Better beware of notions like genius and inspiration; they are a sort of magic wand and should be used sparingly by anybody who wants to see things clearly.(José Ortega y Gasset, “Notes on the novel”)

Does one have to be a genius to do mathematics?

The answer is an emphatic **NO**. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does **not** need some sort of magic “genius gene” that spontaneously generates *ex nihilo* deep insights, unexpected solutions to problems, or other supernatural abilities.

The popular image of the lone (and possibly slightly mad) genius – who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts – is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics. We do have spectacular, deep and remarkable results and insights in this subject, of course, but they are the hard-won and cumulative achievement of years, decades, or even centuries of steady work and progress of many good and great mathematicians; the advance from one stage of understanding to the next can be highly non-trivial, and sometimes rather unexpected, but still builds upon the foundation of earlier work rather than starting totally anew. (This is for instance the case with Wiles‘ work on Fermat’s last theorem, or Perelman‘s work on the Poincaré conjecture.)

Actually, I find the reality of mathematical research today – in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck – to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of “geniuses”. This “cult of genius” in fact causes a number of problems, since **nobody** is able to produce these (very rare) inspirations on anything approaching a regular basis, and with reliably consistent correctness. (If someone affects to do so, I advise you to be *very* sceptical of their claims.) The pressure to try to behave in this impossible manner can cause some to become overly obsessed with “big problems” or “big theories”, others to lose any healthy scepticism in their own work or in their tools, and yet others still to become too discouraged to continue working in mathematics. Also, attributing success to innate talent (which is beyond one’s control) rather than effort, planning, and education (which are within one’s control) can lead to some other problems as well.

Of course, even if one dismisses the notion of genius, it is still the case that at any given point in time, some mathematicians are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” mathematicians should do mathematics; this is the common error of mistaking absolute advantage for comparative advantage. The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research. As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution. It might not be the most glamorous part of mathematics, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts of a subject turn out to actually be more important than any fancy applications. Also, it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle the famous problems in the area; take a look at the early publications of any of today’s great mathematicians to see what I mean by this.

In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be *harmful* for one’s long-term mathematical development; if solutions to problems come too easily, for instance, one may not put as much energy into working hard, asking dumb questions, or increasing one’s range, and thus may eventually cause one’s skills to stagnate. Also, if one is accustomed to easy success, one may not develop the patience necessary to deal with truly difficult problems. Talent is important, of course; but how one develops and nurtures it is even more so.

It’s also good to remember that **professional mathematics is not a sport** (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.

Further reading:

- “How to be a genius“, David Dobbs, New Scientist, 15 September 2006. [Thanks to Samir Chomsky for this link.]
- “The mundanity of excellence“, Daniel Chambliss, Sociological Theory, Vol. 7, No. 1, (Spring, 1989), 70-86. [Thanks to John Baez for this link.]

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22 April, 2014 at 7:04 pm

Anthony Kofi Osei-TutuWow this was most excellent. I’m not even a math person, but your words remind of principles I learned in sales. Some guys will have more talent then others, but even the talented guy will underperform if he doesn’t put in the work.

25 April, 2014 at 11:13 am

AnonymousI need to know maths nd physics the question how?

28 July, 2014 at 10:11 am

IfeoluwaThank You a Ton for the post!…this latest awareness completely decapitates most of my conventional academic beliefs..

9 June, 2014 at 5:13 pm

Biodiversity, morality pills and killer hurricanes | InfoClose[…] we have to be a talent to do maths? Spoiler: […]

2 July, 2014 at 2:58 am

HITCHENS TO THE MATH LESS TRAVELED… | Have Coffee Will Write[…] Does one have to be a genius to do maths? saved 918 days ago. […]

22 July, 2014 at 11:34 am

QuoraIs it possible to raise IQ from 140 to 220 for a 18-year-old person?Leaving aside the problems with IQ as a single measurement of aptitude, we expect there to be 4 millionths of a person with an IQ score of 220, simply going by the definition of the score. The fact that at least one person has scored that high doesn’t…

12 August, 2014 at 9:31 pm

Ivan TellezOnly to say that this is a great post. Once I heard that “the one who wants is the one who does”… part of your main message. I recommend to read the “CODA” at the end of “I want to be a mathematician” by Paul Halmos, a great book.

I’m totally convinced that working hard is the key to success not only at math. Is an ingredient for almost anything in life. Thanks.

