Ask cperciva: Are great mathematicians born or made? | Hacker News Reader

Ask cperciva: Are great mathematicians born or made?

36 pointsdwaters82 comments

Okay, I have been reading quite a bit about this whole nature vs nurture debate. But I want to hear it straight from the horse's mouth. Can people who are not inherently 'gifted', start with an immense interest in math at a mid-point in their life, and still go on to make meaningful and significant contributions to the field? In other words, is precocity a prerequisite to doing good work in Math? I'd be interested to hear from a Putnam fellow like you. Also, what are the rest of YC readers' thoughts on this?

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Can people who are not inherently 'gifted', start with an immense interest in math at a mid-point in their life, and still go on to make meaningful and significant contributions to the field?

Probably not.

In other words, is precocity a prerequisite to doing good work in Math?

Maybe, depending on whether you by "precocity" you mean "demonstrated precocity".

Intelligence is something you either have or don't have -- and while there's still debate about how much of intelligence is genetic (25%? 50%? 75%?) it's clear that the nurture which is most important is that which takes place before age 5, when the brain is still at its most plastic. Consequently, I would say that anyone who is not gifted when they're 5 years old is unlikely to have a significant mathematical impact.

However, not everybody with intellectual gifts demonstrates them early. Mathematics is a field known for child prodigies, not because it's particularly suited to prodigies, but rather because mathematics prodigies tend to get noticed. Some fields, like mathematics or chess, have little knowledge required before intellect can be applied; others, such as chemistry or biology, require years of prerequisite study. Moreover, just like chess prodigies, the abilities of a mathematics prodigy are obvious and unarguable, while a remarkable writer is still likely to be rejected by his first 19 publishers -- and what 9 year old writer is going to send his manuscript to 20 publishers?

In short: You have to be smart to do make a significant contribution to mathematics, and I don't believe that people can "become smart" past a very early age. However, it's possible for someone to be smart despite not having been recognized as such, depending on which fields his intellect is applied against. Finally, it's definitely possible for someone who is smart but has never done well in mathematics to make a contribution to mathematics -- if he can develop the interest which is necessary for him to apply his intellect appropriately.

Is this a useful answer? I have some other ideas about intelligence and IQ and the Putnam and flashes of insight floating around, but I'm not entirely certain how to explain them -- and given that news.yc stories don't stay on the front page for long, I thought I should post what I could promptly rather than waiting for everything else to crystallize.

What is "intelligence?" Can we separate it from society's conception of it? Can we state a computational definition rather than a definition in terms of satisfaction of cultural standards? We undervalue a lot of mental attributes - consistency of thought, sensitivity to subtle differences in concepts, patience, self-reflection, the ability to "go meta" etc. which cannot be detected by the rough and rigid abstract problem solving required by low-level Math (e.g. the Math problems which would qualify prodigies).

I am highly turned off by the idea that we can't "become smart" after an early age. I feel like I "got smart" after I started playing go, I feel like I "got smart" after I learned category theory. I feel like every year I get smarter - and not smarter in the sense of "having more knowledge" but smarter in the sense of being able to solve larger classes of problems, and more complicated problems.

and c.

>I am highly turned off by the idea that we can't "become smart" after an early age. I feel like I "got smart" after I started playing go, I feel like I "got smart" after I learned category theory.

The percentage of the population that can participate in the two activities you mentioned is probably less than 1%. You probably were smart at an early age.

You're justifying a hypothesis circularly.

I was somewhat gifted at an early age, but I consider my intellectual growth over the past 5 years (17-22) to vastly outstrip my growth before that.

Or, you had a certain innate intelligence to begin with, and are only now practicing with it.

At this point it's impossible to know one way or the other. Genetic intelligence versus developed intelligence is something that the top scientists of several fields are still investigating.

I don't think that we can really investigate "genetic intelligence" until we have a definition of intelligence which makes -any sense at all-, which we don't.

I find a lot of the opinions of neuroscientists I've talked to/read articles from to be really dogmatic in their interpretation of intelligence (probably because they aren't regularly faced with Very Hard Problems in the sense that mathematicians/computer scientists are - their conception of intelligence is often a little more superficial - but anyway, I'm massively generalizing).

My entire point is that I don't think it's useful to think about intelligence as a "thing." I think of the brain as a computational structure. At birth, it has certain properties. It changes in certain ways. It can get better or worse at certain tasks. Because Official People have to say Official Things they always treat their own statements about Intelligence as if they're objective and well-informed which, let me tell you, they are not.

Just because people disagree on exactly what constitutes intelligence doesn't exempt the brain from the same general limitations that we see elsewhere in physiology.

Take fast-twitch muscle tissue for one example. Different people have a genetic predisposition towards developing more or less fast-twitch muscle fibers, which gives them the potential to be faster runners or have faster reflexes. Now, that's not to say that someone without that predisposition can't train hard and also be fast; likewise, if someone with that predisposition doesn't make use of it on a regular basis, then they're not likely to be any quicker than anyone else in reasonably good physical condition.

However, assuming the same training regimen, the person with the beneficial genetics will always have an advantage.

There's no reason to think that intelligence -- regardless of definition -- doesn't work the same way. Yes, someone of average intelligence can work very hard and produce the same results as someone who's more intelligent and less motivated. But, you're comparing someone who's operating at their peak potential against someone who isn't.

I think cperciva's original point was merely that due to the nature of the field of mathematics, there's a huge barrier to entry where that genetic advantage becomes necessary. I disagree with that point only a little bit; if I worked really hard at it, I might be able to produce a small handful of exceptions against the very large body of evidence in cperciva's favor.

"There's no reason to think that intelligence -- regardless of definition -- doesn't work the same way."

I agree if what you mean is that there's no reason to believe that there is no fixed, genetically-determined components to intelligence. I disagree if you are making any sorts of claims about what these components are (as I said, I don't even think we can make gross claims like people are born with "good memories").

I also agree with the point that cultures look for certain intellectual traits in young children. I cannot say whether these particular traits are determined genetically because it is highly possible that we learn a great deal (and abstractly) even from the first day of our lives.

