the math requirement because no one ever tried to use them to satisfy it.
At Big US Engineering School, many people are done with their prerequisites in a year.
Unless you're talking about painful core classes like "compiler design" and "networking", which I would say is a different conversation.
Being specific, it's not school, it's what school grants you i.e. a paying job. The higher paying, thusly more coveted jobs, generally filter against good grades which then the requirement pushes downwards into schools because, at scale, it's a decent system; leveraging the schools to help decide who is good.
> Being _really_ good in one thing should allow you to make up for being subpar in other areas, but it doesn't.
I counter with, if you are "_really_" good, it shows because you truly are a genius and you get fast tracked on that subject, but I think your "_really_" is actually just "_pretty_" and you're trying to include more than the 1 in 100 million.
To directly answer your point, for the "slightly smarter than everyone else" my middle school allowed kids to attend highschool in specific subjects and then highschool into the nearby community college and considered "harder/more prestigious" than the AP programs - admittedly only in math for this latter part. The school was in a more affluent neighborhood so I recognize the privilege.
I don't quite understand your point. Pretty good still puts you far ahead of the average. I could easily handle second year college maths and computer science while in high school. And I couldn't hold a candle to Ramanujan.
I still needed to do well in my other courses in order to be able to get into my chosen college.
Whereas if someone was Ramanujan-level, their raw talent would be so apparent they wouldn't have this issue and would clearly stand out.
It is not enough to stand out in the current system.
The parent was saying selecting the 50 kids who can handle it is a much better approach than just taking the highest overall grades.
The average A's across the board high school student can't handle second year college maths. Yet they will be placed ahead of the observably better at math kids.
Imagine if jobs worked like this - "Yes, we know you are a great developer but you don't really understand economics. Sorry".
Being well-rounded and having exposure to a bunch of topics is valuable to an extent. However, in my experience most of the people making a real difference in the workplace and academia are not particularly well rounded.
Thankfully in tech there are alternative pathways. However, for many professions there aren't and these high performers are simply excluded to societies detriment.
This is exactly the route to exploitation by MBA managers.
Great developer, loyal, doesn't understand the need to change companies to get paid a competitive salary. Perfect hire.
The person who doesn't understand economics pays the price themselves.
You can only progress so much in a field of expertise before hitting diminishing return.
At some point it makes sense to broaden your knowledge and skillset.
I suppose how long you can progress for and how far you can progress depends somewhat on the breadth and depth of the field of expertise.
Many fields of expertise are so broad and deep that they have their own sub-fields just to make them manageable.
So you would probably be in a sub-field and then broaden your knowledge and skill-set in a related sub-field of the overall field that you are well suited to.
I'm betting it's likely you can see how your own particular field, as you are on HN, replicates this pattern.
What I'm saying is in some hypothetical system which places a great emphasis on specialization, people who are a few years ahead of curriculum are a dime a dozen and will not stand out. There were 50 kids in my high school, so how many is that nationwide?
Particularly in math, it is straightforward for an exceptional talent to stand out. Competitive math is a clear pathway/credential. If someone is not able to achieve meaningful results then it's probably just the case that they aren't as talented as they believe
> Imagine if jobs worked like this - "Yes, we know you are a great developer but you don't really understand economics. Sorry".
There are plenty of software devs who are not getting hired to the potential of their raw development skills because they cannot communicate or collaborate productively. There are also plenty of software devs who are not getting hired to the potential of their raw development skills because they don't have as much domain knowledge as other devs.
I'm not sure what you mean.
High school has a much lower, but broader, bar.
The number who can handle second year college math are a small fraction of those who can get straight A's at high school.
> There are plenty of software devs who are not getting hired to the potential of their raw development skills because they cannot communicate or collaborate productively.
These are behavioural issues and not knowledge issues. We do not address these kinds of issues at high school at all.
Getting an A in English doesn't prepare you in any way to be a team player.
In my ~15 year career the only people I've seen not hired because of a knowledge gap in this space have been non-native speakers.
But he ignored all subjects besides math and lost his scholarship within a year. He later enrolled in another university, this time in Madras (now Chennai), the provincial capital some 250 kilometers north. Again he flunked out.
Maybe it would be different now?
Most of these universities look at an applicant's grades for just six courses. After looking at the courses required for certain programs (such as calculus and physics for certain programs), the remainder of the six courses comprise the student's top grades for any courses at the Grade 12 (final year) level.
So, a high school student aiming for a top engineering or mathematics program will not be hamstrung by a poor grade in Grade 12 English, nor will a student aiming for a top international relations program be hamstrung by a poor grade in Calculus. At the same time, the student going into a STEM program will have an exposure to Shakespeare, which can provide inspiration and a rich set of works to explore later in life. The student going into international relations may later be inspired some years later to study mathematics for its beauty as a hobby, some years later.
I remember the feeling that I was wasting time with many of my courses in those years, despite having good teachers for many of them—I thought my time spent on mandatory humanities courses like music took time away from more practical subjects, and I wish I took a programming course (though I did love my English classes). Perhaps this remains true for many students, but I personally took an interest in music performance as a hobby years later in life, and the years-old lessons in music theory came back to me. My English classes also introduced me to literature, which has remained a very important part of my life that has guided me through highly consequential life decisions for the better. It is unlikely that I would have taken an interest in literary works without my exposure to English in school.
Of course, if English is not your first language then you’re not required to take this course. You have an alternate path which may be a lot more work for an English-language-learner but it doesn’t demand the critical reading and writing skills you would need for grade 12 English.
However, what I found is that even if the professor disagrees with your paper you’ll usually get an 80% if it’s well-written, well-researched, and well-argued. To get above that on something they disagree with requires both a highly open-minded professor and an argument they can’t find fault with.
Though Computer Science and Physics are distinctly different from the mathematics courses, these are still directly useful for a mathematics student to learn—the problem-solving skills should also carry over. Key mathematical discoveries have been inspired by problems in computer science and physics, and many rigorous university-level mathematics books still draw from problems in these fields to motivate certain problems. At the least, they are less laboratory-heavy than Biology and Chemistry (the student could still attempt these subjects, though, and choose to omit the grades for university admissions).
That leaves a couple of other classes—or just one if English is required, as noted by another commenter. My school offered subjects like Grade 12 Drama, Visual Arts, and Music, where much of the grading was effort-based. In my school, most students in my classes saw these courses as a break from other intensive courses, with grades not being as much of a concern. This would allow the student to avoid using a grade for History, Economics, French (or another foreign language), or another subject.
The English requirement would then be a difficult challenge for the mathematics-focused student. I wish I could speak more about what it was like for most of my classmates who went on to study engineering, as many of them took the standard English course (I took a more demanding version of the course, due to personal interest). My classmates at the time did not seem to have an issue with university admissions to competitive programs despite not enjoying the subject at the time, but the other commenter makes a good point that minimum grades for admission standards have increased greatly since then.
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[1] As an aside: a past classmate—who was brilliant at mathematics and also great with people—later poked fun some years later about the Ontario government's naming for math courses. He said, "there's Grade 11 Functions... and then in Grade 12, there's Advanced (!) Functions." The last I heard, he went on to work as an investment banker at a top hedge fund by profitability in the United States.
Then in university, there’s Elementary Functional Analysis!
Really? What were the axioms?
As I said it was literally one of the first math courses we did, so it wasn't super rigorous with all the technical and logical details.
So responding to OP, you indeed must be an expert in all subjects to have a chance to study in your field of expertise.
In practice, I believe every single Ontario university program lists English one of the required courses so it will always be included in your top six average.
[0]: https://uwaterloo.ca/undergraduate-admissions/admissions/adm...
(Mind you, your teachers might not have approached your classes in that way, and high-end literary criticism tends towards performative nonsense°, so I can understand and sympathize if they were a waste of time, but they didn't have to be.)
°It's exactly analogous to Brain Fuck: huge fun if you have the background, intellectual ability, and are in on the joke. Both are mostly pursued for the opportunities they provide to show off How Smart™ you are.
I go to a mid-tier university (Toronto Metropolitan University) and the admissions average for CS was 97% last year.