17 August, 2014 at 4:02 pm

Does one have to be a genius to do maths? | What’s new | maryble's Blog[…] https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/# […]

10 September, 2014 at 10:44 am

Math for Liberal Arts: Don't Be Afraid! - Math Help Blog[…] math. Terence Tao, a Fields Medalist (that’s the Nobel Prize for math) has a really fantastic blog article about the taboo of genius in math. Furthermore, this research study suggests a link between […]

3 October, 2014 at 1:46 am

Paul CerneaOne question we can start with is, where does the word “genius” come from? According to wikipedia, originally it meant a guiding spirit. In English, I think it’s an old-fashioned way of saying “genie”. So who do we consider a genius? Somebody possessed of extraordinary intuition, creativity, insight, or analytic faculty.

Note that we don’t think of a computer as a genius, even when it outperforms humans, because it takes orders and strictly adheres to formal rules. If we arranged a bunch of computers in parallel, say to take up an entire city, that supercomputer could probably prove theorems synthetically faster than most humans. But if it had some concise elegant heuristics to go by, then we would see it more as a genius.

So I think genius is the ability to find structure in what would otherwise be a jumble of dry formal statements. As such, it’s simply talent (the capacity of neurons to fire) consistently applied to some subject (like math). And it’s something that can (and needs to be) honed. I think beauty is probably a secondary attribute that our psychology imposes, just as we acquire tastes for various foods and activities.

That’s probably not the whole story (talent, multiplied by consistent application) because human beings naturally try to do what they enjoy. And they enjoy what they’re good at, in a (hopefully) virtuous circle. So education is a big factor, whether it comes from outside or within the self.

Thus genius (and the brain) can grow in a lopsided fashion to where somebody might be a genius in math but mediocre in art or psychology. Even within math, someone might not be good at all kinds of math. Gauss was a polymath, but would Galois have made contributions outside of abstract algebra? Was Galois’ genius of such a nature that it also compelled him to seek trouble?

That said, what can one do to hone one’s “genius” for math? I think that, just as in any other mental arena, one of the best things you can do is exercise the power of observation. Step back and ask, “what’s really going on?” “What can I see that hasn’t been yet observed?” Also, focus and ask, “What are the things that need to be done?” and then do them (say, carry out a calculation). Double-check. If you have a hunch or an intuition, follow it through and see what comes out.

Also, it’s important to go through what others have done. This is something I personally don’t enjoy (even dread) but eventually it must be done, and it increases one’s “culture” so one can springboard to further contributions.

It’s an interesting exercise to contrast “math genius” to other mental skill sets. For instance, there is the skill of splitting hairs (finding faults in something) which is indispensable for writing a computer program in practice or making sure a machine works. I don’t think it’s the primary skill used in, say, geometry or design of programs. I think the primary skill is a striving, a reaching out creatively, seeking a paradigm that might work, just as in art or writing or mystery-solving. Only afterwards is the hairsplitting put into place. With practice the initial model is more likely to conform to the desired result. However, I believe some new theories came about by hairsplitting: namely the theories of Cantor, Godel, and Turing.

What about business or social skills? I think there the primary skill is to think about what are the motivations or desires of the other people one is interacting with. Also think about your motivations and desires, and then act accordingly.

What about telling jokes? I think there the key is again practice along with having a subconscious store of stock phrases and ideas to draw on. Not too different from math, but the emphasis is on spontaneity rather than depth.

24 October, 2014 at 2:09 am

On Ed Witten and innate talent: a winding digression | The Daily Pochemuchka[…] (Spearman’s g, the notion of IQ in general, etc). (So does e.g. Terence Tao in his blog post here.) This is rather counterintuitive when you consider that Ron found Math 55 boring (it’s […]

6 November, 2014 at 1:17 pm

Radicals, Improper Ideals, & General Abstract Nonsense: A Look at Mathematics | Let's win college.[…] your mathematics level is below where it should be. Most math majors were not prodigies and are not geniuses! Colleges typically have degree programs designed with the assumption that students have completed […]

6 November, 2014 at 5:54 pm

Ngoc NguyenProf. Tao’s response to whether genius is requisite to do (paradigm-shifting) mathematics is accurate to the best of his knowledge and experience–no one here doubts that–but one cannot help detect a lingering dose of PC behind his answer, as anyone familiar with E. T. Bell’s “Men of Mathematics”–which is 580 pages full of historical mathematicians of nothing less than acknowledged genius, such as Archimedes, Gauss, Newton, Leibniz, Euler, Pascal, Fermat, Poincare, etc.–knows. Bell’s illuminating tome is virtually a written monument to these men because of the groundbreaking leaps in mathematics they made due to their nigh-superhuman mathematical brilliance (and insight). From Archimedes of Greek antiquity to Cantor of the dawn of the twentieth-century, the reader will be hard-pressed to find in Bell’s classic book a single non-genius (mathematician) among them.

