> If Mozart could hear in his head how the music ought to go, Leonardo, judging from his sketches, could simply see in his mind’s eye how the machine should work or the painting should look.
There seems to be some thing that allows creative construction of these things, in mind, as opposed to simple recall of the sounds or images, in mind.
For example, Mozart might have been able to creatively "construct" the sound of his composures in his mind, but just because someone has heard a Mozart composure performed and then can recall it later, in mind, that fact doesn't make them a genius.
It's both creating and hearing/seeing it first that makes it special.
Ja that is kinda a cool way to think about it. It's like they had a better ability to "simulate" the model/music/problem in their mind. By "better" one can argue it was faster, more accurate and more detailed.
Excellent point about what is "better". Being faster, more accurate and more detailed is the "map" problem, where efficiency and interestingness intersect.
My theory is this is how animals arrive at consciousness.
> he quant types (mathematics and science majors) thought genius was due to natural gifts; parents and teachers had told them that they’d been born with a special talent for quantitative reasoning.
I don't think this is because people told them anything. To anyone who is good at maths it is painfully obvious that some people get it much easier than others. I started out thinking that people just did it wrongly, but the more I tried and the more I watched people do things the more I realized that they simply were bad at maths. There is nothing I can do for them. Some people just lack practice and motivation, true, but most lacks the brains to do it, since you need to have at least some level of intuition and some level of rational reasoning capacity to get anywhere in maths.
For those who disagree, how do you explain the very large number of people who spends tons of time and energy trying to get good at maths but fails to even grasp simple concepts like calculus? Contrast that with the much smaller but still considerable number of people who aren't really passionate about maths, don't put in the work yet still aces the class? Many on the "nurture is everything" side simply believes that those differences doesn't exist, the "geniuses" must have learned this beforehand or something else silly. But if you are one of those who aced the classes without even trying and see others try really really hard and still don't even pass, how could you explain that without considering talent as a variable? That person doesn't need to be told that talent is important.
Not that you're making bad points, but there's a big difference in acing a math exam vs actually assimilating the various underlying concepts, at least at a university level IME. I know plenty of people who would pass as 'geniuses' by your definition that just cram past problems and get high grades. And honestly if you've been to college you'd know that it's almost impossible for a novel problem you've never encountered before show up on an test.
A lot of otherwise intelligent people fairly despise such practices so they adopt inefficient studying habits , failing to complete a set syllabus and achieving subpar results.
I didn't say people who ace math exams are geniuses, just that maths makes it very clear that talent is a huge factor for success in the field. So it isn't that STEM students were all told by their parents that they are intelligent and therefore believe that genius is about talent, it is that it is pretty hard to study STEM without noticing the huge difference in talent between people in the field.
> talent is a huge factor for success in the field.
Success in the field as in research? Most certainly so.
However I think you're overestimating the 'math IQ' of STEM (under)grads especially considering the large discrepancy among its different subfields. Engineering students (EE/ME/Civil) are systematically 'dumber' maths-wise compared to their science (Maths [duh] or Physics) counterparts. Still, it's quite likely that you do not have to be a genius to be successful at landing a highly-paid, mentally-challenging related job.
About parenting, smart math kids may not have had their egos boosted however their parents almost certainly exposed them to science from an early age. This holds true especially for children coming from 'STEM families'.
I disagree because math is too vast and ill defined a field for any claim that anyone can be good or bad at all of it to be taken seriously.
In high school, I was one of those people who spent tons of time and energy on it, and failed miserably, barely graduating as a result. It wasn't until I was about 25 when I picked up a basic algebra and trigonometry book on a whim and started self-teaching, that it began making sense to me. I eventually went back to school for pure math.
In my case, I think what helped me as an adult was the freedom to learn on my own terms without the pressure of deadlines, and also a strong support network of friends encouraging me (compare with my teachers in high school who would actively embarrass me in front of the class or joke about their low expectations of me). And of course, some years of life experience in the real world doing things that were actually hard compared to (elementary) math as someone of limited means helped a bit too.
So, all of that is to say that people who struggle in math classes struggle for different reasons. I suspect that much of the time the dominating forces are psychosocial, cultural, or plain & simple lack of interest (which itself might be driven by traumatic experiences with shitty teachers or parents with weird expectations)
So you failed, then took your time and practiced more to succeed. I don't see how that contradicts anything I said. Circumstances matters, effort matters. Talent being a huge factor doesn't contradict that.
Sorry for the lack of post history: I made an account to jump in.
You asked..
> how do you explain the very large number of people who spends tons of time and energy trying to get good at maths but fails to even grasp simple concepts like calculus?
One answer, as vector_spaces said, is that the learning environment can completely change the outcome of this effort. You agreed:
> Circumstances matter
It's true that talent being a factor doesn't contradict that, but given how homogenous early-life learning environments are, it's -extremely- difficult to tell whether a person "failed" because of lack of talent, vs. lack of the right environment to experiment and learn.
Basically, because a "wrong environment" fail looks like a "lack of talent" fail, I believe a lot of talented people don't find out that they have the talent in the first place, as they believe that an attempt in the wrong environment is as good as any attempt, and never try again - particularly if they've been chastised and humiliated -during- that learning process, as vector_spaces was.
In any case, the reason I wanted to jump in was because of your original statement:
>I don't think this is because people told them anything
I would just ask that you reconsider that position, based especially on your agreement that "circumstances matter." Thanks for reading.
Learning environment matters, but I wouldn't say that it would completely change the outcome. If you are smart you don't have to work hard or have a good environment to understand high school maths. If you aren't that smart you can still pass the courses with hard work, but from my experience teaching a lot of college students in 1:1 settings people who have to work hard never really grasp the subjects. And some are so lacking that even with hard work they still can't pass.
So basically, just because someone managed to improve their grades and pass classes with hard work doesn't really say anything about my points. There are still people who pass easily without any of that. In short, my view is that hard work and environment can help improve your results, but unless you are completely prohibited from attending school you wont need that if you are smart enough.
For example, I skipped a lot of classes, was abused, was suicidal, never studied for any tests and didn't do any homework assignments in high school. I still got perfect math grades. Most people have better environments than that and spend more time on maths than that yet still do worse. I spent a lot of time doing 1:1 teaching in college, explaining things in intuitive ways etc helps, but ultimately I had to conclude that I can't make others understand things the way I do. They simply lacked something.
The way you talk about the 'easiness' of high school math makes me think you've received education in the US, where the exams, and even some part of the curriculum, are laughably easier compared to what students from EU countries have experienced.