The admissions process for universities in Ontario is a joke at this point. Imagine if students going to Harvard instead of SUNY Plattsburgh differed by 2 percentage points in their high school averages.
This means you must be absolutely perfect to have a shot of getting into a good school. In reality, teachers overlook mistakes and just give you the marks needed to get into top school if they think you deserve it. But because everyone does that, you also need to farm extracurriculars.
Maybe I'll write a blog post about this since it sounds like people are interested.
UCs historically admitted using a mix of class rank, GPA, and test scores, but the number of seats at UCs didn't really increase in the past decade+ despite a small baby boom in the 2000s, and the growing prominence of STEM in the 2010s, so the average GPAs and SAT scores for UC admissions skyrocketed.
Plenty of Californians have anecdotes of getting rejected from mid-tier UCs but getting into MIT or Stanford. It's had a downstream impact out-of-state as well, as plenty of Californians now attend out-of-state STEM programs for that reason (played a major role in upleveling UT Austin/UW/UIUC/GT/UW Madison's reputations among STEM-targeting HSers ime) and make STEM admissions harder in out-of-state colleges as well.
That said, education quality for STEM majors is consistent across all UCs so the UC you go to doesn't matter as much academic quality wise.
A degree from a good university signifies a smart and dedicated student primarily because the school selects the best students for graduation. That occurs during acceptance and by making the program difficult, causing bad students to leave.
The higher level of competitiveness is hurting the best universities during that acceptance phase. Ontario universities are no longer able to differentiate between the best and average. Waterloo is an exception because it has introduced math competitions across the province as a way to identify "A+++" students, but only Waterloo benefits from that.
I'm noticing that many Ontario schools are now ignoring the acceptance phase and focusing on the weed-out phase. UofT accepts students into a common math/CS program, then only accepts the best students into CS for second year onwards. Queen's University has a common first year for all engineering majors.
Even so, because the acceptance phase no longer differentiates, a lot of good students that would beat the second phase are caught in the first filter.
In Engineering+Accounting+Actuary+Nursing I agree.
> The higher level of competitiveness is hurting the best universities during that acceptance phase. Ontario universities are no longer able to differentiate between the best and average.
Same thing in California, and that's largely because faculty hiring and infrastructure just didn't keep pace with the amount of students declaring Engineering+Accounting+Actuary+Nursing majors (Accounting+Actuary+Nursing face the same problems as STEM fields), which meant admissions need to be much more competitive because you can only teach so many students.
I assume it's a similar story in Ontario due to decades of austerity in the province.
> Even so, because the acceptance phase no longer differentiates, a lot of good students that would beat the second phase are caught in the first filter.
Yep. Because infrastructure didn't scale.
> Waterloo is an exception because it has introduced math competitions across the province as a way to identify "A+++" students, but only Waterloo benefits from that
Yep, the Euclid is basically a soft requirement now for Waterloo CS admissions.
Honestly, Ontario should just ditch "autonomous" universities and merge them under a single "University of Ontario" system and simplifying cross-system course transfers.
Ontario should also force Colleges to stop giving Bachelors degrees and convert them either into Community Colleges to transfer to a University or convert larger Colleges into Universities.
This is what Quebec does, and most state systems in the US (California, Texas, New York, Minnesota, Wisconsin, Massachusetts, etc).
There is no reason for UWaterloo or Wilfrid Laurier to be two different universities despite being a couple blocks away from each other.
Same for UT, UT Scarborough, York, TMU/Ryerson, etc all in GTA
The main advantage that UofT has for its undergraduates is related to funding. UofT’s engineering design teams tend to be better-funded than TMU’s, though TMU’s engineering design teams still perform very well—proudly outperforming UofT’s teams in certain years despite the funding gaps. In addition, some TMU students I’ve spoken with mentioned that UofT has more research opportunities. The name of the university also has a positive impact on admissions for students applying to graduate school. These advantages are not a reflection of the students who attend, but rather UofT’s ability to raise funds.
There is also a disadvantage I’ve seen at UofT. The TMU engineering students I’ve known have mentioned getting extensions approved easily for assignments and homework, to pursue professional opportunities such as hackathons and conferences. That is generally unheard of, from my experience in a STEM program at UofT. Policies are strictly outlined (with some leniency in many courses, such as dropping the lowest assignment mark), and I generally have not seen professors grant exceptions to these.
But the main difference I’ve seen between TMU people and UofT people is university pride. I’ve met several people who were proud to go to TMU and succeed, whereas I haven’t seen that at UofT (with the exception of UofT’s engineering department). I’m satisfied with the opportunities I’ve gotten due to attending UofT—especially as I was involved with its on-campus work program and an engineering design team there—but I haven’t met many UofT people fiercely proud of their school, in contrast to the TMU people I’ve met.
In any case, I am happy to work with people from either school. Work experience and personality has mattered more to me than the name of the university that a person went to, and both TMU and UofT offer great opportunities for students to gain relevant experience—though these are up to the students to pursue, outside of their required classes.
This would also help the case of people like Ramanujan, he might score perfectly on the math portion of the standardized tests, and despite poor scores on everything else, he'd be distinguished.
There's major inequity with the lack of standardized testing.
So all mediocre kids with money have better chance than a brighter kid who couldn't afford coaching. Of course genius could still make it through this system but genius can also make it thru school grading system. There are enough programs to help out them.
Problem here is that barely above average students who wouldn't want to study all subjects per curriculum start to think of themselves as ignored geniuses crushed by the system.
The fact that there are a bunch of asian kids from poor socioeconomic backgrounds who make it into stuyvesant https://www.refinery29.com/en-us/2019/05/233354/stuyvesant-a... is one example of how standardized testing can make it so that hard work can make the difference. I'm not saying that rich kids don't have an automatic leg up, but that's always going to be true regardless. Standardized testing (if anything) gives everyone a shot.
Realistically, it's much easier for a person of average means to study for a standardized test than it is to buy the transport necessary to an international math/science/business/water polo competition.
It's fundamentally impossible for a person to do well in ice hockey if they cannot afford the hundreds of dollars in skates and protective equipment needed to play the game. Taking a standardized test is usually free. Does money give an advantage in both scenarios? Yes, but at least it's not an absolute barrier in the second.
The exams were made by a small panel of teachers and nobody else knew the exam contents until the day of the exam. Everyone across the province takes the exam on the same day and time. Exams were proctored by people external to the school. Exams are again marked by a panel of teachers outside of your school.
Cheating is nearly impossible outside of someone leaking the exam early.
This is basically what Waterloo math competitions are, though. Except only Waterloo can see the results.
School education standards are the barest minimum and anyone of IQ > 85 can make them.
Some of what's taught in English classes is about clear communication, and some of it isn't.
I think learning the 5 paragraph essay structure was very useful. But that's maybe 3 months worth of learning. The rest was English major stuff. Which is fine, but please don't pretend that it has a lot to do with "good communication".
> Even if all English does is force you to read more, it's probably a win in this regard.
It's not enough to say people had to read for English classes. You have to compare it to the counterfactual. In that regard, I don't think it stood up well.
1. I was already reading a lot. I just read different things.
2. I came to hate pretty much everything I read in class. It's only decades later that I've been able to appreciate some of the classics that we read.
On this point, I think English class was a net harm, at least for me. Of course, everyone has different circumstances; I'm sure there are people for whom a similar program as what I went through would have been a benefit.
Of course you can BELIEVE that English classes didn't help you learn how to read and write more effectively or how to better understand what was left unsaid or unwritten, but believing something doesn't make it so.
And yes, as a student reading the classics often sucks. After all, you're not reading them for pleasure - you're reading them to learn how to identify and discuss their themes. But more than that, you see how themes are repeated throughout history, and how the author's experiences changed how they illustrated those themes. English classes taught you plenty of history - not so much in rote facts, but rather by illustrating parts of the cultural zeitgeist of different eras and how authors reinforced, protested, or recorded what was happening at the time.
I doubt many children appreciate education while they're learning, but adults certainly can be thankful that they weren't left in the dark.