9 November, 2014 at 6:30 am

Mike Ross“Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.” Albert Einstein

24 November, 2014 at 2:27 am

AnonymousVery good advice sir. I appreciate it.

30 December, 2014 at 9:39 pm

Does one have to be a genius in order to be a mathematician | MScMathematics[…] To me, the biggest misconception that non-mathematicians have about how mathematicians think is that there is some mysterious mental faculty that is used to crack a problem all at once. In reality, one can ever think only a few moves ahead, trying out possible attacks from one’s arsenal on simple examples relating to the problem, trying to establish partial results, or looking to make analogies with other ideas one understands. This is the same way that one solves problems in one’s first real maths courses in university and in competitions. What happens as you get more advanced is simply that the arsenal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try, perhaps making better guesses about what is likely to yield progress. Sometimes, during this process, a sudden insight comes, but it would not be possible without the painstaking groundwork [https://terrytao.wordpress.com/ca… ]. […]

30 January, 2015 at 8:41 am

Anshul Singhi wanna a competitor to challenge me

8 February, 2015 at 7:01 am

BogdanBut how do we explain phenomena such as Ramanujan?He, to my knowledge, didn’t go to college until Hardy noticed him, and he still made sone remarkable results.He wasn’t as academically trained as other mathematicians, but sonehow ideas came to him.

[See my previous comment at https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/#comment-3277 – T.]11 February, 2015 at 8:08 am

Requirements: “Genius” | Social Mathematics[…] his predecessors, Terrence Tao doesn’t believe in the cult of genius. Here is the evidence from his own blog which very clearly says […]

1 March, 2015 at 12:57 pm

KarlI’d just like to tell you how much I appreciate this post. Sometimes when I struggle with a difficult problem in math, I start to worry that I may never be able to do math on a higher level. This post has helped, and encouraged me, numerous times, and for that I’m grateful.

12 March, 2015 at 3:48 am

AnonymousWell, a lot of very intelligent people, all over the Earth, earn Ph.D. degrees in mathematics every year, but very few go on to become prominent mathematicians. It’s plausible that to become one of the greats (like the owner of this site) one needs two things: very high talent and tremendous perseverance (not to mention sheer luck sometimes). Just one is not enough. Anyway, tremendous perseverance may be considered a talent too, so it may be all about talent after all!

Does anyone have a counter-example? Someone not very highly intelligent that became great after much struggle? Or maybe, someone of very high intelligence that made important contributions without putting much effort? I do really doubt it, but I’m listening!

Probably all examples of people that became great are of ones that had intelligence very much above average and were very obsessed about math (this quote of Wiles comes to my mind: “I was so obsessed by this problem that for eight years I was thinking about it all the time—when I woke up in the morning to when I went to sleep at night.”).

Seriously, the level of competition in math is so insane that many frustrated researches could have been very successful in many other careers requiring related skills had they invested the same effort on them… Let’s be realistic: people of average intelligence can’t reach the level of Gauss, Euler, Newton, Poincaré, Hilbert, Fermat, Abel, etc. no matter how much effort they put forth. Analogously, one can go and watch those long jumpers from the Olympiads and then go and try to jump similar distances… After a few tries one just concludes that it’s impossible for him (and giving up is the best idea sometimes…), but for some reason, when the talk is about “intellectual possibilities”, the reluctance to say it’s impossible is much higher than for “physical possibilities”. Many would like to become like those men, but almost always the case is: to remember your limitations and give up the impossible (otherwise you will be a sad and very unhealthy person).

Just my two cents (since I’m a person of low intelligence, I may be very wrong about all of this, but I was sincere exposing my opinion — it’s what I think, no demagoguery on it).

22 April, 2015 at 8:16 am

high school student interested in mathsHi!