That is not to say that some people didn't have to put as much effort as others - learned faster. But when I come across people, mostly online, brushing off math at or above high school level as easy have either received subpar/simplified math education (happens almost exclusively in the US and some shitty EU countries), or have too much confidence from their positive experience with low-difficulty exams.
> I had to conclude that I can't make others understand things the way I do.
If you've had this happen too often you're probably just bad at explaining things, or even worse you haven't fully understood them.
I took the most rigorous college degree in Sweden and did pure maths up to masters level. High school maths included calculus, although not as rigorous as you learn at college.
> If you've had this happen too often you're probably just bad at explaining things, or even worse you haven't fully understood them.
I understood the subjects well enough too solve problems on exams I hadn't seen before. That was the main way I managed to get good grades without doing practice problems or studying. I didn't manage to get others to understand things that well. I don't think that failing to get people to that level means I am bad at explaining as there is nobody in the entire world who can get average college students to that level.
For my masters thesis I solved a problem that had stumped one of the more renowned mathematicians in Sweden for decades. I'm not sure if that was hard or not, but to me it didn't seem hard. I didn't continue doing maths as I didn't really have enough passion for the subject to continue doing research, it is pretty boring. After that I realized I needed to earn money, so I learned programming, started getting top 100 spots and sometimes top 10 spots in world wide programming competitions and joined Google. That didn't seem particularly hard to me either, as it took about half a year just doing things on my own.
Why aren't I famous etc? Because I just don't care that much about things. I get bored and stop doing them long before I have worked enough hours on things to get any notable results. I aren't bragging here, I know my value to society thus isn't that high, I am just giving my perspective on things. There are tons of people online arguing that I can't exist, I don't like that.
No offense but that's exactly what I felt you were doing on your first comment when you were talking about people having trouble understanding math as easily as you can. We are on the Internet so it doesn't sound as condescending as it would IRL, but it still makes you come off as trying to convince us that you're superior. I'm certain you're not doing it intentionally, I'm just trying to explain why as you said, are there tons of people online arguing that you can't exist. People with academic achievements akin to yours tend to remain humble when similar topics about intelligence and genius are brought up. I don't think they act this way because they legitimately do not consider themselves smart, they just know it's quite socially inappropriate to boast about it. On the contrary, again on a systematic basis, those with negligible life achievements almost always try to inform everyone in the room that they feel superior in terms of mental capability. Based on your academic biography above you seem to be an exception to what I just described and have observed both online and IRL.
> For my masters thesis I solved a problem that had stumped one of the more renowned mathematicians in Sweden for decades.
Sounds impressive but at the same time many would be quite skeptical about a really challenging problem being solved in the short, in terms of time, course of a Masters degree. Again, not attempting to undermine your achievement, but you usually hear about hard unsolved problems being a topic of a PhD thesis which almost always includes a number of publications being done during its course as part of it. I'd appreciate a few more details here but I understand you may hesitate stating more because of potential privacy concerns.
It could be, but it doesn't really matter. There are people who can think circles around me (and probably you). And I don't think that they ought to have some flaws just to not be totally superior out of some universal law of fairness.
What do you even mean with talent? Specifically when it comes to mathematics, that is. Some astrological alignment of circumstance such that someone can realize their potential unimpeded by their economic class, physical or mental health status, abilities in the language of instruction, lack of interest in the material, or innumerable other such challenges? Or are you referring also to some eugenicist fantasy of a person whose neurochemistry/physiology happens to be optimized for that specific subset of human activities that we call mathematics?
The latter assumes the former, since if such a thing exists, you can be sure that external factors would limit such a person's ability to succeed in traditional settings.
In either case, I don't understand why such a notion would be useful to anyone except as a justification for gatekeeping and maintaining a culture of elitism in math, which is very much not needed. I'm genuinely curious as to why you find the concept useful
If you have enough talent for maths you can get good results in every maths course up to graduate level without studying or doing practice problems outside of just listening to classes. I know I did that, so I know it is possible. I don't particularly like maths courses, I just took a lot of them since they were free credits.
Is this useful information for how to treat typical students? No. However, talent having such a profound impact on peoples ability to learn, understand and apply mathematics is something most people are aware of and hence it isn't strange that people who studied maths thinks that genius requires inborn talent. The author of this piece instead takes a jab at it and tries to say that they just believe that since they got told by their parents that they were smart, I don't think that is true. Most people studying maths doesn't view themselves as good at maths, so what is more likely is that they take their own struggles despite hard work and passion as evidence that genius requires inborn talent.
So for you, there was a royal road to math? No homework necessary? If so, you're one in a billion. It's no wonder you don't share the perspective of others who must practice in order to learn.
It wasn't that I didn't need practice, its just that the practice you get from sitting and listening in classes and doing problems in exams was enough practice. It wasn't as if I could just go to the exams without doing anything else.
The trick was mostly that most things just clicked for me, and I could use that to figure the rest out and fill in the gaps during exams to solve the problems they didn't explain in any classes.
And I tried and tried, but to no avail. I can understand what formulas describe, if I take time to remember definitions. But my mind refuses to grasp the reality behind those formulas, if that reality doesn't map well to everyday things like 3D space, objects, movement and so on.
It could be possible that I haven't tried long enough, but at about 35 I decided to put less effort into it and to prioritize other things.
I'm not encouraging anyone to avoid trying, of course. It was a very useful experience. Knowing that seemingly different things may have unifying mathematical structure (like monad in programming) can expand your approaches even if you aren't good at recognizing and exploring those mathematical structures.
>For those who disagree, how do you explain the very large number of people who spends tons of time and energy trying to get good at maths but fails to even grasp simple concepts like calculus?
Calculus isn't simple, calculus is an abuse of notation that works by anomalous cancellation in most of the cases you see in university.
Even for something as simple as arithmetic our notation is terrible and misleading. The people who are 'good at maths' are actually good at making unwarranted assumptions and extrapolations from incomplete data. Drawing the actual parse tree for expressions let my 7 year old cousin work at the level of a 12 year old even though she was failing the maths class before I helped her.
It isn't simple to a computer, no. But humans have intuition to help guide us which is what makes maths simple. The more of it you have the simpler maths become, the less of it you have the more you have to approach it like a computer as you describe. Being able to approach something like a computer doesn't mean that you understand it though, otherwise you could argue that anyone being able to follow the instructions of a Turing engine understands basically all knowledge.
Were that the case we could use human brains in jars to solve np problems. We can't. Human intuition is a poor thing that got us through the middle ages. It should stay there.