At 45 years old I can confidently say that anything I've "learned" on my native language classes through 12 years of having them was totally useless garbage and I'm using none of it in my writing, reading and culturural appreciation or understanding of the world. Everything I use was self-acquired in the time I had that was not spent in native language classes. Ideas of education are great, but the implementation is terrible to the point of being useless.
On the other hand math, physics and chemistry I learned at school has been immensely practically and culturally useful and I wouldn't know a fraction of it if I wasn't taught it in school.
Even if the goal is a cultural appreciation, Shakespeare's plays will tell you as much about England in 1585-1613 as Terry Gilliam's filmography does about the Anglosphere in 1971-present.
It's more than zero, but it's also missing the overwhelming majority of the context and the world in which it exists.
Modern readings treat Shakespeare with excessive reverence: not just "a playwright" but "The Bard".
Those plays were made to be performed with very short rehearsal time before performing, outside, with no lights (at most fire, but it was wood and thatched and burned down from a theatrical cannon), in a crowded venue where audiences would be expected to jeer and cheer, whereas today it's a finely rehearsed performance by people who take it seriously performed for an audience who consider it high culture.
Monty Python's Gumbies aren't well understood by new viewer today, as modern news treats "the man on the street" somewhat differently than in the 70s. How wrong do modern viewers comprehend Shakespeare's characters, considering that "The Taming of the Shrew" is classified as a comedy?
That said, I was also busy teaching myself a lot of maths and science ahead of the classes; what I learned from school but would not have taught myself was the basics of German and French (though only the former stuck with me), the absolute basics of music notation, some metalworking and woodworking, and PE.
Oh, and the practical experiences in the chemistry lab, though I'd have still done the theory myself without that.
Why, given the vast ocean of possible knowledge, would you assume I or most people would have any interest or benefit in that? Over something like geometric algebra, soldering or solving quadratic equations?
Human creations can be roughly divided in two categories, tools and content. Teaching content is pointless. Its selection is arbitrary and its value is roughly same and miniscule. Millions will choose Frotnite streamer over Shakespeare any day of the week. Math, physics, chemistry, foreign language are tools. Tools that the knowledge of can help you create whatever you desire, both content and whats more important and rare new tools. Native language classes, history, geography, even biology are very content heavy and very poor on tools. Thus they are mostly useless.
I was also saying that even though the lens of culture, the generally mandatory Shakespeare is overrated.
I have a similar problem with a few words in German, despite being in Berlin for 6 years now — "Es war krass" sounds like it should be a bad thing.
Even as someone who went full 'STEM' for both my education and career, English is probably among the most professionally useful courses I did in high school. A surprisingly large part of my job involves reading things, understanding them, and then writing a clear and reasoned response to them, all skills I first learned and really got to practice in English class.
How to do things you don't want to do is a fundamental requirement for adulthood, and probably the most important skill anyone learns in school
Freedom? That has not been my experience of adulthood. To me adulthood is about making responsible choices based on their consequences for you and the people around you ... which means that the majority of my time is spent doing things that other people need rather than because I want to do them
What is freedom, if not the ability to make those choices?
If your answer is no, you're currently doing things you'd rather not.
It requires a very small success on a very basic level. It is not good to be a super math genius and know nothing about geography and history.
Now, that’s not to say the only issue is someone’s curiosity. Traditional teaching methods make it very hard to be interested in some topics (history and language comes to mind), but barring that, I’m not sure I accept “it’s not interesting” as a reason not to explore a subject.
The rest of us don't live long enough compared to our reading speed.
- How and when would you know a subject doesn't click? In my case German (my first language) and English sucked for roughly 8 years. I ended school with A in both and was the only student in my school without a single mistake in my exam.
- School gives you good grades if you managed to learn the topics at hand. Being able to learn what is needed or what you are bad at — not what you want and are good at is a skill in itself that can be important in life. I'd argue unless you are an exceptional genius (unlike 99,999% of pupils) you gain more from pulling through than you would if you called yourself a genius and focused on a single topic. Schools goal is to educate the majority
- pupils (and often also their parents) are utterly unable to judge which bits of school will be essential to their later life. I had many collegues who utterly hated every second of a multitude of subjects, only to years later tell me how glad they are now to have had been subjected to it (which brings me back to my first point)
We could (and should!) argue on how school works as a system of grades, teachers and pupils — ideally teachers would motivate students to become curious about and proficient in subjects without the motivation of good or the threat of bad grades. But if my experience as an educator at the university levels (without grades in mh case) shows one thing it is that those first semester students who are really able to judge what will be useful to them later on are not many. Many of them are more like the dog in the meme: "Only stick, no take" — they want to be able to do the cool thing without knowing what is needed to do the cool thing.
Imagine that all we were was measured by the salary that we make and not humans and human civilization. Might as well just hand over our lives to the robots now.
But I wouldn't be able to tell if some of the mental agility I have today (beyond languages) could be attributed to the fact that I was exposed to this language which differs from my first languages in multiple ways. The lessons you may take away from any given subject is not necessarily limited to its raw content.
This is a quite known topic in educational sciences, e.g. when answering the question why it still makes sense to teach kids how to do maths without a calculator in an age where you have it always in your pocket — that is because it (A) forms an intuition about mathematical correctness and (B) logical and numerical thinking gets trained as an side effect.
Ofc all of that doesn't happen when you are unlucky enough to have had bad teachers, but that wasn't the argument.
I would argue that sometimes students are right when they complaint about a subject being useless or obsolete (e.g. our network professor told us everything about the OSI protocol stack in great detail, and barely touched TCP/IP), but most of the time they don't know what will be useful later in life.
On the flipside you can take the most essential, practical and important topic and deliver it with such dullness that everybody falls asleep.
The first step in teaching (IMO) is to make people interested to hear what is being said. If your subject is useless and boring to most people you have to find a way to make it useful and interesting to them. This requires rethorical and social skills, good presentation skills and a deep understandingnand interest for the subjects to be thought.
In reality most subjects have interesting bits, but is the task of the teacher to make them relate to the world. If your prof thougth you about the OSI protocol stack so long everybody forgot its practical applications and how it binds together with higher level stuff, that is certainly knowledge, but it doesn't help you making it part of your understanding of the world.
So you also have to wonder how many potential math prodigies we've missed out on, simply because they had a bad maths education at school.
In my country, there are several students (12-18 years old) who can mix sport at high level (national championship) and a lighter school activity. They work like mad but they do it. But they have to prove there are good enough to get it, which is OK to me.
Being really good is not something you appreciate yourself, it's the others that notice.
You usually need to pass English and maths GCSEs, but universities mostly care about the subjects you do in the last two years (except for very competitive courses).
This can be a problem for those who want to keep options open until they are 18 (like my younger daughter).
Even the system up to 16 is pretty flexible. There is a huge range of available subjects - although most schools offer only a limited selection (my kids were out of school by secondary school age so we had a huge choice and did some less usual subjects like astronomy and Latin).
I feel like the IGCSE times were more than enough exposure to those other subjects to give me a reasonably well-rounded image of those subjects. Now, a decade later, I'm definitely glad I went the STEM route rather than ever touching on any humanities subjects in any amount of detail.
British universities do often (usually?) require English language (or an equivalent) and maths (I)GCSEs, as do British employers and further education colleges etc. so its a really bad idea to skip those.
I do think its good to give kids a broad education, and for some it would be good to continue that for longer. For others (like you) its just boring.
I also think there should be more opportunities for adult education, so if you decide later in life that you would like to learn about English literature or physics or whatever you should have a chance to do it. AT the moment in the UK things seem to be going the other way, with an over expansion of universities sucking up money, and further education colleges that used to offer adult education putting the resources into "16 to 18" courses.
But yeah, until that age it sucks and a lot of people struggle because they have to do classes they aren't at all interested in.
What does an A++++ mean? I guess it's equivalent of publishing a novel result/idea, solving a long standing conjecture sort of ability? If yes then I'd say the system does accommodate them. Terrance Tao for example was fast tracked and didn't get lost in the system.
Though it probably requires some effort from parents to figure out the right path for their prodigies.
It doesnt, at least in my country.
Here you have a few school leaving exams that you need to perform at
Math, native lang and english.
And one at advanced level, but its your choice when it comes to subject.