I would really appreciate if someone with knowledge of the subject could tell me, if there can be a significant change between child (8-10 years old) IQ and adolescent/early adulthood IQ. I am worried that I have caused such a thing with quite excessive substance abuse in my teenage years. I have been tested in a comprehensive overall intelligence examination in the 150s (age 8-10), but I feel like that brightness I can recall isn’t there anymore ( I am 20 now).

Though I feel like this could be mostly due to high anxiety, OCD and other emotional instability which I feel expends a significant deal of my processing power, I am worried about the possibility of having dumbed my self down. How volatile can one’s IQ be? (I know that IQ is only a measure of intelligence, but since it would seem absurd to compare one’s actual intellect as a child to that of one’s adulthood… :)

Thanks

24 September, 2015 at 3:57 am

someguyI had the same exact worry when I had anxiety/depression. I have recovered from this and assure you such thoughts never enter my mind now, when once they dominated it. Focus on the emotional instability; your primary concern should be living a healthy and happy life. Everything else goes on top of this.

18 May, 2015 at 10:31 am

Aaron GoldsmithDear High School Student,

I wouldn’t worry. It would be good let the substance abuse alone, but worrying won’t add anything to your life. I’ve had very similar thoughts after a car wreck, but I’ve decided that it’s okay. If you still like intellectual activity, keep doing it. If not, find something else. I think a point of this thread is that IQ is much less valuable than it’s cracked up to be.

I wish you the best.

12 June, 2015 at 10:31 am

Peter OlanipekunGood one Terence Tao.

This is very much the case. Many limit themselves by believing that Mathematics is for geniuses only. We grow mathematically as we put in more work. Start from where you are and see how much you will cover after sometime.

24 June, 2015 at 7:32 am

EruditeI myself have been guilty of pushing difficult questions away&and taking simpler roots,just because i belive that the “Hard” is for priest like province in marh,#Thanks teo

26 June, 2015 at 8:52 am

Turakimaths is the most interesting subject on earth as my lecturer always says

28 June, 2015 at 11:55 pm

“But aren’t maths and writing worlds apart?” | Cassandra Lee Yieng[…] require talent but only time, I would want to delve into it and see if it paves a way for me. Terence Tao, who was known as a child prodigy in maths, mentions hard work. Even Paul Erdős, one of the finest brains in maths, is claimed to have as many unknown maths […]

29 June, 2015 at 12:08 am

“But Aren’t Maths And Writing Worlds Apart?” Thoughts On Getting Known | Cassandra Lee Yieng[…] require talent but only time, I would want to delve into it and see if it paves a way for me. Terence Tao, who was known as a child prodigy in maths, mentions hard work. Even Paul Erdős, one of the finest brains in maths, is claimed to have as many unknown maths […]

1 July, 2015 at 9:21 pm

Jaime Maldonado LondoñoPor que no puede una persona común hablar con un matematuco destacado. Yo tengo algo muy importante relacionado con los números primos, su distribución y la formación de progresiones aritméticas. Yo teate en mi último intento de hablar con Torence Tao en su oficina de UCLA y fuí tratado descortesmente y me despacho wn 5 minutos. Reconozco que yo soli hablo español y un muy poquito de inglés, pero en cuestion de números y un poco de voluntad dos personas de diferente lengua se pueden entender, mas aún con el Doctor Tao y su calidad de genio.

English

Why can not an ordinary person talking with a prominent matematuco. I have something very important related to prime numbers, distribution and training of arithmetic progressions. I teate in my last attempt to talk to torence Tao in his office at UCLA and I was treated rudely and office wn me five minutes. Soli acknowledge that I speak Spanish and very little English, but in a matter of numbers and some will two people of different language can be understood, but still with Dr. Tao and quality of genius.

13 July, 2015 at 12:34 am

Fracking and the climate debate, universal daemonization, mathematical rigour, the cult of genius and the ban on trolling | scotchverdict[…] https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/ […]

15 July, 2015 at 3:06 am

StrauszInteresting point of view, coming from a genius…

Dear Terence, you reminded me the paradox emerging from the sentence: “I do not exist”

Receive my best regards!

