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Ask cperciva: Are great mathematicians born or made?

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Re: Ask cperciva: Are great mathematicians born or made?

#61

Post by gambling8nt »

cperciva wrote: post #14

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 -- a…

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.

Re: Ask cperciva: Are great mathematicians born or made?

#62

Post by byrneseyeview »

DaniFong wrote: post #25
narag wrote: post #18

Earlier quoted context omitted.

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.

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.

Re: Ask cperciva: Are great mathematicians born or made?

#63

Post by mark-t »

maximilian wrote: post #8
mark-t wrote: post #3

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…

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 h…

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.

Re: Ask cperciva: Are great mathematicians born or made?

#64

Post by maurycy »

cperciva wrote: post #34
gtt wrote: post #26

Earlier quoted context omitted.

"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 tho…

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 no…

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?

Re: Ask cperciva: Are great mathematicians born or made?

#65

Post by lolb »

yummyfajitas wrote: post #45
yelsgib wrote: post #17

Earlier quoted context omitted.

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 pro…

"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…

"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.

Re: Ask cperciva: Are great mathematicians born or made?

#66

Post by yummyfajitas »

lolb wrote: post #65
yummyfajitas wrote: post #45

Earlier quoted context omitted.

"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…

"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 int…

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.

Re: Ask cperciva: Are great mathematicians born or made?

#67

Post by emmett »

yummyfajitas wrote: post #66
lolb wrote: post #65

Earlier quoted context omitted.

"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 int…

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 disc…

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.

Re: Ask cperciva: Are great mathematicians born or made?

#68

Post by emmett »

mynameishere wrote: post #11

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 sha…

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.

Re: Ask cperciva: Are great mathematicians born or made?

#69

Post by yummyfajitas »

emmett wrote: post #67
yummyfajitas wrote: post #66

Earlier quoted context omitted.

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 disc…

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).

Re: Ask cperciva: Are great mathematicians born or made?

#70

Post by lolb »

yummyfajitas wrote: post #66
lolb wrote: post #65

Earlier quoted context omitted.

"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 int…

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 disc…

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.

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