What you are talking about is GPA and GPA is meaningless tbf
To actually be meaningfully better at math you need to go far, far further than the coursework presented in schools, at which point it becomes an extra curricular which is already considered. If you’re a math savant proving novel theorems, they will notice that.
But nobody gives a shit if you’re just really, really good at factoring polynomials from Algebra 2 tests.
After graduating High School, I couldn't really see myself sitting through 4 more years of basically the same subject material - history, English, social studies, etc... when all I really wanted to do is use my talent in electronics and computers that I'd been accumulating since I was 5 years old. So I went to a trade school for electronics and computers instead of a college, this was back in the late 80's.
I can't say that not going to college hurt my job prospects at all, I've been highly paid for many years for doing something I love. Jobs hired me based on experience, not because I went to a college. That may have changed a bit now, since some tech companies favor college graduates, but I've worked with some college graduates that are just dumb as shit and couldn't get anything done - not all are like that, but going to college isn't an indicator of success at all.
But a 780 on the math SAT isn’t likely to put you in the company of people whose math is sufficient to ignore poor grades otherwise.
Could you share your source for statistics on "eccentric geniuses"?
For college and life in general, I think main skill needed is emotional regulation. Everything else flows from that.
As a child I used to get all As and even got into a Stanford pre-collegiate program as a kid where I learned C++ and geometry.
Unfortunately after a surgery in 9th grade that left me unable to attend school for 3-4 months and just terrible QOL for about a year my grades slipped (went from A+ studen to C grade student) and I basically became average. I lost all interest in most subjects at school due to depression and other things.
My goal as a child was to get a Stanford JD/MD MBA (lol I know..), and today I have only a bachelors from a low ranked state college in business.
I enjoyed programming so much as a kid that one summer, so later in life I ended up going back to it. Taught myself enough in a month to get on some projects as a swe. Later I got lucky working at a unicorn company that IPO'd.
Now I am trying to build my own company and see how far I can get as a solo founder. Sometimes I wonder how my life would have turned out if it wasn't for that injury, but oh well. Shit happens, right?
Jeez sorry for the sob story but it feels good to get it off.
I believe that any person here with an inkling of relatable technical experience can greatly appreciate the work you've been doing. Software development can be complicated and frustrating, especially when things don't work as you expect them to (but then, you learn and become better). Leading a business is very difficult, often due to sources of problems you don't expect (such as regulatory and legal requirements, accounting, and publicity).
Some people cruise on to great careers without facing many barriers. But many others face unexpected setbacks and have to manage them. A close friend of mine was living an overall good life until it was profoundly disrupted by a civil war in his home country. But he made it to my country where he began his undergraduate degree at a great university that he loves. A past colleague of mine spent much of her early twenties managing physical disability, but successfully received treatment and went on to graduate with an engineering degree. She has since landed a position at a top aerospace company that she really wanted to work at.
You are setting up a good life for yourself. Many people lack that kind of drive or struggle with executing ideas; several people I personally know would be very proud to one day experience just a small part of your successes so far.
Life is truly a journey unique to each one of us.
I’ve found peace and I carry a signed index card in my wallet on which I’ve written my “ethos”. It took me a long time to come up with it and I’m sure it’s common, but those 5 points are something I try to remain true towards.
Cheers mate, best.
I found that I have a certain knack for it and really enjoyed performing.
In Europe maybe, but in America a lot of students receive their general purpose liberal arts education in College, and will then specialize later with a post graduate degree.
Also does the above apply also to the most selective and renowned institutions, or only to community colleges?
I'd say applies to most colleges, of any size. Community colleges tend to be more like trade schools
I studied computer science (M.Eng.) and I had to provide an English certification as a foreign language, but that was it. The only non-computer science, non-math, non-electronics courses were one semester of chemistry, one of economics and three of physics (which you could say is related to electronics though). So basically 15-20%.
Having to study more history or biology would have been a huge waste of time in a CS university, and I say that as someone who did more Latin and Greek than math and physics in high school.
Being forced to do subjects that you hate is not exposure, it is being forced to do things which you are completely unsuited for.
I would go so far as saying that being forced to take music until 7th grade put me off any musical pursuits for the next 20 years. The less said about the torture disguised as education that is PE the better.
If anything, current (and this is valid globally) school system is not designed at excellence at its core, its about raising an army of obedient but not too stupid citizens. And its not missed expectation, just look at what type of work they expect to fill in. Don't expect massive changes unless society changes itself.
Some narrow excellence is not what our society at large values, those few that made it through made it despite their environment.
Personally I think that we could do better by tailoring every student’s education to their abilities. Put in simplest possible terms, we could arrange classes by complexity rather than by year. Have one class for addition and subtraction, another for multiplication and division, then geometry, algebra, etc, etc. Then let students graduate from one to the next based on proven ability rather than by age. Do the same for language, history, etc. Let every student proceed through the courses at their own speed.
https://en.m.wikipedia.org/wiki/AME_School
The kids that went there … some succeeded, some really struggled to adjust to other schools and environments.
Now there’s a depressing thing to read. Teacher’s unions are the worst.
then why does this 'discovery process' have to continue into college? That was the OP's point. When money and time is on the line, let adults decide what they want to study. An 18-year-old is no longer a child.
While some exposure is probably better on average than none, in some instances bad experiences can trip the fuse on developing an interest.
The rote nature of canned education, bad teachers, bad parents, or bullies can turn kids off of subjects they might otherwise come to love.
It's not good at all that school reward being OK in every subject more than being really good in a handful of them. The school system is a mediocrity factory.
Unless they are born wealthy or find a benefactor so they can go to private school or a gifted person school, that caters to their specific needs, it's the best we can do to serve everyone.
https://www.rfcafe.com/miscellany/factoids/Jean-Shepherd-Cla...
on youtube as well
Standardized tests are screwing everything up. People who learn on their own might stumble upon the entire alphabet except for the letter "B," but standardized tests want only the first 5 letters. Hence the incredible efficiency of knowing the entire alphabet is thrown under the bus in favor of making sure none of the 5 are missing.
You can't teach someone to play, and there is no way to play systematically, at scale, and with guaranteed results. All the incredible people I know have some hole in "basic" knowledge, and if it is revealed nobody cares about them being miles ahead elsewhere. "Their basics seem lacking, in the name of stability and norm, throw them back to square one."
Following standards never produces something new, but the world is so afraid of failure and lack of definitions in "messing around" that they are willing to trade their souls for it.
Take any hacker here on HN, and ask how much they learned in CS class vs. how much they learned messing around with Perl on a weekend.
Excellence requires individual attention and cannot be so readily mass produced.
I dispute this on the grounds that students are going through American schools and many of them don't even know how to read.
The number, while high, is not satisfactory. Clearly, we also want adults to be functionally and not pass a super low bar of being able to read a sentence which 92% does not care to distinguish, but it is not fact true that "many of them don't even know how to read".
1. https://en.wikipedia.org/wiki/Literacy_in_the_United_States
Even within academic stem fields you have people who know how to promote and speak and they have the most influence.
I guess what I’m trying to say is the system is mostly selecting for what it wants.
Sadly the later part of your comment may hold - I don't remember what ended up happening with him, whether he graduated high school or what. Hopefully at that level you just disappear into academia and not off the face of the earth in general.
The school I went to grades 1--12 tried to be especially good so taught Latin, French. Some of the girls were in ballet. MIT came recruiting. The year before me two guys went to Princeton and ran against each other for President of the Freshman Class (whatever that meant!). In my class, one guy (did nearly as well on the SAT Math as I did!!!) went to MIT.
In one of the early grades, I got dumped on (adenoids, couldn't hear well until that got fixed). Apparently the teachers talked to each other and had me with a dunce cap until I proved otherwise. In 1st algebra, discovered math: I liked it, was good at it, was the best in the class, proved myself, got sent to a math tournament, couldn't get dumped on, etc. Continued that way: Was so good at math that I got an unspoken but powerful by in any subject, e.g., English literature, I didn't like.
Got sent to summer math/physics enrichment programs.
So, for that example, for
> But what if those just aren’t interesting to someone?
some schools will let a student who is good at some one subject get a by in other subjects.