The thing is that (as another poster pointed out) there IS a salient statistic (g - for general intelligence) which we can be "better" or "worse" at - but its value in a field like math which requires highly specialized mental strategies is questionable.

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I think that you should look at the example of the polgar sisters - they were raised to be grandmasters in chess and 2/3 of them did (the other one become an international master). I think this puts a bullet in at least one interpretation of your theory.

I now think you're allowing your own prejudices on this subject cloud what you're reading, and what you're saying in response. I think I was fairly clear in my position that genetics plays an important role in intelligence, specifically because we have no indication that the brain is exempt from the same physiological rules that the rest of the human body obeys.

| I think that you should look at the example of the polgar sisters...

I specifically covered edge cases in mentioning the impact of training on innate ability.

What does it mean that the brain "obeys the same physiological rules" that the body obeys? This point is incoherent.

My point is that yes, there is a genetic aspect to intelligence (duh), but that doesn't mean that there is a genetic component to how "fast" your brain is or how much "memory" you have. These could be emergent phenomena - not directly determinable.

Please tell me more specifically how my prejudices are clouding what I'm reading/saying. You didn't follow this point up.

I am highly turned off by the idea that we can't "become smart" after an early age.

I find this notion rather peculiar. Do people get "highly turned off" that they can't "get tall"? I've known for my entire life that I could never become a professional athlete -- I simply don't have the physique.

What is it about intelligence which makes people get so much more upset than they get about physical attributes?

I think you're being presumptuous. I'm not saying that I'm "turned off" by the -fact- that I can't get tall(1). I'm saying that this is not a -fact- at all. Even if it -is- a fact, it certainly is much more uncertain/open for debate than the question of whether you can get tall at a late age. I'm saying that the fact that you -claim it is a fact- is what turns me off. Perhaps I was slightly imprecise. I should have said "I am highly turned off by assertions to the effect that we can't become smart after an early age." This turns me off because belief/assertion creates cultural structure, and cultural structure changes the world. This point is too complex to go into here, sorry.

My claim is even slightly deeper/more provocative than that - I don't think that efforts by a lot of academics (including a lot of my friends) to propagate an idea of "general intelligence" which solidifies at a young age are entirely benign. I think that they are largely founded in worry and attempts to create an elite academic class. Once you're in, you're in (since getting in means that you have a "good" brain which is not really subject to change). It's self-reinforcing and (in my experience) always based in self-doubt.

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As an aside, the idea that "people get so much more upset" about the intellectual than the physical is completely absurd. I know a ton of people (girls and guys) who freak out constantly about their appearance. I know like 3 guys (including myself) who are the least bit worried about our intellects. Admittedly, most of my friends are academics, but my interactions with people outside this sphere indicate that the trend is not peculiar to my social group (and is probably even more physical-attribute oriented, in general). Seriously, I can't believe that you wrote this. This isn't meant to be mean or like ad hominem - but seriously? Do you actually believe what you wrote? Do you know anyone who isn't an academic?

To answer what I think you might have meant (namely: why do _I_ get much more freaked out about my intelligence than my physical attributes - which is also presumptuous because you have -no- idea how much I care about my physical attributes) I would like to observe that in modern western society "we are our minds." Your mind is essentially the limit of your capabilities in the sort of world I'm sure both you and I live in (the quote unquote information world). The mind is connected with the ability to create beauty (art), the ability to connect with other human beings (sex), the ability to experience spirituality (love), etc. In the academic world, at least, our physical bodies are a burden. They wear out. They get cancer. They die.

Did you seriously ask this? Were you being rhetorical? I can't tell. I don't mean to be mean, but sheesh.

(1) and I accuse you of attempting to employ an underhanding rhetorical trick in making this implicit comparison, I might add.

There are basically 3 requirements for intelligence that surpass the basic need to survive:

DNA, Diet, and Stimulus.

DNA limits what structures can develop which is the basic foundation of intelligence.

Diet provides the raw materials to crate the structures that DNA is trying to build. Poor diet causes DNA to sacrifice specific objectives to insure survival. Specific toxins like lead also limit the body's ability to create structures. At the same time short term diet inhibits performance.

Experience refines specific structural elements. Without early experience in specific areas it becomes harder to develop efficient systems for dealing with those situations. It's not impossible to learn French at 50 if you only know Spanish, but it's far easer to learn it at 5 than 50. Part of this has to due with the brain ignoring sounds that it finds unimportant.

All of the above statements are well supported by a huge body of research being annoyed by them is like getting pissed off at gravity while building rockets.

I don't have an answer to this question, but this seems to be the same attitude that got Lawrence Summers into trouble!

My guess is that "intelligence" can be largely hidden, whereas your physique is immediately apparent even if you just sit and do nothing. That it is hidden then, gives it a mysterious feeling of potential, and nobody wants that put in a box. I think it is a cultural phenomenon that necessitates something like unconditional encouragement.

Intelligence hits closer to home because intelligence is more fundamental to who you are. Children who want to insult each other rarely say "You're weak!" They're more likely to say "You're stupid!"

Would you rather be a brain without a body or a body without a brain?

For a heartrending story about the role intelligence plays in shaping personality, read Flowers for Algernon by Daniel Keyes.

I'm not sure that your experience of feeling smarter contradicts cperciva's point about intelligence being determined early in life. I can see that there could be a difference between intelligence as some kind of innate capacity and how that capacity is applied to various tasks.

You can certainly learn more stuff and so be able to do more difficult things, but your ability to learn and apply knowledge may not be changing.

This is a good point. I agree that my feeling smarter does not actually mean that I am smarter. I -do- produce a lot more, but this could be due to other factors.

I think you're hitting very close to the important issue here which is that all of this is very difficult to define/think about without a really good idea of what sorts of computations our brains perform.

My point isn't really so much that I think it can change, even, it's that we should be way more skeptical when it comes to making claims about subtle concepts like these. Usually I find that those who say "we are only really growing intellectually until we're 5" have a very limited and kind of rigid conception of intelligence.

"Intelligence" is a hidden variable g, which comes out of statistical tests of various mental examinations. Basically, give a bunch of people a french test, a math test, a physics test, a driving test, etc.