Human intuition is what lets humans solve problems we still can't solve using computers. It is therefore highly relevant until we have invented general purpose AI.
I'm old enough to remember that being said about chess and go. We do not need general purpose AI to solve either of those, and neither do we need it for solving mathematical problems.
Chess and go aren't solved in the proper sense. There's no mathematically proven optimal strategy. What we have is a sort of artificial intuition engines.
> For those who disagree, how do you explain the very large number of people who spends tons of time and energy trying to get good at maths but fails to even grasp simple concepts like calculus?
Are there really a lot of these people? Everyone I know who is good at math likes it and spends a lot of time on it.
In school I was one of those kids who got it pretty easily. I skipped a year in math, I did my exams a year early, I won a math contest, and I went to top uni. But then I also went out of my way to buy extra math texts (not cheap for me) and had extra lessons to learn more stuff.
The kids who didn't do so well were the ones who just saw it as a bunch of riddles to be memorized. That works briefly for an exam or two, but if you don't pick up the ideas properly it gets pretty easy to run into a question that can't be answered.
> Contrast that with the much smaller but still considerable number of people who aren't really passionate about maths, don't put in the work yet still aces the class?
My brother told me he never saw me studying. This is true, why would I have him around when I was studying? I did a fair bit at night when he was asleep. Or I'd think about things on the train.
Also at uni there was a culture of pretending you made no effort but did well anyway. It was childish. Clearly people were studying, because there are very simple things you won't know in any other way, such as the names of concepts or people associated with them.
First is that math builds upon itself. You can't do algebra without understanding operators, you can't do calculus without understanding algebra etc...
Second is that many feel math is boring, and they just zone out during the critical years. To get good at math, you need to assimilate concepts by playing around with them and doing the homework.
Combine both these and you really have no way to tell if a person is lost because they are not good at math or because they've done #1 and #2 their whole lives.
This is a good piece that goes into the complexity of the idea of Genius. We’re taking individuals who’ve had a large impact on society and then saying that this is caused by some hidden factor called genius. We then ask if this genius is present in all of us or can it be developed. Now this obviously has a lot of limitations.
I doubt my own opinions on genius will have much credibility given I’m not not one. But studying geniuses it seems to me that developing an intuition for the subject is what truly sets them apart. Einstein had the impact he did because somehow he knew what areas to work on. In the sea of confusion and subjects floating around at the time he zeroed in on Brownian motion, Relativity, Photoelectric effect and Lasers. Any one of these is enough to go down in history as one of the greats.
I have a friend who’s extremely intelligent. What I’m often surprised by is the speed of thinking. It seems way too quick to be the result of any rational deliberation. Kind of reminds of the policy and value networks in alpha go. Their brains are wired to search much more effectively through the search space.
The other aspect is meta cognition. They understand their own abilities and how they improve . Which means they are able to train themselves with an efficiency we cannot.
To mention something he left out: first mover advantage. Consider the great number of geniuses in tiny, mostly-illiterate Ancient Greece, for instance. Consider the simplicity of Beethoven's fifth symphony's motif, or Dijkstra's algorithm. You have to be there first. (It's not that simple, natch...)
This rankles though:
...no absolute pitch – all things very necessary to a professional performer
I wonder why someone with so much musical education would think that.
Does anyone else feel like you're at a weird in between with intelligence?
I have been developer for 15 years, and have probably put in over 15-20K hours of very heavy learning. I've produced well over 100 projects, and have had some very huge wins. I have an absurd work ethic.
In my small peer group, people constantly talk about "youre smartest person I know" and this makes me somewhat cringe on the inside, because in my mind.. the people who _I_ consider are actual geniuses, who have such an insane otherworldly level of learning ability, math ability, talent, programming ability (ie like John Carmack).
So, I constantly am called smart, smartest, etc, yet in my own head I often feel incredibly dumb because of the people who I look up to, and don't even remotely hold a candle to. It feels like being stuck in a sort of no man's land... where I have enough IQ to realize how little IQ I have.
I always feel like certain subjects often lie JUST outside my mental grasp and I must spend considerable effort to visualize it before I can understand it. For example, I'm starting to build an algorithm for trading and it involves bayesian statistics, and calculus, linear algebra, with a dash of machine learning. All things which have long been forgotten. I spent about 60-70 hours over a few days re-learning a lot, and making huge strides in understanding, feeling pretty good I just got such a huge chunk of learning done quickly, and then I go on youtube videos of math geniuses who probably accomplished what I did effortlessly in the blink of an eye.
I suppose I have no other question, other than does anyone else feel like this? It really kind of sucks, as I do highly value intelligence, and it would be different if I weren't 1/4 or 1/2 of the way there.
It is relative, many people who never attended college thinks that all engineers and doctors are geniuses. If you were less talented you'd likely call the current you a genius as well. If you were more talented maybe you wouldn't consider Carmack a genius, who knows? He probably doesn't consider himself a genius at least, he can point to the likes of Einstein and Newton and say that he is very lacking in comparison.
Not genius. What you're describing is the difference between a problem and a mystery.
If you have some basic skills in a domain, when a problem arises, you usually have some idea about how to solve at least part of the problem. But if you have no skills, every part of even the most basic problem is a mystery that you have no idea how to begin to solve. Then when you see someone solve your mystery, especially if they can do it adroitly, to you they appear to be a genius. Mystery solved.
On that basis, a genius is someone who can solve a problem that other experts had no idea how to solve. A supergenius is someone who solves a problem that other experts didn't realize even existed. (AKA, they hit a target others could not see).
I assume you're on the upper end of the bell curve. Well, unless you're the furthest to the right (in some combination of talents), then there's someone further out there than you. You can beat yourself up about not being as smart as they are, but you'd have to be the smartest to avoid it. So, maybe you should stop expecting that you should be able to keep up.
What's more, for some of these people, it wasn't easy for them, either. General relativity wasn't easy for Einstein. He literally asked someone to "come help me with the mathematics before I go mad". That is, Einstein thought he wasn't smart enough to understand it!
Comparing yourself to others on smarts is a game you can only lose, because someone else is always smarter. Even if you're the smartest (Einstein, maybe), there's some related area or different combination where you're not. Someone else is better at math... or has more background in algebraic topology... or Hilbert spaces... or...
So don't be so hard on yourself. Stop playing the comparison game, because you can only lose. Instead, be who you are, and make the most you can of who that is.
Thank you, I really appreciate that reply - it makes a lot of sense. I think I've come a long way being nicer on myself, I suppose there are just remnants that are triggered by certain events or things. Thanks again
You sound as if you're doing well, professionally.