Really, schools, K-Ph.D., have a tough time finding any students really good in even just one subject, are thrilled when they find one that is, and don't want to block him/her because he viewed fictional literature as a not very credible presentation of common reality?
That by pattern continued: In grad school, they insisted that I take their computer science course. My background in computing was already nicely above that course, and I'd already taught a similar course at Georgetown. Soooo, mostly laughed at the course: E.g., they had a test question about Quicksort (very common topic then), and I answered with material they didn't know.
The best case of by: Took a reading course; decided to address a question in the pure math of optimization; two weeks later had a surprising theorem and from that an answer to the question. The work, clearly publishable, was instant news all over the department, some profs angry that I had done well, others pleased. Angry/pleased, the work got me a general purpose by, a gold crown, immunity from any criticism, and an unspoken, implicit, easy path to the rest of the Ph.D.
Zero? If you qualify as a prodigy, it is apparent from a young age. Maths prodigies are especially easy to distinguish. Given a little time, they will self-learn, grok and innovate on anything you throw at them and will likely attend higher education early unlike "the brilliant kid"s who will struggle with advanced concepts all their lives.
There are children with exceptional mental abilities in a limited range of topics or even a single topic, such as math, music or drawing, They may struggle to various degrees with subjects or life skills that don't correspond closely with their specific topics of interest and ability.
How would you characterize R's master and the "normie" master?
It's the crazy ones that push humanity forward. We lose far more than we can imagine by not enabling even just one of them. This is one of the most important problems for us to fix.
People who are "weird" and yet are entirely functional are the best of both world and a much rarer combination.
Well norms were in place when R did his work. Even the most strict systems have made concessions for extraordinary people. It is just that mostly average people go around claiming they'd be genius, had system not smothered their creativity.
> I wonder how many geniuses we skip on because doing the chores of homework..
I think from not many to hardly any as I can't believe if kids who are really genius can just go on for more than a decade of primary schooling without ever finding outlet for their creativity.
Generally I think: "Unless you're Ramanujan, then you should probably have some breadth to your knowledge rather than pure depth" is not a terrible policy.
It may now be possible with Videos, Games and VR.
Instead of giving kids grades with a ceiling, each subject would have (unlimited number of) levels of proficiency, and to attain a level, kids would have to pass a test (demonstrate certain skill). The choice of subjects and levels to attain would be up to each kid, but they would have to choose to do something (working at getting next level of something would be mandatory). (Although perhaps they should be encouraged to explore different subjects and attain some minimum of levels.)
Also, I would group kids by subject, and not by age. So kids of slightly different levels would train together, and the higher level kids would be obligated to help kids on lower level to learn, while lower level kids were taught to be respectful of higher level kids.
I would be cautious with this, they may have the same academic ability but a large gap in social skills.
We probably aren’t going to get it quite right for that 1, either. Extreme outlier geniuses are extreme outliers and there’s no easily generalizable pattern around them. The ideal education for a young Ramanujan is probably different from the ideal education for a young Von Neumann. Of course in an ideal world we would give an extremely individualized education to every child, but that’s much easier said than done. Failing that maybe we could identify and then invest in the extreme geniuses, but that’s what we already try to do.
I got mediocre marks in high school but thankfully did well in JEE. Now I have a decent PhD in math with an extremely mediocre school record.
I don’t think you can say, even all these years later, that Ramanujan, that mathematician, made the economy more productive, but he certainly increased the high watermark of human civilization and created an inspiring story for individual achievement, creative realization, and artistic and mathematical expression. There’s something sublime and transcendental—no pun intended—in the kind of truths that he was able to tease out and the unique, idiosyncratic way that he expressed them. Sort of like a Basquiat of mathematics, I suppose. Or probably better than that.
That aside, I think it’s unfortunate that he died of cholera or something, isn’t it? I mean, he apparently didn’t think it was unfortunate that he was going to die. And certainly, the formal education system didn’t necessarily fail him, in that a professor at a university recognized his genius and sponsored him to the UK.
But I think, in a sense that you identify, there is this general failure of the education systems in the human civilizations on this planet to foster perhaps the best thing that they could be fostering. They’re more like a manufacturing assembly line to produce cogs as part of the economic machine.
Not that there’s anything necessarily wrong with that. I think it’s good that people can have a role to play in the larger economy and that there are pathways to bring people to the level of capability where they can contribute like that. But the lack of pathways that these systems provide—those that could contribute to the creation of the full realization and expression of individual potential—I think is sad. And I think that’s what you’re kind of identifying.
You seem like someone who thinks things through, so I suspect you’ll know what I’m about to say, but given the sentiment of your comment, I think it’s worth explicitly sharing this:
The fact that your class valedictorian went on to be a doctor is great. Not everyone needs to push the boundaries. Your classmate may end up saving/helping countless lives.
He is saying that the people who add most value to science isn't always the ones who are at the top of the hierarchy in the school system.
Performing well in school is like a F1 racing car: very fast, but can only go on paths very well trodden already, i.e. paved road.
This system is meant for a worker that no longer exists. If someone wants to specialize at the age of 13 and has shown reasonable aptitude in that subject, then let them do it. Sure, give them a well rounded education, but don’t weight those grades anywhere close to equally.
I think in this example you are vastly overestimating the “average genius”, in comparison to those rare few who truly push the boundaries. We tend to do this because of the way our brains estimate things of significant scale, like the fact that keeps floating around social media about how bad most of us are at having a concept of the difference between thousands, millions, and billions.
There are many valedictorians (of the order of some per thousand) but few Ramanujans (of the order of tens per billion?), and gearing an overall education system specifically for those few could do a disservice to a great many others at every level below. Ramanujan was not the result of an effective education system anyway: like a lot of other world changing minds he was largely self-taught. Perhaps there is room to encourage more investigation to the side of the curriculum a lot more than we currently do, but the problems that stop other "Ramanujan"s, that could so easily have scuppered Ramanujan himself, are usually not caused by the education system but by other societal problems (death & disease, racism, sexism, caste or class biases, etc.) not giving them a chance to explore & self-learn or have useful access to education & other resources at all. Addressing those problems will help a much wider chunk of the population, as well as reducing constraints for the truly exceptional geniuses¹ amongst them.
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[1] And of course those non-geniuses who luck out and have that one brilliant idea. Their contribution is often vital for progress too, and I'd wager that there are orders of magnitude more of them than there are true geniuses!
That doesn't matter. A student is unable to evaluate the value of unknown subjects and the synergy between subjects. People are often unmotivated to do important things, because we are just lazy s**.
> I wonder how many geniuses we skip on because doing the chores of homework and getting through boring classes is busywork and memorization for the sake of getting an A.
And how many lunatics making wrong decisions did we prevented by forcing them to learn necessary things?
> Meanwhile, hardly anyone actually remembers anything about those topics
They are engraved in your mind. Even when you don't remember every little detail, they still helped you in forming your understanding of the world.
Two major caveats. The first is that Berlin never wanted the essay to be taken all that seriously. To him, it was an intellectually parlor game. But if TvTropes is any indication, people are addicted to classifying things into simplistic categories, so the fox and hedgehog game took off.
Second, the rest of the essay actually challenges the binary by focusing on Leo Tolstoy, an unimaginably creative thinker and writer who, according to Berlin, was a fox by nature but wanted to be a hedgehog.
Why did I share all of that? Well, without taking the categories too seriously I do think some people are more foxy by nature and others are more hedgehog-esque. I've always thought of myself as a fox (which to be honest, might limit me), and I had no interest in focusing on a single subject in school. At worst, I'm a dilettante who can have a surface level conversation about any intellectual topic at a party. At best, I can muster up a real enthusiasm about trying something entirely new.
I say all this not just because I'm running on 4 hours of sleep, but also because I think foxes get a bad rap. Not everybody should strive to be Ramanujan, because it isn't in everybody's temperament.
Being a genius is a phenomenon of the individual, but being able to pursue one's talent is a social aspect of the phenomenon of civilisation.
The US political system is currently in a state of utter dysfunction because people have decided that facts "just aren't interesting."