Run statistical tests for hidden variables, and you will discover this mysterious hidden variable 'g'. It is relatively independent of cultural knowledge: i.e., french is nearly independent of g (1), while physics, plumbing and loading irregularly sized boxes into a truck are highly correlated to g. This all comes out of the statistical analysis, and is not assumed apriori.

Then look for correlations between g and career outcome (and other such things), you'll discover strong correlations there as well. Physicists tend to have high g, janitors low g, etc. For instance, I've never met a math/physics/eng faculty member with an IQ below 120 (though mine is below 100).

So this statistical measure g fits very closely with the intuitive picture of intelligence. It's not cultural, but we don't know what it means computationally either.

(1) Amoung frenchies, people with higher 'g' will score better on french tests. But a low g frenchy will beat a high g brit.

I agree with this point, but I think that what you run into is that there are certain important cross-cultural trends which might cause g to be pertinent. What I mean is that perhaps the set of tasks which we "regularly perform" are governed by g - these tasks usually involving "problem solving" of the you-give-me-a-well-defined-problem-and-I'll-solve-it variety. It's a kind of reciprocal process - society will produce lots of jobs/roles which fit mass statistics it can determine, and it will also screen for that mass statistic.

What interests me, and what I'm worried that focus on this sort of gross statistic deprives us from thinking about, is the non-regular. I'm interested in the philosophical, the revolutionary, the artistic. I'm interested in what made Einstein Einstein. I don't think it's "g."

Slightly more abstractly/hypothetically: The way I kind of see intelligence is that we can function within contexts (e.g. functioning within the "Math context" would be doing math problems) and "g" corresponds to our ability to manipulate symbols generally, so that we are generally good at working within contexts. This is a really abstract way of saying g corresponds to our ability to solve problems. However, I don't think g says anything about long-term problem solving (involving abstraction, self-reflection), novel content creation (involving cross-context thinking), etc.

Does this make sense?

You are aware that Einstein's brain had some very unusual features, right? I don't pay much attention to this, but I remember it is unusual enough that it is safe to say most people's brains don't have those features.

Relativity was not a well-defined problem when Einstein came up with it. You can analyze this intelligence in however many ways, but it is highly possible that "quantum leaps" in scientific advancement come from unusual thinkers -- unusual in a way that you can't just train very hard in and attain.

"When I use a word, it means exactly what I mean it to mean, no more and no less." -- Humpty Dumpty

"Intelligence is what intelligence tests measure." -- Edward Boring

I'm actually surprised by how few comments have been made regarding the actual nature of intelligence in this thread. yummyfajitas' parent comment is one of those few, and he equates intelligence as we understand it with g, the hypothesized general intelligence posited by psychometricians.

y.f. makes the same two arguments that psychometricians make for the validity of g as a representation of human intelligence,[1] which are supposed to demonstrate g's "internal validity" and "external validity". The former refers to the positive correlations between various (supposed) intelligence tests (described in the jargon as the "positive manifold"), and the latter to the positive correlations between g and supposed success in life. Broadly, the internal validity argument proves the "general" in "general intelligence", whereas the external validity argument proves the "intelligence".

The first wrinkle in these arguments is the claim that all human intelligence tests positively correlate with each other to give a universal intelligence factor. This is false; there exist demographics where g is _not_ the predominant factor explaining inter-demographic test score differences.[2]

Here is another problem. Psychometricians tend to decide whether some test metric measures intelligence by how well it correlates with existing "intelligence tests" that're g-associated. y.f. does this in his comment: "It is relatively independent of cultural knowledge: i.e., french is nearly independent of g (1), while physics, plumbing and loading irregularly sized boxes into a truck are highly correlated to g". (I.e., French tests can't be real tests of intelligence, since they don't correlate with g, but physics, plumbing, and bin packing must be, because they do.) But this renders the internal validity argument circular: its two prongs now become: (1) g must exist because intelligence tests correlate positively, and (2) and intelligence tests are those tests that correlate positively with g. The circle closes.

The external validity argument supposedly rescues the g concept from this trap, by demonstrating that g correlates vaguely with real-life behaviours. Unfortunately, the correlations are nowhere near perfect, that g correlates with success doesn't suffice to show causality, and pointing out the correlations does not eliminate the underlying circularity in g's definition. All in all, the evidence is inadequate to reject the null hypothesis that there is no causal link between g and life success.

[1] Well, I say "human intelligence", but if I recall correctly, at least one psychometrician even tried to argue that g might well be a cross-species phenomenon!

[2] Dolan, C.V. Roorda, W., and Wicherts, J. M. (2004). "Two failures of Spearman's hypothesis: The GATB in Holland and the JAT in South Africa", Intelligence, vol. 32, p. 155-173.

You seem to misunderstand the definition of g, as g is not defined in terms of intelligence tests.

Take as your statistic the result of many tests: physics, box loading, french, basketball, etc, and then do a PCA or similar test. One of the principal components will be g, provided you have enough data. This is what defines g.

Intelligence tests are simply tests designed to be more highly correlated with g. If we discarded intelligence tests, we could recreate them (or equivalent tests) based on statistical analysis of the other test data. They are certainly not arbitrary measures.

The external argument doesn't rescue g; g is quite safe. The external argument merely claims that 'g' and 'intelligence' are the same thing, or very close. g exists regardless of what you want to call it.

As for the paper you cite, I skimmed it. Unless I misunderstood it horribly, it merely claims that particular IQ tests don't effectively measure g across different groups. That doesn't mean g doesn't exist as a hidden variable, merely that a particular test is poorly correlated with it for some population.

I've posted this at least three times on Hacker News, but I'm just shocked people haven't read it:

http://cscs.umich.edu/~crshalizi/weblog/520.html

This guy is a statistics professor, and has a lot to say about exactly what "g" is. He even runs experiments! I know the article is a bit long but I promise it's worth reading.

Well, I admit, I just skimmed it, I'll try to read the whole thing later. But near as I can tell, he isn't addressing the validity of g at all, merely claiming it is unproven that genetic differences between population groups is due to genetics or other factors (a claim I don't disagree with).