Esteem from your work comrades, who share your work environment and it's challenges, is not to be taken lightly . . . you have excelled, given what you have to work with.
Personally, I thought in my younger years, that I could understand anything, given enough study time.
Further, I thought I could (probably) suggest innovations, if circumstances permitted, after that time of study.
I'm older now . . . and realize that I can both contribute to innovation . . . and stand aside in admiration when others innovate without my input.
There's a certain utility in that. No one can do it all; time is limited . . .
If you're surrounded by people that constantly tell you how smart you are, you should take it as a signal and go find new (more intelligent) people to surround yourself with.
You don't want to be the one-eyed king.
Also, on the subject of "genius", having met a few in my life, many people seem to have a simplistic idea of what it takes to be one. According to one popular model, all that is required is an increase in the magnitude of certain qualities which everyone already possesses in some measure. Make the particular qualities pronounced enough, and you get to genius. But a better way to understand the concept may be that a genius has a particular capacity, which on a certain level can seem obvious or unremarkable, but which no one else has. A genius, on this understanding, is a person uniquely capable of making a leap ‘off the path’. With hindsight the leap may seem simple or obvious, but at the time no one else was, apparently, capable of making it. A potential leap of this kind is made possible by preceding leaps. Nevertheless its actual occurrence may go on not happening for decades. During that time there may be clear pointers towards it. Yet it is not until a genius comes along that the leap actually happens.
A good perspective, I appreciate it. Also as a sidenote, just because I don't want to give the impression of any hubris, I always immediately dismiss it directly, and it's also not "constantly," (will edit where I used that word) as if I somehow get enjoyment out of it. I hear it enough though, for it to be noticeable (objectively).
It’s a good attitude to have unless you feel bad about yourself and it becomes a problem (like a crushing self defeating impostor sindrome), being modest helps keeping the ego in check. You can self compete and aspire to whatever minds you keep in high regard and it’s likely you’re on the right path. Just keep at it
If you go through life comparing yourself to people who are "better" than you, you're bound to be constantly disappointed by your relative lack of success.
The only person worth comparing yourself to is yourself.
Comparing yourself to those better than you doesn't automatically lead to disappointment, it is about perspective. You could say "I'll never be as fast as Usain Bolt", or you could say "I'm just 15% slower than the worlds fastest man, great!".
Genius is relative. For most of the world any dev is a genius. For devs, john carmack is a genius. For EE Shannon is a genius. For academics Galois/Gauss/Grothendieck are geniuses.
True genius as in having a burst of epiphany, an indescribable out of control impulse leading an eureka might not even be tied to a person..
It's something that I wonder often about because as a kid I had a few eureka moments and this was a rare bliss. Since college I lost it. I failed hard in maths yet with time I finally clicked .. and I wonder if the smartest people also have similar struggle or if everything is always lightweight inspiration after inspiration finding partial solutions then improved solutions.
I think the problem is you place too much value on intelligence, since other people told you you are smart, you consider it core part of your identity, which makes it feel bad when you realize you are not the smartest, you felt like a fraud, that you don't belong to the group of smart people, which sounds quite like imposter syndrome.
Those folks you look up to, by its definition, is quite often people who you think are better than you in one way or another, that's why you look up to them anyway, there is nothing wrong looking up to those people. However, it is necessary to build your identity around more than just intelligence, that way you should not feel so anxious about this intelligence superiority.
Intelligence is tricky in itself, likely no one came up with relativity theory before Einstein, but I'm pretty sure someone before Einstein is smarter than him, we just don't know who, but that someone didn't come up with general relativity, because he would not have had the kind of tools and access to knowledge Einstein have. Intelligence doesn't mean you already understand how the world works when you are born, it's not like it's burned on a disk, it means you have the ability to understand and the creativity to come up with ideas. What should matter more is always what you do with your intelligence, rather than what you have.
Hmm that makes sense. I have always felt it is more damaging to hear that than anything (I'd rather people not tell me that) because you're right, it starts to seep in my subconscious. Thanks for your perspective, I like seeing fresh perspectives here to keep my mental space in check.
Reminded me of a passage from a DFW short story called Good Old Neon [0]:
"He asked if I ever played chess, and I told him I used to in middle school but quit because I couldn’t be as good as I eventually wanted to be, how frustrating it was to get just good enough to know
what getting really good at it would be like but not being able to get that good, etc."
You might like the story, it's pretty well-written and maybe touches on something similar to what you're feeling.
When it's yourself, you see how hard you've worked. Time spent going down the wrong path, learning a lot of incidental things like how some lib works, reading papers.
When you meet someone else, you see the product of that work, a summary of their training that seems to have an answer to whatever comes up.
Genius = Various permutations of intensities of { Innate Talent, Hardwork, Imagination, Opportunity, Self Confidence, Curiosity, Environmental Context in Time } tipped over by "Chance".
Yes, sometimes all the needles tick in the right quadrants but other times not. There are plenty of tortured geniuses out there who had one extremly developed facet to the detriment of others, but did register as ‘genius’ after all. I wonder if inner happines is to ever be considered part of genius
Apparently Shannon described genius as a response to discovering a source of irritation. If that's universally true of genius, then their ability to reduce their own irritation would imply increased personal happiness. If that irritation were shared by others, reducing it should increase happiness in others, throughout those afflicted.
Certainly medical geniuses increase happiness to others through decreasing disease and the loss of happiness it causes. Don't artistic geniuses do something similar, by elevating their art form to new heights, instilling wonder and awe in their audiences? Perhaps increasing happiness in others is essential to genius. If the genius cares about others, that service to others ought to please themselves too.
Of course, some examples of social psychopathy might also be called genius: Hitler, Stalin, Henry VIII, Pol Pot and many successful politicians possess(ed) a perverse genius for rising to power and wielding it effectively, often at great expense to the happiness of their people. These people did so much damage to so many lives, it's certain they cared little about the happiness of others, nor what their Machiavellianism said about themselves or their impact on history.
What does the latest research in neuroscience say about the nature of general intelligence, nature of talent and geniality? Are there any physical markers, I do not know, connections in the brain are established faster and more brain areas are triggered while thinking? Are there any more attempts to produce geniuses after famous László Polgár?
I can't speak to brain wiring, but we do know geniuses almost always share a number of distinctive behaviors: a capability for intense focus, persistence, passion for solving (or understanding) a problem, and the ability to learn efficiently and deeply.
An inclination to think and work independently seems essential too. A disinterest in the opinion of others, a disinterest in pleasing others, especially those who don't share their perspective or priorities -- these are common traits.