I have a family member who scored perfectly in math on the final exam of middle school but failed other subjects so badly that he couldn't advance to high school. What frustrated us even more was that the next year, he achieved a perfect score in math again but still failed overall, leaving him no choice but to drop out of school.
This story has stuck with me for years, highlighting how our education system focuses solely on maintaining an overall average score above a certain threshold without considering how individual subjects are mastered.
I can’t help but think that if he had been allowed to pursue mathematics, he might have accomplished something great. It’s really disheartening.
I don't think we can or should design a system optimized for that one-in-a-million genius. It would be worse for everyone else, and ultimately for society.
But I always wondered how polymath like Leonard davinci and Isaac newton that are excellent in many area are possible.
Maybe there are other topics where the same phemonen exists. I'm not sure if off the top of my head I can think of one where it gets so intense so quickly - long, detailed, emotionally charged anecdotes immediately when school comes up! [I love a good yarn, and am not judging or belittling people for expressing themselves here, I must stress].
I have speculated in my comfy armchair as to why our education structures seem to make venting such a need for people. Anecdotally, there does seem to be a suggestion of some sort of lingering and powerful malaise around the whole thing, in any case.
I wonder if it's like an abusive relationship - the emotional work required to move on and have better relationships (i.e., different educational structures) gets so large at some stage that all you can focus on is "coping". And you're sort of stuck, and the cycle more or less continues.
EDIT: may I just add - I read that whole article, and am very fond of Ramanujan. He made university much harder for me when I learned about his existence, as my maths classes seemed so far removed from whatever it was he was doing.
(Venting is a need that everyone has when under pressure not sure it has anything to do with our education structures)
Edit: or happy, one-parent homes. I just have an inkling it is good for children to see a contrast of parents, not just one parent.
Attempting to scale something that large forces people into boxes that fit inside the framework - one must necessarily ignore little detailed differences between humans.
But from the perspective of those individual humans, ignoring those little details doesn't really work. Those things are important, and tend to ping our egos, so we react accordingly, and are eager for opportunities to give voice to this frustration.
a.k.a. sampling from the distribution of interest. Once you have enough anecdotes, you actually have a representative distribution of the real thing. Selection may be biased but you can weight those data points as you like, it's still good data.
Teaching has not incorporated morality or ethics as strongly as some other professions where the need is more obvious. This is combined with the way teaching "quality" is measured. The result is that your average school teacher often uses tactics and methodologies that border on straight up public bullying of their students in order to "achieve results."
> I wonder if it's like an abusive relationship
Neither the teacher or the student engage in a process which selects each other nor is there any process to manage particularly bad combinations of such. They are assigned to and stuck with each other. The ethical failures in teaching abound.
Not sure if you're in the US.
Here, ALL non-technical topics, e.g. nutrition, healthcare, immigration are just a few of the big ones, devolve into a discussion of anecdotes and "feelings."
Maybe it's worth a bit of perspective.
I found it quite ironical how the whole topic has been hijacked into this side discussion. For the record, Ramanujan is NOT a product of the Indian education system. In fact, that system was quite cruel & off-putting to him. He was a self-taught math prodigy. Apart from the well known 2 big biopics, there are numerous TV soaps telecasted in India in various languages that regularly emphasize this point. He taught himself mostly using Inequalities by GH Hardy, & several other books, all of which are freely available right now at the click of a button. Nobody is stopping anybody from studying math, and education or the lack of it has nothing to do with any of this.
Indeed. I submitted this post, because I found the article's discussion of the math involved fascinating, along with biographical details of Ramanujan I hadn't heard about before. I'm surprised to see how popular the post is, but also that there's maybe 5 or so posts that engage with the actual mathematics.
I probably shouldn't be surprised - generating functions pointing out unexpected connections between different restricted integer partitions is pretty niche.
[...]the day I started reading educational research was the day my life changed.
It all began with Daniel Willingham’s Why Don’t Students Like School, which led me onto his ridiculously good series of Ask the Cognitive Scientist articles for the American Federation of Teachers and the associated research papers he quoted. I devoured them all with eyes wide open and jaw firmly on the floor. Next came Cognitive Load Theory, which I was introduced to via the incredible blog of Greg Ashman and our subsequent conversation on my podcast. I was familiar with the work of Dylan Wiliam through the development of my Diagnostic Questions website, but I read and re-read everything I could get my hands on from the great man. I didn’t think life could get any better — and then I came across the work of Robert and Elizabeth Bjork. Good God.
One thing led to another, and before I knew it I had read well over 200 hundreds books and research papers. I would wake up in the middle of the night, my head buzzing with ideas[...]
Ramanujan short life itself is a loss to the world, Imagine how many Ramanujan's were ignored where there is no G.H. Hardy and what about Ramanujans in the other 95%?
The other version in the West is he moves to rural Montana and sends bombs through the mail.
There might be a middle ground between the two?
I think that if you tried you could imagine a world where Ramanujans are both supported and respected with nice lives, while also contributing to the future of the species rather than helping dig our own grave.
> The other version in the West is he moves to rural Montana and sends bombs through the mail.
Why in the world would those be the only two options?? What a profoundly strange thing to say.
That brilliant geniuses have chosen corporate America and to not go into academia is yours and mine and the world's loss, but a defining feature of the Western world is that they get to make that choice for themselves.
... As if people outside the West have no self-determination? Pretty questionable, on multiple levels. And another indication of absurdly binary thinking/rhetoric. Good luck with that.
I for one would ask him what we could do to make it easier for him to listen to the voice of god(s) rather than say "well akshually..."
I'm not sure there's much you can do to help someone at this level, other than perhaps take away outside distractions.
It's not remotely clear how Ramanujan's mind worked, or whether he really knew himself. He seems to have been a lot more highly functioning than an idiot savant, but still operating at a level that is hard to comprehend. Was this an expression of humility or devotion by Ramanujan, or was he just expressing his own lack of understanding of how his mind worked?
Feynmann, like von Neumann, also seems more approachable/understandable in that he wasn't just operating at genius level, but was also a regular fun loving guy (strip clubs, bongo drums, safe hacking and all). Von Neumann and some of his Hungarian peers were famously referred to as "The Martians", but it's really Ramanujan who seems like the alien.
Einstein made a few extremely valuable insights in theoretical physics, but I'd say the sheer breadth of comparable work that Von Neumann did across multiple fields makes him more outstanding.
It was intended to apply to oneself, but seems just as true when appreciating the genius of others. No need to diminish von Neumann, etc., AND doing so just diminishes your own credibility.
“I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops.” ~ Stephen Jay Gould
This is similar to another popular narrative that no girl received an education until some sympathetic Englishman in partnership with a local woman allowed girls to go to school. What a load of BS.
But people in the upper classes don't like that, so they're telling working class people that people even worse off are after their jobs.
Tall claim.
[1] https://www.hup.harvard.edu/books/9780674987883 [2] https://en.wikipedia.org/wiki/Caste_system_in_India
I know more than 3 people in the US, who talk about caste all the time (as in that's their job now) but grew up in very privileged families.
Only the target has changed. The land grabs and slurs and dehumanization are essentially the same, if sometimes with a modern twist.
divinely inspiring
I'm curious to hear how others think about this phenomenon. Do recent advances in neuroscience, AI, or cognitive psychology offer any clues about how innovators like Ramanujan access these hidden sources of insight? Or are we still stuck in the realm of "genius is mysterious"?
One guy able to tap into this knowledge in dreams is an indication that it is possible. Now, how do we make this the default for everyone is the question I wonder about
The way we found one variant of wheat in Mexico that was resistant to bacteria, and replicate that to the whole world -- can we do something like that for humans ( even I don't like the sound of it, but I hope you get the feeling )
Great analogy.
Borlaug's famous Mexican dwarves.
According to the Wikipedia page of Norman Borlaug, he _developed_ that variety of wheat.
You delete stupid prejudices from society, then it allows itself to do that. One group of people forcibly practicing it for idiotic reasons now plagued the whole field with its label, amplified by religious ideas. The irony is, it’s that same limitation that disallows to address itself and continues to propagate.