You're right; sorry, wrong essay. The other one, by the same guy, addressing that point:

http://cscs.umich.edu/~crshalizi/weblog/523.html

Are there any studies showing that basketball performance, French fluency, bin packing, and physics knowledge, all taken together, will produce a meaningfully large general factor? Intuitively it seems unlikely, and it doesn't jibe with what I understand g's definition to be based on the comments of people like Eysenck and Jensen, who justified g on the basis of intelligence test correlations, and only later tried to tie it to biological functions (evoked brain signal potential in Eysenck's case, reaction time in Jensen's) and metrics of social, sports, and job performance.

Regardless, I agree that g exists in at least a limited sense, but I maintain that there is insufficient evidence to demonstrate that it emerges from innate, immutable physical properties of the brain, and you still can't attribute life success to it.

I've never met a math/physics/eng faculty member with an IQ below 120 (though mine is below 100).

Yummy, you have to be kidding us. Either that, or you were drunk when you took the test, or we really need to rethink what the tests are measuring. There is no way somebody with subnormal intelligence is involved in exchanges like this:

http://news.ycombinator.com/item?id=216701

Apparently someone decided the score a bunch of the Caltech faculty on an IQ test, and a surprising number of them turned up with scores < 100. This doesn't prove that the concept is untenable, but clearly some of our tests are missing things.

That's interesting... any source?

I'm afraid not -- it was just an anecdote circulating around the astro department at Princeton. Feynman was known to have an IQ of 125, and I met a grad student there who admitted to an IQ of around 80 -- a perfectly bright person, to be sure. Again, I don't have the ability to prove these things to you, but I think we should recognize that, occasionally, our methods for measuring intelligence are very very broken.

>Feynman was known to have an IQ of 125

That's still more than 1.5 standard deviations above average. If Feynman's IQ was exactly 125, he would be ahead of "only" 95% of the population.

"not because it's particularly suited to prodigies, but rather because mathematics prodigies tend to get noticed. Some fields, like mathematics or chess, have little knowledge required before intellect can be applied"

it is not true. imo modern math requires years of study to reach the border of known.

to topic starter: i know pretty good mathematicians who think that geniality exists and who think it does not.

i thought about it for a while and come to conclusion that it does not matter.

The right question to ask: Am i enjoy (fall in love with) math?

p.s. those mathematicians who believe in existance of geniality think that "love math or not" is a good test.

imo modern math requires years of study to reach the border of known.

You're right -- and modern mathematics requires more study than a field like computer science, simply because computer science as a field hasn't had enough time to accumulate very much content. (This is one of the ideas I mentioned as floating around but not fully crystallized earlier.)

But this doesn't contradict what I was saying: My point was not about doing original research, but merely about crossing the threshold to being recognized as a prodigy. A 13 year old attending university is certainly noticed -- but that doesn't mean he's doing original research (I didn't until I was 15).

15? wow! may i look at your papers?

My publications are at http://www.daemonology.net/papers/, but this doesn't include the work I did when I was 15 -- that was a novel algorithm for computing polynomial GCDs over algebraic number fields, which I never published due to unresolved loose ends involving high-degree fields (professors encouraged me to publish it anyway -- but I didn't want to published something "unfinished").

It turns out that those "loose ends" are rather mixed up with the problem of integer factorization, which might be why I couldn't manage to tie them up. :-)

damn good for 19 years old, but as i see there aren't groundbreaking works i hoped to see. =(

damn good for 19 years old

I'm 27 years old, actually...

there aren't groundbreaking works

Ouch. I'd say that my shared caches side channel attack work was groundbreaking (although Shamir and his graduate students were only a few months behind me). I'd say that my projective algorithm for matching with mismatches is groundbreaking. The sqrt(5) \epsilon error bound on complex floating-point multiplication shouldn't have been groundbreaking, but apparently was -- I've never seen numerical analysts get so excited about a ~30% reduction in an error bound.

But how do you quantify groundbreaking? A lot of research can't be qualified as such until many years later, after a field has built on the foundation laid by the originator.

If math's content grows expotentially, it might explain why even one hundred years ago it was much easier to come up with serious work before studying for years.

Side note: it is interesting to precise what is the content actually. Besides the obvious definition, like theorems and proofs, I also see hidden content: paradigms and problem solving techniques. What else?

I don't agree with your choice of cutoff age--certainly it is correct that the historical figures who had made the largest contributions to the field of mathematics have done so while quite young--indeed, the prime ages for developing new mathematics seem historically to be between 17 and 23--but there is nothing to suggest predestination regarding the matter at age 5. Also, given the level of specialization involved with some open problems (and the overwhelmingly large amount of data involved in all of mathematics), it is certainly possible that someone deeply specializing in an aspect of mathematics not "popular" at the time could make a non-negligible contribution to mathematics even if they started late in life, and were less "smart" than their "prodigy" peers.

If you start mathematics late in life, you are extraordinarily unlikely to find your name on a list with names like Gauss, Newton, and Archimedes. Nor will you likely reach any sort of par with Euler, Hilbert, or Russell. Indeed, even the levels of Conway, Shannon, and Wiles are probably unattainable. But math is a huge field now, and there are parts of it that most young mathematicians find boring, or simply aren't exposed to. If you find some such part that you like, you can probably make some progress.

Hi dwaters,

I know you are waiting for cperciva to give you reply. But I want to tell my own experience.

I was very bad at math when I was in high school. I figured out algebra without problem but failed pretty bad in classical geometry when I was 9th grade. From my 10th to 12th grade, I just have dyslexia to what math textbook says.

Then I studied with 2 teachers for a year to prepare taking another entrance examination in my country (Taiwan). Both explained clearly two things to me that I like most later, one is physics and another is math. The math teacher taught me calculus in 2 weeks and I just started to become literate to meanings in those symbols and after that when I look back analytic geometry and classical mechanics in our high school textbooks, I just felt my mental block was removed! I did not feel any trouble in understanding the problems and I later chose my major in physics just because I like to learn more of it and went through my undergraduate pretty ok and later I found computer science and I started to work in the trench for 16 years.

I did not make any contribution to math and physics, but I still like to use the training that I received in my daily life and I still sometime open books from Dover Publishers to read and feel interested. I like the feeling in figuring out problems and find ways to solve them. And that is the joy I'm glad to had a chance to acquire due to two best teachers that enlighten me.