Excellent memory and the mastery (and advancement) of skills in the domain seem to be essential too. Probably that's where the 10,000 hours of practice comes in. And learning from exploring possibilities and testing whether each new idea or method does or does not add value, that's typical too.
I think the brain wiring that enables these behaviors is what you seek. But I doubt we understand the brain well enough to suss out which wires enable which essential behavior.
When Lenin died of a brain hemorrhage in 1924, his brain was preserved in formaldehyde, where it remained for two years. In 1926, [German physician and neurologist Oskar] Vogt was recruited by the Soviet government to help establish Lenin's genius via histological investigation of his brain.
In 1927, Vogt gave a preliminary report on his findings in Moscow, concluding from his histological observations that Lenin must have been an athlete in associative thinking ("Assoziationsathlet") - a conclusion deemed farfetched by some of his neurologist colleagues and adversaries.
I had a high school teacher who used my IQ test results to justify my 'old soul' status enough to sexually assault me. As a working class person, most of my friends did not finish highschool or go to college and I did but I hated the people there. My father skipped to grades and got a full scholarship to the University of Toronto quit and joined the Hells Angels and died at 40. I found my groove (my partner is a finnish physicist) but the ability to read and think fast in the context of female and working class can be hell - at least in one's youth.
It reads like a rationalization for being mediocre to me, to be honest. The definition of genius by "impact on society" is obviously wrong. And it's easy to selectively quote people who have something bad to say about a person to make them seem like awful people. Also doesn't seem rigorous: are ALL geniuses awful persons? That seems very unlikely. Especially as there may be geniuses in the field of being "good people".
I know well the ex-wife and children of the inventor of an influential computer language. According to them, this language is his obsession to the exclusion of all else. They describe him as "an asshole". He habitually berated his children for not being passionate about anything. He is famous, but lost his family.
I think about this fellow often, when I worry that I'm not fulfilling my potential. If genius means emotional misery for my family, then I am fine with more pedestrian accomplishments
We all have fairly similar hardware, brains don't vary that much between people. I think what unites a lot of "geniuses" is actually monomania. Whether it be athletes, scientists or musicians, being obsessed with a single activity will of course help you excel, but to the detriment of everything else in your life.
I suppose the answer to that is whether some 'geniuses' also are known for treating others well. As I recall, Shannon was described as a nice guy who seemed to care about others or at least respected their feelings. Perhaps technical genuises are better about this than artistic geniuses?
I suspect the majority of genuii are self absorbed. Certainly that aspect of (mis)behavior is a mainstay of the genius stereotype. Or maybe that portrayal in movies is just the conceit of lazy scriptwriters...
John von Neumann, a genius if there ever was one, was apparently a hoot.
"Von Neumann is remembered as a man of warm personality: courteous, charming, and jovial, with an often ribald, sometimes wry, sense of humor that made him excellent company and gained him a reputation as a bon vivant. He was fond of limericks and practical jokes and hosted frequent Princeton parties. He was also well known as a reckless driver, once emerging, so the story goes, from a smashed car with the explanation: "I was proceeding down the road. The trees on the right were passing me in orderly fashion at 60 MPH. Suddenly, one of them stepped out in my path. Boom!""
Interesting, but ultimately overly self-centered essay lacking real substance and evidence.
1. Gripe: Why do authors seek to write auto-biographical introductions? I count 5 full paragraphs of intro into this self-admitted "plodding" professor, who is not a genius, nor is he describing the "nature" of genius in those paragraphs. It comes off unnecessary and self-centered to write 5 paragraphs about yourself at the start of an essay on the nature of genius, does it not?
2. Mozart is held up as his first great example of Genius. Mozart had numerous advantages when young, though: he grew up in an intensely musical family, his brain was adapting to music in his first 3 years and then he was professionally trained beginning as a toddler. If you'd been playing piano for 3 years and your brain was specially adapted to it, yes you could be a wonder of Europe if you were also still only 5-6 at the time. The exponentially faster learning that young children exhibit is well known: their brains are extremely neuro-plastic.
3. Further down the essay, once more, a few autobiographical paragraphs about his class and Yale, and waxing poetic on gender discrepancies, not about geniuses. The essay seems to lack focus on the stated topic, the Nature of Genius, not the Nature of Yale or of the writers' life.
4. After more waxing poetic and little hard evidence except for cursory mentions of the famous geniuses, I find myself waiting for the simple point: genius is a social construct. We are all cooperatively working together, and geniuses are those who are labeled geniuses based upon the impact of their output. To explore the "nature" of genius is therefore a confluence of exploring psychology, human networks, history, and epistemology, and how socially geniuses excel inside the network of thought.
5. IMO, modern schooling is quite time-consuming and limiting. I find myself wondering, didn't the apprentice system where children had time to work on topics for a full decade longer, while younger, doesn't that aid in the creation of greatness? Rather than our painters painting from age 12, they don't begin working in earnest until age 22, up until which time their "work" consists of theoretical assignments, problem sets, and doesn't touch the real world applications which greatness necessitates.
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[ 3.4 ms ] story [ 72.2 ms ] threadThere seems to be some thing that allows creative construction of these things, in mind, as opposed to simple recall of the sounds or images, in mind.
For example, Mozart might have been able to creatively "construct" the sound of his composures in his mind, but just because someone has heard a Mozart composure performed and then can recall it later, in mind, that fact doesn't make them a genius.
It's both creating and hearing/seeing it first that makes it special.
My theory is this is how animals arrive at consciousness.
I don't think this is because people told them anything. To anyone who is good at maths it is painfully obvious that some people get it much easier than others. I started out thinking that people just did it wrongly, but the more I tried and the more I watched people do things the more I realized that they simply were bad at maths. There is nothing I can do for them. Some people just lack practice and motivation, true, but most lacks the brains to do it, since you need to have at least some level of intuition and some level of rational reasoning capacity to get anywhere in maths.
For those who disagree, how do you explain the very large number of people who spends tons of time and energy trying to get good at maths but fails to even grasp simple concepts like calculus? Contrast that with the much smaller but still considerable number of people who aren't really passionate about maths, don't put in the work yet still aces the class? Many on the "nurture is everything" side simply believes that those differences doesn't exist, the "geniuses" must have learned this beforehand or something else silly. But if you are one of those who aced the classes without even trying and see others try really really hard and still don't even pass, how could you explain that without considering talent as a variable? That person doesn't need to be told that talent is important.
