Math is still catching up to the genius of Ramanujan (opens in new tab)

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.

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.

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.

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 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!

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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."

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.

https://www.rfcafe.com/miscellany/factoids/Jean-Shepherd-Cla...

on youtube as well

But I always wondered how polymath like Leonard davinci and Isaac newton that are excellent in many area are possible.

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.

How would you characterize R's master and the "normie" master?

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.

Whether they are successful and whether they are mostly a bunch of snake oil is another question...

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.

But none of them work as well as sleep and exercise.

https://www.psypost.org/new-intranasal-rna-therapy-shows-pro...

Wonder why more people aren't using it.

You're perpetuating "be ashamed of not knowing" instead of encouraging "learn by asking questions".

If you agree, consider upvoting downvoted questions.

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.

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 )

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.

_______

* 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.

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.

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.

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.

[...]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[...]

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."

(Venting is a need that everyone has when under pressure not sure it has anything to do with our education structures)

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.

Maybe it's worth a bit of perspective.

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...

I imagine most people won't recognise the name. But everyone's heard of a boson. So he's somewhat immortalised — more than most.

National Book Trust also has several books on Indian scientists.

[1] https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama

And Mani Chandy for computer science.

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%?

https://i.imgur.com/AzL6hVQ.png

https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity

[1] https://www.imdb.com/title/tt0787524/

[1] https://www.imdb.com/title/tt0787524/