Short answer: nurture, very early on, because children have a fantastic curiosity and ego-less stubborn determination.

Long answer: From my personal experience, a lot of what some claim to be 'nature' is actually very early-age 'nurture'. So it's not that children of academics are genetically smarter, it's that they are raised by people with whom dinner or a walk in the park ends up being as educational as a university lecture. On top of that, if you consider that knowledge is like money under compound interest, I think it's possible to explain people that seem smart beyond comprehension as just having gotten a good head start on learning.

Math is also a special case for two (related) reasons: (1) emotions, especially ego, has a lot to do with learning math, and (2) math is filled with "aha" moments, where one thing which seemed incomprehensible one moment is obvious the next. Little kids have an advantage related to both of these - they (hopefully) don't yet have an ego to build or protect when it comes to knowing things, and they take a lot more pleasure from the "aha" moments.

So to answer your question - I would say it's nurture, but the kind which is (most of the time) limited to very young ages. On the other hand, given the determination and stubbornness of a 5 year old facing a challenge, it's nothing you can't do at 40. But I worry that I don't quite have that determination at 22, and will only have less of it as I am starting to worrying about becoming financially secure, starting a family, etc.

I whole-heartedly agree with everything you wrote.

I competed in Math competitions while in high school but never made it to the IMO. My friends who went to the IMO were doubtlessly very smart, but they had something else: they had many years of training under their belts. I failed because I was an amateur, they succeeded because they trained like professional athletes.

To do Math, intelligence is required, but not sufficient. Tenacity, patience, stubborness, an obsession to figure things out, and the willingness to work very hard are also required. Prof. Terence Tao knows much more than I do on the subject and he explained things clearly:

http://terrytao.wordpress.com/career-advice/does-one-have-to...

When one is young, one does not mind feeling stupid. Understanding something in Mathematics takes a lot of time and effort. After a certain age, one does not want to feel that stupid anymore. That's why many great mathematicians did their best work before they were 30. It's more psychological than intellectual, I'd say.

That's right on the spot, and a much better way of saying it than I did when I talked about ego.

All the easy math has been done. People have been working on Mathematics for a long, long time. To "make meaningful and significant contributions to the field" is one of the hardest intellectual tasks a person can do. It is probably impossible for someone at a midpoint in their lives to suddenly develop an interest in math and make a meaningful contribution. Even people who have been immersed in Mathematics their whole lives often cease to make major contributions in middle age.

You do have to be inherently gifted to be a good Mathematician. You even have to be really smart to be a mediocre one. This is hard stuff, the Olympic Marathon of intellectual pursuits. Do you have to be gifted to run in an Olympic Marathon? Someone that was merely determined might be able to qualify for it, but the ones that win are definitely genetic freaks.

You can certainly enjoy math without being a super-genious. If someone were dogged and creative enough and focused on a new enough subfield, they might even be able to contribute a little, provided they've kept their analytical facilities sharp in a field like Physics or Engineering. But then again, anytime a new field opens up there's hundreds of PHD students across the world that jump into it looking for a new bit of math to write their thesis on. It gets fleshed out pretty quick, and the low hanging fruits are the first to go, and the people picking the fruits are PHD math candidates.

Please note that this is just my opinion from being around Mathematicians. I'm not a Mathematician myself, but I was around them when I got an undergraduate degree in Mathematics. That degree taught me that Math was really hard and I would probably have a more significant impact on the world if I pursued something else.

I disagree with most of your points, though you're more qualified to talk on the subject than I am.

| All the easy math has been done.

How easy are we talking about? While I was in high school, I remember a news article about some other high-school-age kids that happened to a solve a long-standing and relatively simple problem in geometry involving triangles. I can't remember the specifics, nor can I find anything about it now, but I think there are still plenty of relatively entry-level problems to work on.

| It is probably impossible for someone at a midpoint in their lives to suddenly develop an interest in math and make a meaningful contribution.

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

It's a pretty crappy list, and most of the names are from centuries past, but there are a couple of interesting entries there.

| Do you have to be gifted to run in an Olympic Marathon?

Your example of Olympic marathons is specifically a zero-sum game, whereas mathematics is not.

| But then again, anytime a new field opens up there's hundreds of PHD students across the world that jump into it looking for a new bit of math to write their thesis on.

This seems to contradict your previous point that mathematics has become inaccessible. There are a few differences between a grad student and a sufficiently dedicated hobbyist, and none of them are genetic, nor are any of them necessarily bound by a particular age bracket. Indeed, someone in middle age could attend university on a specific curriculum, and in a few years be looking at the same problems as the PHD students. The older person might have some advantages in self discipline or experience in tangential fields.

> There are a few differences between a grad student and a sufficiently dedicated hobbyist, and none of them are genetic, nor are any of them necessarily bound by a particular age bracket.

This raises an interesting empirical question: is there a significant deterioration in brain performance after about 40 years of age or so? Anecdotally it would seem to be the case. The Fields Medal is the highest award in Mathematics and it has an age cap of 40, but the rule hasn't raised controversy because almost all the worthy contenders have been under 40 anyway. However, some people have hypothesized that the dominance of the young is the result of other career and family concerns distracting people in middle age. That could well be the truth, and I hope it is. It would be interesting to know the answer.

http://www.timesonline.co.uk/tol/news/world/article1149615.e...

:-)

Few notes.

The whole discussion whether intelligence is born or made, misses the fact that motivation to work hard for years is unfrequent. Most people give up quickly, and are not able to commit themselves without immediate reward.

I easily imagine a mathematican without great intelligence, yet with high motiviation and ability to stay focus for hours. I can't imagine a mathematican who is smart, yet is not able to concentrate.

Another thing is that what seems to upset most people about genetic intelligence is limitation of our free will. It goes something like this: "I realize I want to become a mathematican and environment limits me; I can't afford this thought."

The problem is that between born and made there is a lot of grey area. What if, for instance, you do not realize it?

You might grew up in a suburbia environment that encourages you to become a MBA or a lawyer. If so, non-zero chances are you'd simply don't want to become a mathematican, or a quant. (no money, no chicks). Is your free will limited, or no? In different environment you'd dream about becoming an Erdos.