A lot of otherwise intelligent people fairly despise such practices so they adopt inefficient studying habits , failing to complete a set syllabus and achieving subpar results.
Success in the field as in research? Most certainly so.
However I think you're overestimating the 'math IQ' of STEM (under)grads especially considering the large discrepancy among its different subfields. Engineering students (EE/ME/Civil) are systematically 'dumber' maths-wise compared to their science (Maths [duh] or Physics) counterparts. Still, it's quite likely that you do not have to be a genius to be successful at landing a highly-paid, mentally-challenging related job.
About parenting, smart math kids may not have had their egos boosted however their parents almost certainly exposed them to science from an early age. This holds true especially for children coming from 'STEM families'.
I never said otherwise. However you can't be dumb to land those jobs.
In high school, I was one of those people who spent tons of time and energy on it, and failed miserably, barely graduating as a result. It wasn't until I was about 25 when I picked up a basic algebra and trigonometry book on a whim and started self-teaching, that it began making sense to me. I eventually went back to school for pure math.
In my case, I think what helped me as an adult was the freedom to learn on my own terms without the pressure of deadlines, and also a strong support network of friends encouraging me (compare with my teachers in high school who would actively embarrass me in front of the class or joke about their low expectations of me). And of course, some years of life experience in the real world doing things that were actually hard compared to (elementary) math as someone of limited means helped a bit too.
So, all of that is to say that people who struggle in math classes struggle for different reasons. I suspect that much of the time the dominating forces are psychosocial, cultural, or plain & simple lack of interest (which itself might be driven by traumatic experiences with shitty teachers or parents with weird expectations)
You asked..
> how do you explain the very large number of people who spends tons of time and energy trying to get good at maths but fails to even grasp simple concepts like calculus?
One answer, as vector_spaces said, is that the learning environment can completely change the outcome of this effort. You agreed:
> Circumstances matter
It's true that talent being a factor doesn't contradict that, but given how homogenous early-life learning environments are, it's -extremely- difficult to tell whether a person "failed" because of lack of talent, vs. lack of the right environment to experiment and learn.
Basically, because a "wrong environment" fail looks like a "lack of talent" fail, I believe a lot of talented people don't find out that they have the talent in the first place, as they believe that an attempt in the wrong environment is as good as any attempt, and never try again - particularly if they've been chastised and humiliated -during- that learning process, as vector_spaces was.
In any case, the reason I wanted to jump in was because of your original statement:
>I don't think this is because people told them anything
I would just ask that you reconsider that position, based especially on your agreement that "circumstances matter." Thanks for reading.
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So basically, just because someone managed to improve their grades and pass classes with hard work doesn't really say anything about my points. There are still people who pass easily without any of that. In short, my view is that hard work and environment can help improve your results, but unless you are completely prohibited from attending school you wont need that if you are smart enough.
For example, I skipped a lot of classes, was abused, was suicidal, never studied for any tests and didn't do any homework assignments in high school. I still got perfect math grades. Most people have better environments than that and spend more time on maths than that yet still do worse. I spent a lot of time doing 1:1 teaching in college, explaining things in intuitive ways etc helps, but ultimately I had to conclude that I can't make others understand things the way I do. They simply lacked something.
I appreciate you sharing your story - I'm sorry you had to deal with that experience.
That is not to say that some people didn't have to put as much effort as others - learned faster. But when I come across people, mostly online, brushing off math at or above high school level as easy have either received subpar/simplified math education (happens almost exclusively in the US and some shitty EU countries), or have too much confidence from their positive experience with low-difficulty exams.
> I had to conclude that I can't make others understand things the way I do.
If you've had this happen too often you're probably just bad at explaining things, or even worse you haven't fully understood them.
> If you've had this happen too often you're probably just bad at explaining things, or even worse you haven't fully understood them.
I understood the subjects well enough too solve problems on exams I hadn't seen before. That was the main way I managed to get good grades without doing practice problems or studying. I didn't manage to get others to understand things that well. I don't think that failing to get people to that level means I am bad at explaining as there is nobody in the entire world who can get average college students to that level.
For my masters thesis I solved a problem that had stumped one of the more renowned mathematicians in Sweden for decades. I'm not sure if that was hard or not, but to me it didn't seem hard. I didn't continue doing maths as I didn't really have enough passion for the subject to continue doing research, it is pretty boring. After that I realized I needed to earn money, so I learned programming, started getting top 100 spots and sometimes top 10 spots in world wide programming competitions and joined Google. That didn't seem particularly hard to me either, as it took about half a year just doing things on my own.
Why aren't I famous etc? Because I just don't care that much about things. I get bored and stop doing them long before I have worked enough hours on things to get any notable results. I aren't bragging here, I know my value to society thus isn't that high, I am just giving my perspective on things. There are tons of people online arguing that I can't exist, I don't like that.
No offense but that's exactly what I felt you were doing on your first comment when you were talking about people having trouble understanding math as easily as you can. We are on the Internet so it doesn't sound as condescending as it would IRL, but it still makes you come off as trying to convince us that you're superior. I'm certain you're not doing it intentionally, I'm just trying to explain why as you said, are there tons of people online arguing that you can't exist. People with academic achievements akin to yours tend to remain humble when similar topics about intelligence and genius are brought up. I don't think they act this way because they legitimately do not consider themselves smart, they just know it's quite socially inappropriate to boast about it. On the contrary, again on a systematic basis, those with negligible life achievements almost always try to inform everyone in the room that they feel superior in terms of mental capability. Based on your academic biography above you seem to be an exception to what I just described and have observed both online and IRL.
> For my masters thesis I solved a problem that had stumped one of the more renowned mathematicians in Sweden for decades.
Sounds impressive but at the same time many would be quite skeptical about a really challenging problem being solved in the short, in terms of time, course of a Masters degree. Again, not attempting to undermine your achievement, but you usually hear about hard unsolved problems being a topic of a PhD thesis which almost always includes a number of publications being done during its course as part of it. I'd appreciate a few more details here but I understand you may hesitate stating more because of potential privacy concerns.
It could be, but it doesn't really matter. There are people who can think circles around me (and probably you). And I don't think that they ought to have some flaws just to not be totally superior out of some universal law of fairness.
Obsessing about it is obviously not healthy.
The latter assumes the former, since if such a thing exists, you can be sure that external factors would limit such a person's ability to succeed in traditional settings.