My guess would be by filling the reservoir first. After filling it, then you can stir it and extract new knowledge. In other words: you get better at math by studying, reading, and thinking a lot about math. I don't think there has been a brilliant mathematician who has not immersed himself in math at first.
And, it works the same with us 'normal' people, even though the results are less spectacular. Want to have a good idea? You need to feed your brain everything you can and obsess over it, then let it stew for a bit. Again, and again. Only then do you stand a chance for a really good idea to just pop out.
From Karma Yoga, Chapter: I, Karma in its effect on character—
What we say a man “knows”, should, in strict psychological language, be what he “discovers” or “unveils”; what a man “learns” is really what he “discovers”, by taking the cover off his own soul, which is a mine of infinite knowledge.
We say Newton discovered gravitation. Was it sitting anywhere in a corner waiting for him? It was in his own mind; the time came and he found it out. All knowledge that the world has ever received comes from the mind; the infinite library of the universe is in your own mind. The external world is simply the suggestion, the occasion, which sets you to study your own mind, but the object of your study is always your own mind. The falling of an apple gave the suggestion to Newton, and he studied his own mind. He rearranged all the previous links of thought in his mind and discovered a new link among them, which we call the law of gravitation. It was not in the apple nor in anything in the centre of the earth.
Whenever I read about Ramanujan having divine revealing formulas in his dreams, I remember Swami Vivekananda’s quoute on consciousness and mind.
edit: found another relevant quote from Upanishads on tapping the infinite knowledge:
Mundaka Upanishad 2.2.9:
“Eṣa sarveṣu bhūteṣu gūḍhātmanā prakāśate, dṛśyate tvagryayā buddhyā sūkṣmayā sūkṣmadarśibhiḥ”
Translation: “The Self hidden in all beings does not shine forth, but it is seen by subtle seers through their one-pointed and subtle intellect.”
Explanation: The ultimate knowledge or truth is hidden within all beings and is revealed through subtle inner perception. The idea is knowledge is latent within the mind, and it is discovered, not externally found.
That doesn't look sound to me. If the "seers" are seeing "The Self", are they beyond and separate from "The Self"? If they do so with their "subtle intellect", is that intellect outside of "The Self"?
If affirmative, then "The Self" is something external to the seer, making the term a misnomer. And furthermore, there is something outside of The Self (that which is seeing The Self), which remains to be explicated.
Quoting one from Katha Upanishad 1.3.10:
"Indriyebhyaḥ parā hy artha, arthebhyas ca param manah, manasas tu parā buddhir buddher ātmā mahān parah"
Translation: "Beyond the senses are the objects, beyond the objects is the mind, beyond the mind is the intellect, beyond the intellect is the Self."
This quote emphasizes that the intellect is still a part of the illusion. Beyond even the intellect lies the Self, which is one and undivided.
In case I mistook it, ignore the above.
All knowledge that the world has ever received comes from the mind
Doesn’t it come from the mind’s interaction with the world? I don’t think that any important knowledge comes from doing nothing (apart from psychology due to its nature). It all sounds like a typical religious salad, tbh. “You” discover from your “mind”, but then some seers understand that “you” and “all” and “mind” are the same, etc. Well, you don’t have to be a seer to see it. “You discover from your mind” is a useless semantic loop that creates all this unnecessary complexity, imo.
> Doesn’t it come from the mind’s interaction with the world?
There is no difference between these concepts, is there? In both cases knowledge comes from the mind.
Also, you can come up with ideas without interacting with the outside world, can you not? This feels like nitpicking.
What physically happens is that a mind accumulates and (sometimes) systematizes information from outside, and knowledge is the result of this concentration process. Akin to a puddle collecting water from higher grounds. But it doesn’t synthesize water by simply being concave.
Blowing ideas out of poor metaphors is a regular thing in religious texts, so be careful with that if you want to build some practical world view.
Maybe a better term would be information vs. knowledge.
That's not to say there isn't something spiritual and mystical in knowledge. I think the fact that he was from a different background gave him a different perspective. But yeah you cannot skip all the math studying.
The core premise of the book is to describe how mathematicians work and think and to show that this is a process that everyone can do (although some will be better at it than others). It includes interesting accounts of Grothendieck, Bill Thurston, and Descartes as well as from the author's own research career at Yale and École normale supérieure. The book is targeted at the general reader and at times reads a little like a self-help book, especially in the first third or so. However, I found it to be an enjoyable and fascinating read. It provoked a lot of interesting questions about the nature of learning and provided a framework to begin to answer them (e.g. "How can I have proved something and yet feel no understanding of it?", "How can some people solve problems orders of magnitude faster than other smart people, as if they don't even have to think about it?", "Why do I sometimes watch a presentation on a new topic, follow every step, and come away feeling like I've learned nothing?"(* see except below)). I don't think I'm doing it justice here, so I'll stop by saying I highly recommend it based on your comment.
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* I'll use this as an excuse to provide a related excerpt featuring Fields Medalist and Abel Prize winner Jean-Pierre Serre:
One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. I didn't have substantial new results to announce, but it was an opportunity to make a presentation even simpler than usual. [...] A couple of minutes before the talk was to start, Serre came in and sat in the second row. I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things.
What I didn't tell him, of course, was that his presence was intimidating. Still, I didn't want to raise the level of my talk only to keep him interested. I just kept an eye out to see if he'd taken off his glasses, which would mean he was getting bored and had stopped listening. No worries there—he kept his glasses on till the end.
I gave my presentation as I would have without him there, speaking to the entire audience, especially the students seated in the back, whom I was pleased to see listening and looking like they understood. It was a normal presentation, fairly successful, not very deep but well prepared, clear, and intelligible. At the end of the seminar, Serre came up to me and said—and here I quote verbatim: "You'll have to explain that to me again, because I didn't understand anything."
That's a true story, and it plunged me into a state of profound perplexity.
Apparently, Serre wasn't using the verb to understand the way most people use it. The concepts and reasonings of my talk couldn't really have caused him any difficulty. I'm sure he wanted to say that he understood what I had explained, but he hadn't understood why what I had explained was true.
There are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it's correct. Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it's natural.
In thinking again about Serre's comment, I realized that my presentation had too many “miracles,” too many arbitrary choices, too many things that worked without my really knowing why. Serre was right; it was incomprehensible. His feedback helped me become aware of a number of very big holes in my understanding of the objects and situations I was working on at the time. In the years that followed, research into explanations for these various miracles allowed me to fill in some of the holes and achieve some of the most important results of my career. (However, some of the miracles remain unexplained to this day.)
But the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension.
My intuition is that some people are able to develop abstractions on top of abstractions (compression is a key part of intelligence) that allow them to traverse the search space much, much faster than without these, and with enough repetition this becomes routine, even faster in the brain.
I don't think we have a good theory of how this works, or at least I haven't come across it.
Neural network guided search is somewhat similar but I think we are missing several key pieces
1) Mathematics Wizard Srinivasa Ramanujan : Some glimpses into his Life and Work by two Indian Mathematicians Narendra Kumar Govil and Bhu Dev Sharma is a good biography with an introduction to his Mathematics and links to further resources. Good complement to Robert Kanigel's book The Man Who Knew Infinity.
2) In order to understand the fascination that Mathematicians have for Ramanujan see this and other lectures by Prof. Ken Ono who credits Ramanujan as his inspiration in becoming a Mathematician; Why Does Ramanujan, "The Man Who Knew Infinity," Matter? - https://www.youtube.com/watch?v=7ynhiZJUMzA
3) Mathologer on Youtube has good walkthroughs of some of Ramanujan's most famous identities (eg. 1+2+3+... = -1/12) - https://www.youtube.com/results?search_query=mathologer+rama...
4) All of Ramanujan's published papers and unpublished notebooks can be found online at - http://ramanujan.sirinudi.org/
PS: In the submitted article, George Andrews is wearing a Ramanujan tie :-)
The Man Who Loved Numbers - Srinivasa Ramanujan documentary (1988) - https://www.youtube.com/watch?v=PqP2c5xNaTU
Srinivasa Ramanujan: The Mathematician and His Legacy (2016) - https://www.youtube.com/watch?v=K5jsgBvJMUc
Watch particularly Prof. Ken Ono's videos where you can literally feel his Admiration and Reverence for Ramanujan. He credits his Career/Repairing relationships with his Father/Family to their shared love of Ramanujan and his Mathematics (his autobiography is named My Search for Ramanujan: How I Learned to Count). I found it highly inspiring and motivating and encourage all students/teachers to watch them.