15 July, 2015 at 2:30 pm

2 – Terrence Tao: Does one have to be a genius to do maths? | Offer Your[…] Source: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/ […]

18 July, 2015 at 8:47 am

shireenReally I got too much inspired as I love mathematics and I have a great interest in it

20 July, 2015 at 12:05 pm

Does one have to be a genius to do maths? | No. Betteridge’s Law[…] Does one have to be a genius to do maths? […]

25 July, 2015 at 11:51 am

AnonymousVery happy to see such an article, I learned this fact from a friend who i considered ”genius”. very encouraging :) Before that i actually believed that talent was 99% responsible for being a good mathematician. I did not believe i would be able to grow in understanding. Sure i thought i could learn more and more things, bu i did not expect to get better at easily grasping new concepts and solving problems. Discovering this was not the case was a great milestone in my intellectual life.

10 October, 2015 at 11:39 am

gninrepoliAccording to your post can be seen that genius does not exist, and we are all equal (I would like to believe, but it is not). Throughout the whole of humanity can be seen as one knowledge gives support to create the next. What sort of confirms the above described. But I think that the notion of a genius you have perverted and in general (probably).I think the genius of this level of analytical abilities, which are transmitted from generation to generation. The question is how to raise this level, I do not know exactly, but expect that not through work, and for the supervision of how the other person.Another person is working and you are analyzing. And the opening of something new on that there was no support of an accident.The world is cruel.

P.S. Sorry for the Google Translator.

18 October, 2015 at 5:00 pm

mustafa12131Thank you professor Tao!

Reminds me the talk of David Hilbert… http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.pdf

2 November, 2015 at 9:43 pm

ayilara dejiThis is not easy to know

16 November, 2015 at 6:06 pm

Amit Kumar "Vaishampayan"Footnote: The author of this article – Professor Terry Tao – has the reputation of being a “genius”

4 December, 2015 at 3:47 am

afiodorovThe subject of innate vs learned ability in mathematics fascinates me and I really wish there was a simple answer. But there isn’t, such is life. After reading more on the subject, I came to the conclusion that Terry Tao is right in certain things, but the following statement is not something I can defend:

> But an exceptional amount of intelligence has almost no bearing on whether one is an exceptional mathematician.

————————–

I think we all agree that some kind innate talent is necessary and just having a lot of innate talent is not sufficient. But can we actually quantify this sentiment?

This is called threshold theory. Once you are above a certain threshold in raw intelligence, other factors start to play a role. Factors such as being studious and creative. Or just being in the right place at the right time. And, of course, deliberate practice makes wonders. This is fairly uncontroversial until we actually get round defining this threshold.

Now what do we actually mean by raw intelligence? No one really knows what it is, but one thing is clear: clever people do well on a lot of seemingly unrelated tasks. And this is highly frustrating for some of us who invest a lot of time in the aforementioned deliberate practise only to be outperformed by “genius amateurs”. The correlations across different fields are referred to as a g-factor. Colloquially, we refer to people who possess “much raw intelligence” as high G people.

So one way to measure G is by distributing an IQ test. The role and value of IQ tests is the fact that people who do well on them *tend to* do well on other tasks, such as mathematics. Bear in mind, we are talking about a correlation, which is not *1.0*. The are and will be people who are not so good at maths yet score high on the IQ test for various reasons.

———————————

So what should the threshold be? How high?

If you are aiming for the Fields medal, then, yes, it actually has to be very high. But may be, you just shouldn’t? Why is this a worthy goal? I want to congratulate Terry for pointing this out as he gets this point absolutely right. Repeat after me: *your IQ is fixed for you and it doesn’t define your worthiness as a human being*.

Now, going back to the claim that IQ has to be high. Anne Roe wrote in “The Making of a Scientist” eminent scientists in some fields have average IQs around 150 to 160. She measured 64 of them and you can find a summary of her findings here: http://infoproc.blogspot.co.uk/2008/07/annals-of-psychometry-iqs-of-eminent.html. Please note that Feynman’s low score is addressed there too.

Hence if you are aiming for eminence in some crowded field like mathematics or theoretical physics then spare yourself the hassle and take an IQ test first. It might force you to set more realistic targets. And if you only score 120 this is only *evidence* that becoming eminent will be unlikely. But you could still do it, against all odds.

So it appears to me that only 1 in 10000 of us even has a chance of becoming eminent and that’s not taking into account other necessary factors. The 150 threshold is quite hard to accept, but I am coming to terms with it.

——————————————-

So what should you do if you have reasonable doubt about being above a certain threshold? It’s complicated but it is ultimately tied to your life goals. If you love the subject, why does it all matter? Oh, of course, you still have to get the funding… Such is our sad reality at the moment. However, we are talking about different threshold now. My guess would be 130, but this is a baseless hunch: please don’t change your entire career plan based on my hunch.

