An important thing about math is that it's so ordered and logical that you don't need a textbook to learn it. Every theorem, every proof, is available to any mathematician through logical analysis provided that the prerequisite material is well understood. And historical math prodigies generally had access to most or all of the contemporary published works on math as well as private tutors, themselves excellent mathematicians, to help them absorb it as quickly as they could. Leaving them time to advance the field while still young.

Almost no one discovers much math on their own, especially in the modern era where most people are exposed to it in school and then quickly ascend to higher levels as they demonstrate ability. Even fewer people have the talent, interest, and dedication to become a named figure in math history. The people you are describing are a handful out of billions across all of human history.

If you are phenomenally brilliant, mathematically inclined, and have the time to invest you can become a math prodigy. But again, this describes virtually no one.

Well Einstein needed help with his maths in the general relativity period (tensors I think).

Einstein was not a handsome fellow.

Nobody ever called him Al.

He had a long moustache to pull on, it was yellow.

I don't believe he ever had a gal.

One thing he missed out in his theory,

of time and space and relativity.

Is something that makes it very clear he,

is never gonna score like you and me.

He didn't know about quark, strangeness and charm.

(Copernicus had those renaissance ladies crazy about his telescope though, check it out)

One thing that helps is not to have parents who hold you back.

Many parents purposely (and misguidedly) prevent exposing their children to too much information in order to not “burn them out”. This concept is utter rubbish. You couldn’t burn out a normal child’s mind if you tried.

For example, I often claim that if I had a kid, she would learn C++, piano, at least 2 languages, and chess starting from a very young age (starting at birth for languages and no later than 5 for the rest).

Someone told me that that would be ridiculous and overload the child’s mind. “The idea that a child’s mind can be overloaded is what is ridiculous. They learn best the younger they are, so that is the time to be teaching for best (and prodigal) results.”

A perfect example is a kid I “met” on the plane leaving America when I first left. The woman behind me was a Japanese who married a South Korean and they lived in America. Their 4-year-old son spoke fluently and with no accent in all 3 languages.

Potential prodigies are all over the place.

Actualized prodigies are only rare because they were not exposed early enough. You don’t become a 5-year-old prodigy on piano if you starting playing piano at 8.

L. Spiro

I would expose my kid to many stuff, but I wouldn't force her, that's for sure. So I couldn't say things like "she would learn C++, piano, at least 2 languages, and chess starting from a very young age (starting at birth for languages and no later than 5 for the rest)" because it sounds planned, and that's ridiculous if you force that plan on a kid.

Well Einstein needed help with his maths in the general relativity period (tensors I think).

True, he did need help with the maths, He was always very lose with his proofs and so he had an actual mathematician who would help to make them rigorous.

*stuff*

Cute, except he did get married, twice IIRC, and had children.

Parents usually play a great role in child prodigies and I once read an article claiming that the skills of most of them are just average when they are adults (compared to the other professionals in the same field). When creating lists we are clearly slightly biased to the most successful examples. I don't know if this is true, but I agree with L. Spiro when she say childs are the best learners. They are particularly good at learning new languages, but they can learn basically everything if there is interest.

Also in addition to what everyone else has said, learning isn't something that you stop when you're in your mid twenties. Learning also ISNT something that needs to be done in institutes either, though I hear that magic piece of paper you get from institutes is a great way to convince people you are smart. Adults often aren't a good role model for children, in my family my siblings who are now all 'parents' are lazy as hell, dedicating their entire time to watching TV or something, yet they would happily and persistently degrade their kids for not wanting to study. If a child sees his / her role model learning, he will likely be more motivated in copying, those parents who force their kids to learn would have a higher chance of success if they were learning something too.

Personally I don't want to have kids due to the time consumed by raising one, but if I did I would also likely follow L.Spiro too on teaching them young, but also in an environment where that type of learning was as normal as eating.

I have never really bought that whole "children are better learners" theory, I learn faster and better now than I did as a kid and based on what I have seen even on these forums I don't think this is uncommon either.

What was your roadmap, I mean did you have any set goals or did you just go with the flow and go from one category to the next naturally?I'm trying to learn my way into harmonic analysis, since it looks very interesting.Having some trouble, cause I haven't yet memorized all the trig identities and I keep going back to the written piece of paper to see them.

My 2 cent since I read alot into these topics:

There are made 'prodigies' and then there are real prodigies, and then there are exceptional prodigies.

Made prodigies are those where 'prodigy-ness' have been thrust on them, forced. Once they have complete control over their life (read-reach 18 years old) then interesting stuff happens. Some really do enjoy it and keep it that way, but in less rigorous way, in order to catch what they been missed in their life. Some will rebel.

Real prodigies are just kids who really loves something and totally into it. Just like some are introvert and some are extrovert and its hard to change someone from one to another, some kids are just into stuff - whatever that is. I don't know if you consider Slash (guitarist) as prodigy (music) although he might not known or popular when he was a kid, but for me it doesn't change the fact, but the moment he saw people playing guitar when he was a kid, he built one from his broom and simple string and practiced till his grandmother bought him a real one. And he still playing to this day. The same can be said about Neil Tyson. Saw the stars when he was a kid, and he is who he is now.

However, some kids get better attention than the others, and better resources too, and considered prodigy. Some, are lost. If you are a Aghanistan or Iraqi kids today, how can you access to robotic stuff when you are too busy not getting bombed?

So, some of the kids do get resources. Like chess class, music class, access to professionals, healthy diet, etc. But these kids, when they grow up, will deal with people who try to gain the same access and attention like they too, so finally they will even out.