In either case, I don't understand why such a notion would be useful to anyone except as a justification for gatekeeping and maintaining a culture of elitism in math, which is very much not needed. I'm genuinely curious as to why you find the concept useful
Is this useful information for how to treat typical students? No. However, talent having such a profound impact on peoples ability to learn, understand and apply mathematics is something most people are aware of and hence it isn't strange that people who studied maths thinks that genius requires inborn talent. The author of this piece instead takes a jab at it and tries to say that they just believe that since they got told by their parents that they were smart, I don't think that is true. Most people studying maths doesn't view themselves as good at maths, so what is more likely is that they take their own struggles despite hard work and passion as evidence that genius requires inborn talent.
The trick was mostly that most things just clicked for me, and I could use that to figure the rest out and fill in the gaps during exams to solve the problems they didn't explain in any classes.
It could be possible that I haven't tried long enough, but at about 35 I decided to put less effort into it and to prioritize other things.
I'm not encouraging anyone to avoid trying, of course. It was a very useful experience. Knowing that seemingly different things may have unifying mathematical structure (like monad in programming) can expand your approaches even if you aren't good at recognizing and exploring those mathematical structures.
Calculus isn't simple, calculus is an abuse of notation that works by anomalous cancellation in most of the cases you see in university.
Even for something as simple as arithmetic our notation is terrible and misleading. The people who are 'good at maths' are actually good at making unwarranted assumptions and extrapolations from incomplete data. Drawing the actual parse tree for expressions let my 7 year old cousin work at the level of a 12 year old even though she was failing the maths class before I helped her.
Are there really a lot of these people? Everyone I know who is good at math likes it and spends a lot of time on it.
In school I was one of those kids who got it pretty easily. I skipped a year in math, I did my exams a year early, I won a math contest, and I went to top uni. But then I also went out of my way to buy extra math texts (not cheap for me) and had extra lessons to learn more stuff.
The kids who didn't do so well were the ones who just saw it as a bunch of riddles to be memorized. That works briefly for an exam or two, but if you don't pick up the ideas properly it gets pretty easy to run into a question that can't be answered.
> Contrast that with the much smaller but still considerable number of people who aren't really passionate about maths, don't put in the work yet still aces the class?
My brother told me he never saw me studying. This is true, why would I have him around when I was studying? I did a fair bit at night when he was asleep. Or I'd think about things on the train.
Also at uni there was a culture of pretending you made no effort but did well anyway. It was childish. Clearly people were studying, because there are very simple things you won't know in any other way, such as the names of concepts or people associated with them.
First is that math builds upon itself. You can't do algebra without understanding operators, you can't do calculus without understanding algebra etc...
Second is that many feel math is boring, and they just zone out during the critical years. To get good at math, you need to assimilate concepts by playing around with them and doing the homework.
Combine both these and you really have no way to tell if a person is lost because they are not good at math or because they've done #1 and #2 their whole lives.
I doubt my own opinions on genius will have much credibility given I’m not not one. But studying geniuses it seems to me that developing an intuition for the subject is what truly sets them apart. Einstein had the impact he did because somehow he knew what areas to work on. In the sea of confusion and subjects floating around at the time he zeroed in on Brownian motion, Relativity, Photoelectric effect and Lasers. Any one of these is enough to go down in history as one of the greats.
I have a friend who’s extremely intelligent. What I’m often surprised by is the speed of thinking. It seems way too quick to be the result of any rational deliberation. Kind of reminds of the policy and value networks in alpha go. Their brains are wired to search much more effectively through the search space.
The other aspect is meta cognition. They understand their own abilities and how they improve . Which means they are able to train themselves with an efficiency we cannot.
What is alpha go? Is it https://en.m.wikipedia.org/wiki/AlphaGo ?
This rankles though:
...no absolute pitch – all things very necessary to a professional performer
I wonder why someone with so much musical education would think that.
I have been developer for 15 years, and have probably put in over 15-20K hours of very heavy learning. I've produced well over 100 projects, and have had some very huge wins. I have an absurd work ethic.
In my small peer group, people constantly talk about "youre smartest person I know" and this makes me somewhat cringe on the inside, because in my mind.. the people who _I_ consider are actual geniuses, who have such an insane otherworldly level of learning ability, math ability, talent, programming ability (ie like John Carmack).
So, I constantly am called smart, smartest, etc, yet in my own head I often feel incredibly dumb because of the people who I look up to, and don't even remotely hold a candle to. It feels like being stuck in a sort of no man's land... where I have enough IQ to realize how little IQ I have.
I always feel like certain subjects often lie JUST outside my mental grasp and I must spend considerable effort to visualize it before I can understand it. For example, I'm starting to build an algorithm for trading and it involves bayesian statistics, and calculus, linear algebra, with a dash of machine learning. All things which have long been forgotten. I spent about 60-70 hours over a few days re-learning a lot, and making huge strides in understanding, feeling pretty good I just got such a huge chunk of learning done quickly, and then I go on youtube videos of math geniuses who probably accomplished what I did effortlessly in the blink of an eye.
I suppose I have no other question, other than does anyone else feel like this? It really kind of sucks, as I do highly value intelligence, and it would be different if I weren't 1/4 or 1/2 of the way there.
If you have some basic skills in a domain, when a problem arises, you usually have some idea about how to solve at least part of the problem. But if you have no skills, every part of even the most basic problem is a mystery that you have no idea how to begin to solve. Then when you see someone solve your mystery, especially if they can do it adroitly, to you they appear to be a genius. Mystery solved.
On that basis, a genius is someone who can solve a problem that other experts had no idea how to solve. A supergenius is someone who solves a problem that other experts didn't realize even existed. (AKA, they hit a target others could not see).
What's more, for some of these people, it wasn't easy for them, either. General relativity wasn't easy for Einstein. He literally asked someone to "come help me with the mathematics before I go mad". That is, Einstein thought he wasn't smart enough to understand it!
Comparing yourself to others on smarts is a game you can only lose, because someone else is always smarter. Even if you're the smartest (Einstein, maybe), there's some related area or different combination where you're not. Someone else is better at math... or has more background in algebraic topology... or Hilbert spaces... or...
So don't be so hard on yourself. Stop playing the comparison game, because you can only lose. Instead, be who you are, and make the most you can of who that is.
Personally, I thought in my younger years, that I could understand anything, given enough study time.
Further, I thought I could (probably) suggest innovations, if circumstances permitted, after that time of study.
I'm older now . . . and realize that I can both contribute to innovation . . . and stand aside in admiration when others innovate without my input.
There's a certain utility in that. No one can do it all; time is limited . . .
You don't want to be the one-eyed king.