You might think that you made the decision, while, actually, the socialization process made the decision, and if you want to change it, you have to overcome the defaults. A non-trivial task.

I also feel that, besides the intelectual curiosity, there is something more behind this question. It seems, at least to me, that most people asking it, actually ask whether chances are their effort will be rewarded.

There is nothing wrong with it. One wants to minimize her risks. If investment of 10 years would be known to give 0 output (here: becoming a great mathematican) in some cases (here: lack of genetic intelligence), then, under given circumstances, it doesn't make sense at all.

The point is that intellectual work is highly non-linear and if there are any constant factors, they are highly more complicated than born vs made.

Precocity is overrated. To me, believing that genius is decided at age 0/5/10 whatever means that you are just rationalizing not working hard enough with "I am not a genius. I will not make it. Therefore no point in working hard".

The primary test really is : a) Are you having fun (with maths in this case)?

b) Are you willing to put time and effort into it? Note that the amount of time available may be one hurdle if you are starting at mid-age since its likely that you are pursuing it as a side hobby instead of a full time profession.

See Hamming's lecture on PG's site. Search particularly for "hard work is compounded" and "why is he so much smarter than me" or roughly along those lines.

http://www.jinfo.org/Fields_Mathematics.html

25 percent of fields medal winners vs. 0.227 percent of world population. >100x difference. [discussion of statistical signfigance omitted.] Nature vs. Nurture is a stupid discussion, as "nurture" is costly and transient and a ripe subject for government boondoggles. People are different. No way around it.

On the other hand, a basic command of mathematics can be widely shared, and learned at most ages.

That may well be true, but I think the problem of disentangling the incredibly powerful Jewish cultural influence (or any other cultural, influence, really, as a half-asian I can tell you my heritage helped push me) from whatever biological differences there might be is an incredibly tricky one. So I think that asserting such conclusions here is either chauvinist or premature.

Assuming as little as 10% could be nature, the lack of genetic conditions would mean a lot for top achievements.

Also talking about a certain community doesn't necesarily mean something related to race. There could be some kind of selection for people that become jew marrying or people that quits.

I would also like to ask, how could such conclusions, were they actually true, be put to use by society?

I haven't been able to identify any way to use such information which I wouldn't consider deeply immoral. But I have unusual views here.

"...how could such conclusions, were they actually true, be put to use by society?"

If there's a cause, someone could try to find out what it is and it might be useful. Or not. But knowledge isn't moral or inmoral, it's knowledge. Wishful ignorance is wrong for my sense of morality.

It would help us avoid expensive mistakes to correct non-problems. Imagine a world in which height is as politically charged as race. Every year, short people rail against the overrepresentation of the tall people in sports, in politics, in film, etc. And every year, the tall people talk about how some of their best friends are short, they themselves aren't prejudiced, they understand there's a horrible legacy of discrimination, perhaps short-person-related affirmative action programs are necessary.

But then some annoying researcher points out that of course tall people are better at basketball, because they are closer to the basket (among other things). And that within ethnic groups, height correlates with intelligence because of a confouding variable: malnutrition is known to reduce cognitive ability and height, and cognitive ability correlates very well with income.

So it basically transforms something from a political problem to a scientific fact. There is no longer anything to rectify, because everything is as it should be -- people have characteristics that make them better or worse at various tasks, and that's just the way things are. A short person who tries hard can succeed at tall-person fields, but not to the extent that they could if they were tall. A tall person can live a long time if they try hard to be healthy, but not as long as they could if they were short.

There are systematic environmental differences for different cultures as well. This isn't to say "people aren't different", but rather that until you've estimated environmental differences and gene-environment interactions you know almost nothing about the nature vs. nurture debate.

See http://cscs.umich.edu/~crshalizi/weblog/520.html for a great summary, including specifically questions about Jewish intelligence. I know it's a bit long, but I promise it's worth reading.

Math education is so bad right now that any genetic differences are drowned out almost entirely. Of all the mathematicians I've met (including IMO gold medalists, Fields medalists, etc.), there seems to be little correlation with siblings, at least beyond what you would expect from having parents who emphasize learning math.

To answer your question, if you want to contribute significantly, you will need to study hard for several years, but it is possible. One of the best mathematicians I know well didn't have any interest until he was 16, and he's now 25 or so.

Wow, 16 is sooo old. I'm sure plenty of great mathematicians didn't know what they wanted at 16 either.

25 is pretty young still I think. Thats only 3 years into a PhD, so really I wouldn't expect anyone (except super geniuses) to make many achievements by that age. I'd say 30 between 27 and 30 seems like a pretty solid age, as you've reached a peak with your PhD and you're still young and inquisitive and hopefully humble.

Ask around. 16 is quite late to start taking math seriously and end up at the very top. He wasn't doing math for fun. He wasn't asking himself mathematical questions. He wasn't doing math competitions. He had no knowledge outside of what was taught in school to students at his grade level. This is freakishly uncommon, but he shows it's possible.

25 is young. He is just a few years into his PhD, and he hasn't made many major achievements (he's been a grader for the IMO, written one paper that's still being refereed for publication). The point is that he's noticeably better than I am at almost every branch of mathematics except combinatorics, and he had no interest until he was 16.

"...there seems to be little correlation with siblings, at least beyond what you would expect from having parents who emphasize learning math."

I'd bet for a set of negative factors. For you to be 100%, you need to have full 15% of genes (maybe there are a lot involved, so it could be very improbable to have more than 10% of that 15%), 30% of early nurture and 40% of favorable social conditions (good environment, or just not being poor that lets out a big part of humanity) and the rest of luck, effort and good teachers.

The lack of any of these conditions could explain why the siblings are not so bright. The lack of genetic conditions and the early nurture could also ruin more easily the posibilities.

"Genetic" is also broad. It could be quicker thinking, focus, even a illness that forces a child to stay home reading instead of playing outdoors.

Anyone else see the flaw in asking a gifted mathematician about the human brain, merely because they're a gifted mathematician? cperciva is certainly very smart, and may also happen to know a thing or two about the subject...but mathematical ability doesn't necessarily correlate with knowledge of how humans learn or cognitive development.

