
Ask cperciva: Are great mathematicians born or made? - dwaters
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?
======
cperciva
_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.

~~~
yelsgib
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.

~~~
cperciva
_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?

~~~
yelsgib
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.

\---

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.

~~~
Retric
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.

------
eugenejen
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.

------
andreyf
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.

~~~
Rod
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...](http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-
genius-to-do-maths)

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.

~~~
andreyf
_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.

------
Prrometheus
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.

~~~
thaumaturgy
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.

~~~
Prrometheus
> 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.

~~~
thaumaturgy
[http://www.timesonline.co.uk/tol/news/world/article1149615.e...](http://www.timesonline.co.uk/tol/news/world/article1149615.ece)

:-)

------
maurycy
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.

------
randomhack
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.

------
mynameishere
<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.

~~~
DaniFong
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.

~~~
narag
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.

~~~
DaniFong
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.

~~~
byrneseyeview
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.

------
mark-t
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.

~~~
maximilian
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.

~~~
mark-t
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.

------
SwellJoe
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.

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

~~~
SwellJoe
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.

------
DaniFong
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...](http://web.archive.org/web/20061018164649/http://www.qinfo.org/people/nielsen/blog/archive/tough-
learning/tough-learning-final.html) )

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>

~~~
thaumaturgy
| 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...](http://www.timesonline.co.uk/tol/news/world/article1149615.ece)).

------
selva
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.

------
edw519
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.

------
jhancock
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".

------
Tichy
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.

~~~
jeroen
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.

------
larryfreeman
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.

------
spinor
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?

------
yelsgib
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.

------
mstoehr
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...](http://projects.ict.usc.edu/itw/gel/EricssonDeliberatePracticePR93.pdf))
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...](http://web.archive.org/web/20070630211817/www.rota.org/hotair/hotair.html)),
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.

------
meme
converting -110 degrees celsias into fahrenheit

------
LPTS
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.

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

~~~
cperciva
* 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.

~~~
andreyf
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.