Also, on the subject of "genius", having met a few in my life, many people seem to have a simplistic idea of what it takes to be one. According to one popular model, all that is required is an increase in the magnitude of certain qualities which everyone already possesses in some measure. Make the particular qualities pronounced enough, and you get to genius. But a better way to understand the concept may be that a genius has a particular capacity, which on a certain level can seem obvious or unremarkable, but which no one else has. A genius, on this understanding, is a person uniquely capable of making a leap ‘off the path’. With hindsight the leap may seem simple or obvious, but at the time no one else was, apparently, capable of making it. A potential leap of this kind is made possible by preceding leaps. Nevertheless its actual occurrence may go on not happening for decades. During that time there may be clear pointers towards it. Yet it is not until a genius comes along that the leap actually happens.
The only person worth comparing yourself to is yourself.
True genius as in having a burst of epiphany, an indescribable out of control impulse leading an eureka might not even be tied to a person..
It's something that I wonder often about because as a kid I had a few eureka moments and this was a rare bliss. Since college I lost it. I failed hard in maths yet with time I finally clicked .. and I wonder if the smartest people also have similar struggle or if everything is always lightweight inspiration after inspiration finding partial solutions then improved solutions.
Those folks you look up to, by its definition, is quite often people who you think are better than you in one way or another, that's why you look up to them anyway, there is nothing wrong looking up to those people. However, it is necessary to build your identity around more than just intelligence, that way you should not feel so anxious about this intelligence superiority.
Intelligence is tricky in itself, likely no one came up with relativity theory before Einstein, but I'm pretty sure someone before Einstein is smarter than him, we just don't know who, but that someone didn't come up with general relativity, because he would not have had the kind of tools and access to knowledge Einstein have. Intelligence doesn't mean you already understand how the world works when you are born, it's not like it's burned on a disk, it means you have the ability to understand and the creativity to come up with ideas. What should matter more is always what you do with your intelligence, rather than what you have.
> Praise for intelligence can undermine children's motivation and performance.
"He asked if I ever played chess, and I told him I used to in middle school but quit because I couldn’t be as good as I eventually wanted to be, how frustrating it was to get just good enough to know what getting really good at it would be like but not being able to get that good, etc."
You might like the story, it's pretty well-written and maybe touches on something similar to what you're feeling.
[0] http://sdavidmiller.com/octo/files/no_google2/GoodOldNeon.pd...
When you meet someone else, you see the product of that work, a summary of their training that seems to have an answer to whatever comes up.
Certainly medical geniuses increase happiness to others through decreasing disease and the loss of happiness it causes. Don't artistic geniuses do something similar, by elevating their art form to new heights, instilling wonder and awe in their audiences? Perhaps increasing happiness in others is essential to genius. If the genius cares about others, that service to others ought to please themselves too.
Of course, some examples of social psychopathy might also be called genius: Hitler, Stalin, Henry VIII, Pol Pot and many successful politicians possess(ed) a perverse genius for rising to power and wielding it effectively, often at great expense to the happiness of their people. These people did so much damage to so many lives, it's certain they cared little about the happiness of others, nor what their Machiavellianism said about themselves or their impact on history.
Many have only been discovered after century in oblivion because someone found and promoted their work, think Bach and Mendelssohn, etc.
An inclination to think and work independently seems essential too. A disinterest in the opinion of others, a disinterest in pleasing others, especially those who don't share their perspective or priorities -- these are common traits.
Excellent memory and the mastery (and advancement) of skills in the domain seem to be essential too. Probably that's where the 10,000 hours of practice comes in. And learning from exploring possibilities and testing whether each new idea or method does or does not add value, that's typical too.
I think the brain wiring that enables these behaviors is what you seek. But I doubt we understand the brain well enough to suss out which wires enable which essential behavior.
The masses have more fun, as evidenced by:
https://www.youtube.com/watch?v=sBdo5WPsHM8
FICKÄÄHN, FRÄSSÄÄÄHN, FÄRRRNSÄÄÄÄHN!
EDIT: JETZT ERST RECHT!
I think about this fellow often, when I worry that I'm not fulfilling my potential. If genius means emotional misery for my family, then I am fine with more pedestrian accomplishments
I suspect the majority of genuii are self absorbed. Certainly that aspect of (mis)behavior is a mainstay of the genius stereotype. Or maybe that portrayal in movies is just the conceit of lazy scriptwriters...
"Von Neumann is remembered as a man of warm personality: courteous, charming, and jovial, with an often ribald, sometimes wry, sense of humor that made him excellent company and gained him a reputation as a bon vivant. He was fond of limericks and practical jokes and hosted frequent Princeton parties. He was also well known as a reckless driver, once emerging, so the story goes, from a smashed car with the explanation: "I was proceeding down the road. The trees on the right were passing me in orderly fashion at 60 MPH. Suddenly, one of them stepped out in my path. Boom!""
https://www.ias.edu/von-neumann
1. Gripe: Why do authors seek to write auto-biographical introductions? I count 5 full paragraphs of intro into this self-admitted "plodding" professor, who is not a genius, nor is he describing the "nature" of genius in those paragraphs. It comes off unnecessary and self-centered to write 5 paragraphs about yourself at the start of an essay on the nature of genius, does it not?
2. Mozart is held up as his first great example of Genius. Mozart had numerous advantages when young, though: he grew up in an intensely musical family, his brain was adapting to music in his first 3 years and then he was professionally trained beginning as a toddler. If you'd been playing piano for 3 years and your brain was specially adapted to it, yes you could be a wonder of Europe if you were also still only 5-6 at the time. The exponentially faster learning that young children exhibit is well known: their brains are extremely neuro-plastic.
3. Further down the essay, once more, a few autobiographical paragraphs about his class and Yale, and waxing poetic on gender discrepancies, not about geniuses. The essay seems to lack focus on the stated topic, the Nature of Genius, not the Nature of Yale or of the writers' life.
4. After more waxing poetic and little hard evidence except for cursory mentions of the famous geniuses, I find myself waiting for the simple point: genius is a social construct. We are all cooperatively working together, and geniuses are those who are labeled geniuses based upon the impact of their output. To explore the "nature" of genius is therefore a confluence of exploring psychology, human networks, history, and epistemology, and how socially geniuses excel inside the network of thought.
5. IMO, modern schooling is quite time-consuming and limiting. I find myself wondering, didn't the apprentice system where children had time to work on topics for a full decade longer, while younger, doesn't that aid in the creation of greatness? Rather than our painters painting from age 12, they don't begin working in earnest until age 22, up until which time their "work" consists of theoretical assignments, problem sets, and doesn't touch the real world applications which greatness necessitates.