I definitely, definitely agree. A lot of mathematicians I know are kind of self-appointed "kings of the mind" and so pontificate tremendously on subjects they're not really qualified to talk about. Not saying that cpervia is, but it's definitely a trend.

That being said, mathematicians definitely have (as a class) some good mental attributes - they usually think very consistently and can see very subtle flaws in arguments (especially common ones (it gets frustrating)). They also solve really tough problems which means that their ideas about problem solving (probably pretty central to the question of "what is intelligence?") are probably good, assuming they're self-reflective.

Yeah, but his answer is in line with what's known about cognitive development.

Sure, but anyone with a couple of hours to spend on the Internet can find out what is in line with what's known about cognitive development (certainly more than a few threads at HN will provide). Since it was directed at cperciva specifically, there seems to be a desire to get something more than just knowledge. Permission, maybe? I dunno.

I just found the question interesting. And it's a common phenomenon. Musicians, for example, even not particularly gifted ones, get this kind of question pretty frequently--"can adults really learn to be good musicians?", "it just comes naturally to you, right?", etc. Humans have funny logic sometimes, I guess, including the people on the receiving end of the questions.

You're right, of course, that a mathematician is not the same thing as a cognitive scientist; but it's not completely crazy to ask me such a question. People are naturally interested in topics which touch on them directly -- this happens very frequently in medicine, where individuals having a particular disease frequently know far more about the disease than a general medical practitioner (but less than a specialist studying that disease, of course) -- so it's not unreasonable to think that a randomly selected genius is likely to have spent more time learning about the nature of genius than a random non-genius.

I would strongly suggest that you read the biography of Erwin Schrodinger. Empirically, it seems to be quite possible to pick up mathematics and science at a later stage. But it doesn't happen too often. I cannot tell if this has to do with society's influence (there are many pressures keeping people from switching fields) or something to do with natural talent.

But I do think that success in science, mathematics, or really, anywhere, is much more a product of your habits, discipline, environment, and beliefs than is ordinarily given credit. I don't think 'nature', or 'very early nurture' is unimportant -- for example, it can play a very important role in the sorts of things you get interested in and the preferences, beliefs and skills you acquire. But conditioned on having the same beliefs, habits, and discipline as masterful scientists, business people, or researchers, if you're talking about great contributions, I think the other dimensions of talent truly shrink into irrelevance.

Three terrific links: Michael Nielsen on Extreme Thinking. http://michaelnielsen.org/blog/?p=19

(Edit: Whoops! Michael, the link is broken! Here's an archived version, http://web.archive.org/web/20061018164649/http://www.qinfo.o... )

Terry Tao's Career Advice: http://terrytao.wordpress.com/career-advice/

And the classic, Richard Hamming, You and Your Research: http://www.cs.virginia.edu/~robins/YouAndYourResearch.html

| I cannot tell if this has to do with society's influence ... or something to do with natural talent.

Both. Those with natural talent are more likely to produce notable work at an earlier age. Likewise, for male scientists, testosterone seems to drive creativity, and in later ages that has a tendency to get channeled into marriage and family instead (http://www.timesonline.co.uk/tol/news/world/article1149615.e...).

I just read 'A Mathematician's Apology' where Hardy touches on this at one point. If you want to consider someone's opinion on your question (besides cperciva), consider Hardy's (because he was a great mathematican and a brutally honest person). Briefly, Hardy thought the ability to make deep contributions to mathematics erodes after middle age. He cites Euler, Abel, Ramanujan, et al. Hardy mentions Gauss who was old (55, I think) when he made a major contribution but, Hardy notes, the idea for it had germinated while Gauss was young. I personally think Nature plays a bigger part than nurture when it comes to mathematical ability (of the rarer kind). Ramanujan is a classic example. He had pretty much no nurture. Way before anyone knew or cared, he was producing stunning mathematical insights.

I don't know what the "experts" say, but I've always run my business with a principle that almost no one else agrees with, "Almost anyone can learn almost anything".

I constantly witness failures in business because someone said, "It can't be done," or "He can't do that," or "She'll never be able to do that."

I love proving them wrong. I'm rarely disappointed.

Given the right resources, the opportunity, and most of all, genuine encouragement and support, it's amazing what the human brain, at any age, can accomplish. Sometimes, all it takes to discover it is a little more faith in another than in onesself.

I have had similar thoughts recently. Although my desire to get better at maths is not to "make meaningful and significant contributions to the field". My desire is to make meaningful contributions to other fields. Reviving and extending my dusty math skills may be an essential component. So I think if you realign your question/purpose as I have, the answer is "yes". Otherwise, my instinct says "no".

Wasn't there a story recently on HN about a guy who proved a longstanding open problem as a hobby, and was a security watchman in his day job? (Or some other job, I don't know).

I am surprised so many people here seem to lean towards "nature". I am pretty sure you can pick up mathematics at any age. Sometimes I also wonder: humans manage so many really complicated tasks in their daily lives, can it really be a much farther stretch to reach into mathematics?

Of course experience and knowledge and training help in mathematics, but as in other fields, maybe sometimes an outsider can bring a fresh view into it. It might also depend on the field of maths you choose, some might be more approachable than others.

That said, I agree with the other post that genius mathematicians are probably somehow born or made very early on.

There was such a story ( http://www.haaretz.com/hasen/spages/966679.html ), but the guy was "an accomplished mathematician" before becoming a security guard.

With the well known exceptions of Gauss and Ramunajan, great mathematicians tend to emerge in communities of talent. I would say that community and social network play a very large role in the emergence of mathematical genius.

What about Hardy's collaborator Littlewood, who continued to do excellent research into a very ripe old age?

Hardy was a depressive, and this may have had an effect.

How many of the great mathematicians used Cocaine or amphetamines--not a silly idea, look at how many students use such drugs--to get early success. Recall that Cocaine was LEGAL in the 19th century. Does this explain their early burnout, and the bipolar depression symptoms ( called repeated nervous breakdowns) of so many of the great mathematicians in the 19th and early 20th century?

People vary in mental attributes. Let's get this out of the way right now. It -has to be the case-. We should obviously remain agnostic about what these attributes are (for instance, I would be hard pressed to say that "good memory" is even genetic because it is hard to disambiguate "memory software" from "memory hardware" in the brain).