Also see Ramanujan's wikipedia page which contains lots of details/quotes/links to books/videos/etc. - https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
Quote:
K. Srinivasa Rao has said, "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, J. E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'"
Another marvelous mathematician story is that of Yitang Zhang. The Newyorker had a good article about him if you haven't seen it yet.
https://web.archive.org/web/20220527042904/https://www.newyo...
1. https://arxiv.org/abs/2405.06451v2
2. https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_...
Well, at least to western people. Are Indians more familiar with Indian scientists?
I don't think a non STEM guy would know Ramanujan or C V Raman.
I imagine most people won't recognise the name. But everyone's heard of a boson. So he's somewhat immortalised — more than most.
In order to rectify the status quo;
1) Everybody should get the monthly magazine Science Reporter published by National Institute of Science Communication and Policy Research (NIScPR), Council of Scientific and Industrial Research (CSIR), New Delhi, India. which gives a overview into Indian Science - https://sciencereporter.niscpr.res.in/
2) The two-volume The Mind of an Engineer by Purnendu Ghosh et al. published by Springer contains essays from many of our recent Scientists/Researchers/Engineers etc. - https://link.springer.com/book/10.1007/978-981-10-0119-2
3) Books on Indian Science/Scientists by various authors are available on Amazon India and are worth getting.
4) Also see the books by the great astrophysicist/cosmologist Jayant Narlikar (https://en.wikipedia.org/wiki/Jayant_Narlikar), specifically; The Scientific Edge: The Indian Scientist From Vedic To Modern Times. - https://www.penguin.co.in/book/the-scientific-edge/ and Science and Mathematics: From Primitive to Modern Times - https://www.routledge.com/Science-and-Mathematics-From-Primi...
a) Resonance : Journal of Science Education published by Indian Academy of Sciences - https://www.ias.ac.in/Journals/Resonance_Journal_of_Science_...
b) Science India magazine published by Vijnana Bharati- https://scienceindiamag.in/
And Mani Chandy for computer science.
National Book Trust also has several books on Indian scientists.
https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology
Fairly short and beautifully written.
"An equation for me has no meaning, unless it represents a thought of God" - Ramanujan
I love the Humility and Kindness in Hardy's statement.
AlphaProof does do some kind of neural network guided search and automated theorem proving to validate it https://deepmind.google/discover/blog/ai-solves-imo-problems...
But it's still fairly brute force and inefficient
Not every task ended up being worth doing, but some sure did. And regardless, more important than any one individual tasks, I earned a reputation with management. This in turn resulted in a steady stream of bonuses, stock grants, and promotions. I suppose I was doubly-fortunate to have focused my efforts in the tech industry from the early 90's onward. You have to be really unlucky for nose-to-the-grindstone efforts in FAANG companies to not work out pretty well for you from 2003-ish until today.
With continued good health I can look forward to another 20 or 30 years of life, where I get to do whatever I want with my time because I earned so much money being content to conform to whatever my employers required of me. Nobody will remember my name, but that's okay with me, because I won't either.
This is how I felt in 90% of my theoretical math classes in university.
Whether they are successful and whether they are mostly a bunch of snake oil is another question...
I am naturally so tired around 9pm when I shut the lid of my laptop that I fall asleep within minutes of getting in bed.
On a side note.. Somehow my dreams have been insane and I’ve low key enjoyed the vivid worlds I find myself in over the past few months.
Wake up around 5 or 6, go for a stroll and then eat some breakfast.
Then I can work taking only breaks for lunch and dinner. Sometimes a 30 min nap in the afternoon in the park.
https://en.wikipedia.org/wiki/Straight_edge
> Straight edge (sometimes abbreviated as sXe or signified by XXX or simply X) is a subculture of hardcore punk whose adherents refrain from using alcohol, tobacco, and recreational drugs in reaction to the punk subculture's excesses. Some adherents refrain from engaging in promiscuous or casual sex, follow a vegetarian or vegan diet and do not consume caffeine or prescription drugs. The term "straight edge" was adopted from the 1981 song "Straight Edge" by the hardcore punk band Minor Threat.
Weird. First time I've ever seen that (abbreviation?) For straight-egde. Thought you were talking about some supplement at first.
I'm naturally tired at around 2am. I barely dream, or can't remember my dreams. I struggle to wake up at 6:30 even after 7+ hours of sleep.
I do like an afternoon nap, but Sunday's are almost the only opportunity. As a bad consolation prize, I involuntarily micro-sleep at my desk, working from home or in the office, a handful of times most days, generally at peak afternoon nap times (1:30pm - 3:30pm).
I wouldn't say I'm productive, but I would say that the work I produce is generally of a high quality.
I wonder what academia would look like if adderall, vyvanse, modafinil were just as accessible, or even less controlled substances that are considered to enhance mental performance like L-tyrosine, alpha-GPC, Lion's Mane mushroom, Bacopa, or Ginko.
We really, really know the long term effects.
Modafinil is straight up better than caffeine though, which is a crappy and addictive stimulant.
I'm not sure where the distinction actually lies, but it is also considered a (generally recognized as safe) drug.
As someone who benefited greatly from a few mentors in childhood and adolescence, my goal is to be able to give the same to at least a few other people in my lifetime.
But then, if 'mentors' applies to parents, then I guess I'm saying the same thing.
Having said that, my teenage son wastes a lot of his life playing online games with his friends, but having said that, I heard him say mid-game to a friend of his "Brazilian isn't a language you idiot!". So, I mean it's trivia, but he knows there's no language called Brazilian. He's a smart kid (that's not the only data point).
(I had to look it up: Portugese is the official language of Brazil).
+1. Same for me.
Relationships has been more important than what I could consider ideal or optimal (in terms of how many missed opportunities are there).
Once I was reading a technical paper from a colleague and could not follow much. Spoke to him and he could explain the crux of it in two-three sentences. I was left wondering why that was not there in the paper itself. But that happens repeatedly. People are able to explain much better than the literature they produce*.
This perhaps has a contribution as to why good relationships and mentors end up being so important. But then, this ultimately comes in the way of scalability.
* Of course, it may possibly also be that in a verbal conversation when the author personally knows the reader, they can explain to that reader better. However, I think this effect is less important than people being able to write better.
For instance we banned meth and other drugs that have tremendous productivity effects at the expense of the individual and how we had to deal with them, so it's not a rhetoric question.
Your apparent disbelief in ADHD doesn’t make it imaginary, by the way. Consider yourself lucky that you do not have it; I am unemployable without medication.
I should have been clear I saw it in the "make people smart" light, as doping an already acceptable situation, instead of correcting something perceived as a pathology.
Meth was widely available over the counter at some point, and we made it legally disappear outside of strict medical settings.
But none of them work as well as sleep and exercise.
...or me?
tbf tho, I am somewhat contradicting another comment I made about bioavailability on this very gp thread so suffice to say, I do understand your point.
On the other hand, I just looked at Twitter and saw two different people get ~$3000 payments from a class action lawsuit against Juul, which is by far the biggest payment I've ever heard of from a class action signup, so I suspect they're about to die.
Wonder why more people aren't using it.
That is not totally true for at least three reasons:
1. Modafinil can interact badly with some other stimulants. One must remain careful.
2. Modafinil by itself can cause SJS and similar serious problems in rare cases.
3. Chronic use of modafinil can easily produce anxiety, so much anxiety that it makes using modafinil impossible. This is even in a very low dose.
Because it's classified as an amphetamine in large parts of the western world and would be illegal obtain without a prescription.
You're perpetuating "be ashamed of not knowing" instead of encouraging "learn by asking questions".
If you agree, consider upvoting downvoted questions.
https://www.psypost.org/new-intranasal-rna-therapy-shows-pro...
And somehow this guy is remembered as a 'genius'...