And then there are exceptional prodigies. These are when nature helps them out, and nurture could push things better. I will always use the story of the memory prodigy for this: the guy who can memorize everything people say without knowing it, and thought that everybody can also do the same (http://en.wikipedia.org/wiki/Solomon_Shereshevsky). And the likes with photographic memory. Unlike savants, they have the best of both world.

Some people brains do wired differently (like seing numbers and alphabets in colours). And these are exceptional prodigies who last to their adulthood, unless, they decided to slow down, or burn out due to over exposure or just plan to have a change of life style.

http://en.wikipedia.org/wiki/William_James_Sidis

In short, there are many type of prodigies and you just cannot group them together. Its hard to know which is which, but in the end you will notice.

http://mentalfloss.com/article/27906/9-child-prodigies-who-actually-ended-doing-something

I think the list should be longer, but the names eluded me. So to say that a prodigy end up just like normal people when they are old, is wrong. Some are exceptional when they are kids, and stay exceptional for a long, long time.

edit:

there are many example, but I think Bobbdy Fisher can be used here.

- He was into it himself, not forced.

- He was lucky to be given resources

-

"In May 1949, the six-year-old Bobby, and his sister Joan, learned how to play chess using the instructions from a chess set bought at a candy store below their Brooklyn apartment.[22][23] When Joan grew disinterested in chess, and Regina didn't have time to play, it left Fischer to play many of his first games against himself.[24] When the family vacationed at Patchogue, Long Island that summer, Bobby found a book of old chess games, and studied it intensely.[25]On November 14, 1950, his mother sent a postcard to the Brooklyn Eagle newspaper, seeking to place an ad, inquiring whether other children of Bobby's age might be interested in playing chess with him. The paper rejected her ad because no one could figure out how to classify it, but forwarded her inquiry to Hermann Helms, the "Dean of American Chess", who told her that Master Max Pavey would be giving a simultaneous exhibition on January 17, 1951.[26][27]Fischer played in the exhibition, losing in 15 minutes. One of the spectators was Carmine Nigro, president of the Brooklyn Chess Club, who introduced Fischer to the club and began teaching him.[28][29][30] Fischer attended the club regularly, intensified his interest, and gained playing strength rapidly. In the summer of 1955, the then 12-year-old Fischer joined the Manhattan Chess Club, the strongest in the country.[31][32] Fischer's relationship with Nigro lasted five years, from 1951 to 1956,[33][34] when Nigro moved away to Florida.[34]"

- And he might get tired (being a human) and might need rest.

"After the World Championship in 1972, Fischer virtually retired from chess: he did not play a competitive game in public for nearly 20 years"

- But that will not change who he is.

"After twenty years, Fischer emerged from isolation to play Spassky (then tied for 96th–102nd on the FIDE rating list) to a "Revenge Match of the 20th century" in 1992."

"The purse for Fischer's rematch with Spassky was US$5 million, with $3.35 million of that to go to the winner"

"Fischer won the match, 10 wins to 5 losses, with 15 draws"

Now that is real, exceptional, prodigy,

And oh, old habit die hard.

"On December 10, 2006, Fischer telephoned an Icelandic television station and pointed out a winning combination, missed by the players and commentators, in a chess game televised live in Iceland."

Just a quick question: how long did it take you guys to finish calculus 3?I know it's relative, I mean some people "finish" by just passing the exams, while others study 5 times more in their free time and gain deeper knowledge, but I'm looking for a raw estimate.Also it's a good idea to share at what intensity you learned.I mean some say "took calc 3 in 50 hours", but was it divided over a long period of time or just a week...or maybe even 50 hours straight?(some people can probably can do that).

I don't know the number of hours, but it was a 3 credit semester long class for me. I hated calc 3 because my professor was not a good teacher though, so a lot of it left my brain shortly after that class unfortunately. It's a shame because I really enjoyed calculus up to that point.

It is not entirely impossible that a small number of prodigies develops given the sheer number of people being born. The list on Wikipedia shows 23 names in 400 years, which is a diminuitively small number compared to the living people of Europe and the Americas, and even more diminuitive compared to the births, which, over the course of 400 years is much higher.

It is hard to estimate the number of births (not even knowing the number of living), but given around 150 million people in Europe during the 17th century, at least around 150 million people must have been born during a lifetime (or population would have shrunk). People didn't live awefully long on the average during that time. So if one assumes 50 years (which is optimistic), there must have been at least 1.2 billion births in Europe during that time. Seeing how close to 740 million people live there today, the actual number must be much higher, maybe ten times. Let's say 10 billion.

23 out of 10 billion is well within what natural mutations or "sheer luck" are able to produce.

With that said, many of the most well-known child prodigies are not as great as people would think. W.A. Mozart published around 65 symphonies, of which about a dozen were demonstrably written by his father, and around 10 are being disputed for being "genuine" (nobody knows who wrote those). Einstein was a clever thief working at the patent office who made a lot of money (and lost it due to stupid investments) by stealing ideas from patent applications and patenting them himself quickly. Much of his scientific work could also be considered having had "help". Newton was credited for what was clearly Leibniz' work, but of course Newton was an English gentleman, and nobody liked that queer Leibniz guy anyway.

History is never fair. When Jacques Heim showed the bikini in '46, people were offended in their prudish bigotry, but they still bought them. Was Heim a genius? Maybe, for the choice of name which was as provoking as showing a "naked belly", but other than that he was just being lucky. His predecessors in the late 20s were publicly humiliated and chased out of town for attempting such an obscene thing.