Cultures screen for genetic attributes at a young age. This is not necessarily a good thing, but it just happens to be the case. Attractive children are treated differently. "Bright" children are treated differently. "Slow/inattentive" children are treated differently.

The sum of genetic attribute and cultural reaction to that attribute produces higher-order attributes which are consumed by new cultures, etc.

There is an academic culture of Mathematics. To participate, you need to have had the right experiences (enabled by the right genetics, pre-conditions, and cultural/socio-economic starting-point). People like MIT's Daniel Kane or Reid Barton were made-and-born.

So no, you can't become part of the culture of mathematics if you're not. Sorry, they're very picky. They also have a lot of resources (people to talk to, seminars to attend, journals which are expensive) which you can't have if you're not in the club.

Another unfortunate thing is that new math research really has to be presented in terms that "old math" can understand. E.g. constructivist mathematics struggles to get widespread approval. Adoption of ZFC axioms seems arbitrary in some senses (the C part).

So what does it mean to do "good" research? You need to create a result and argue it. Creation requires resources, argument requires facility with standards and ears to listen. Without acceptance into the community, you have neither.

That's one definition, anyway.

Do you just want to develop truth for yourself? Do you want to understand math deeply?

The fact is that there are many people who are extremely gifted in terms of genetic skills, who can do good math, but who don't quite "fit the mold" (trivial example: people with tremendous visuo-spatial reasoning abilities vs. people with tremendous pure-abstraction symbolic-manipulation abilities). These people can still do good math. They just have to do it largely on their own and they may have trouble communicating it with others.

In short, it's insanely complicated and has to do with the marriage of genetics and -culture-, not nurture, per se. If you want deeper insights into this question, I recommend you look at the history of mathematics and mathematicians.

---

Anyway, if you want a good example of a non-stereotypical mathematician, I recommend you look at the autobiography of Grothendiek. Good read.

I'm surprised at how strongly held most of the opinions are argued in the forum. The reason why is that I know of very little scientific evidence that would lend people such certainty. Firstly, as I understand it, to date, there has been no longitudinal study of mathematics talent in people who develop in interest after adolescence. Secondly, the neurological basis of learning is a rapidly growing field, but the field right now may not have much to say about the development of mathematics talent. The points that have been made: i.e. that you can't do it may be true, it's just that you probably should take what they say with a grain of salt since they are talking about a subject that is not very well understood.

I would wager, though, that going from having a small knowledge of math to going to a level of knowledge of comparable size to a mathematics researcher is going to be very hard. People who focus on math starting freshman year of high school go through 4 years of high school + 4 years of college + 5-7 years of grad school = 13-15 years before they become research mathematicians. So it might take you 10+ years before you know enough to start contributing.

Additionally, if you read psychological research on expert performance (which you probably should, and Eric K. Andersson is one of the experts in that area so perhaps start with an article by him), it normally takes 10 years to develop expert skills in a field, and that expertise comes about from something he terms "deliberate practice". Check out: (http://projects.ict.usc.edu/itw/gel/EricssonDeliberatePracti...) The Role of Deliberate Practice in the Acquisition of Expert Performance.

Most evidence would indicate that the abilities of a Putnam fellow are probably not out of reach to somebody who uses a strict deliberate practice regimen to develop their working memory (google that with 'mathematics') of and the ability to execute the thousands of tricks that mathematicians employ to solve problems. I am not very confident in this particular assertion, and I believe there is only weak evidence for it, however, there is only weaker evidence against it (and most people who argue against it make use of hand-waving has their primary argumentative technique).

On the actual task of becoming a mathematician I suggest you read some essays by Gian-Carlo Rota (http://web.archive.org/web/20070630211817/www.rota.org/hotai...), in particular look at the "10 Lessons I wish I had been Taught" and the reflections on math and mathematicians. He points out that mathematicians seem to have only a couple of tricks up their sleeve which they apply over and over and over again.

  The points made by yelsgib are good since mathematics is about a community of researchers, and most problems have been looked at by hundreds of researchers.  At the very least you will need to attend conferences at Universities.

This brings me to my final point, which is that if you want to make contributions to the field you'll make it a lot easier on yourself if you attend grad school in mathematics. In general people don't have to pay for math grad school, but it will cost you time. The reason why I say that it will make your life easier is that, firstly, math grad school will induct you into the community of mathematicians. You'll have better guidance than I can give (I am a lowly undergraduate) on how to become a mathematician, and when you get to doing serious research you'll have an advisor who will (hopefully!) guide you through it. One of my friends who is finishing up his P.h.D in analysis points out that he would have no idea whether he was making progress or not if it weren't for his advisor (he's doing research on semiclassical wave functions). Secondly, you'll know whether mathematics is right for you.

And, if you plan on making 'significant' contributions you'll have to do more or less the same preparation that a math grad students goes through and you'll have to put in the same, if not more time as grad students do in preparation to do research.

I admit that I am curious to see what happens if you go through with such a project. It would take a great deal of perseverance on your part as you come across all the barriers I am facing (as a math major) as well as those that come about because of your age and station in life.

My main suggestion is to try to get into correspondence with mathematicians to try to get more guidance. This can be difficult because you are probably just starting out, but you will most likely find one or two gracious ones if you start trying to correspond with mathematics departments.

I lack the experience to be a good person to talk to but I am nearly always willing to discuss this sort of thing so feel free to send me an e-mail to [my sn without the 'm'] [the a with a circle around it] uchicago [period ] edu.

converting -110 degrees celsias into fahrenheit

Very great mathematicians are born. You're either born as Ramanujan or Erdos or you're never going to be Ramanujan or Erdos. No way around it.

But, merely pretty damn good mathematicians can work really good and be a thousandth as useful as an Erdos or Ramanujan.

I'm pretty sure most people wouldn't want live the life of Erdos...

* I'm pretty sure most people wouldn't want live the life of Erdos...*

Come on... travelling the world and couch-surfing for years on end, while having no financial concerns at all? I think most people would love to live that life.

For a bit, sure, but I'd imagine sooner or later, most people get tired of doing math 16-20 hours a day and want to start a family or something.

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