——————————————-

Finally there are other consequences of such a stark realisation. May be spotting kids like Terry Tao is a more efficient way for you to contribute to maths than actually doing it. Again, that is if your life-goal is narrowly defined as “contribute to maths” disregarding the entire process: which is equally unwise as setting out to win the Fields Medal. Another roundabout way to go at achieving a Fields Medal would be to address your shortcomings in innate intelligence by conducting research into neuroscience instead. Or, even, Artificial Intelligence. Ultimately, none of us are actually all that smart compared to what we could have been, even Terry.

It’s all too complicated and I am getting too side-tracked, but thanks for reading all of this.

P.S. read more about Terry’s claim here:

http://lesswrong.com/lw/lq3/innate_mathematical_ability/

4 December, 2015 at 12:40 pm

Terence TaoIt appears my previous comment may have have been interpreted in a manner differently from what I intended, which was as a statement of (lack of) empirical correlation rather than (lack of) causation. More precisely, the point I was trying to make with the above quote is this: if one considers a population of promising young mathematicians (e.g. an incoming PhD class at an elite mathematics department), they will almost all certainly have some reasonable level of intelligence, and some subset will have particularly exceptional levels of intelligence. A significant fraction of both groups will go on to become professional mathematicians of some decent level of accomplishment, with the fraction likely to (but not necessarily) be a bit higher when restricted to the group with exceptional intelligence. But if one were to try to use “exceptional levels of intelligence” as a predictor as to which members of the population will go on to become

exceptionallysuccessful and productive mathematicians, I believe this to be an extremely poor predictor, with the empirical correlation being low or even negative (cf. Berkson’s paradox).Now, at the level of theoretical causation rather than empirical correlation, I would concede that if one were to take a given mathematician and somehow increase his or her level of intelligence to extraordinary levels,

while keeping all other traits (e.g. maturity, work ethic, study habits, persistence, level of rigor and organisation, breadth and retention of knowledge, social skills, etc.) unchanged, then this would likely have a positive effect on his or her ability to be an extraordinarily productive mathematician. However, empirically one finds that mathematicians who did not exhibit precocious levels of intelligence in their youth are likely to be stronger in other areas which will often turn out to be more decisive in the long-term, at least when one restricts to populations that have already reached some level of mathematical achievement (e.g. admission to a top maths PhD program).For instance, many difficult problems in mathematics require a slow, patient approach in which one methodically digests all the existing techniques in the literature and applies various combinations of them in turn to the problem, until one gets a deep enough understanding of the situation that one can isolate the key obstruction that needs to be overcome and the key new insight which, in conjunction with an appropriate combination of existing methods, will resolve the problem. A mathematician who is used to using his or her high levels of intelligence to quickly find original solutions to problems may not have the patience and stamina for such a systematic approach, and may instead inefficiently expend a lot of energy on coming up with creative but inappropriate approaches to the problem, without the benefit of being guided by the accumulated conventional wisdom gained from fully understanding prior approaches to the problem. Of course, the converse situation can also occur, in which an unusually intelligent mathematician comes up with a viable approach missed by all the more methodical people working on the problem, but in my experience this scenario is rarer than is sometimes assumed by outside observers, though it certainly can make for a more interesting story to tell.

4 December, 2015 at 6:47 pm

Tetiana IvanovaBest regards,

Tetiana Ivanova

On 5 Dec 2015, at 02:25, Tetiana Ivanova wrote:

I have been thinking for a little while about what made me so uneasy about the whole concept of achievement being caused by intelligence, and I think I finally got it.

Looking at the statement “intelligence causes achievement (as a necessary but insufficient condition)” as a belief, one needs also to look at the practical utility of this belief. What would be the outcomes of holding this belief, assuming a) it is true; or b) it is false. Also, let’s further restrict the investigation to only those people who already have some proven track record in their chosen field- which in this particular case means looking at PhD students at top universities.

Now, the complexity in assessing the outcomes further increases due to the fact that we don’t know for sure which factors have determined the chosen field of research for a given PhD student. Trying to discuss this while introducing additional conditions such as whether the research field was selected out of sheer passion for the subject, chance/luck, status/glamour, convenience, etc, would complicate things way too much. Every single of these cases would require a different approach, so let’s just assume that our student is simply in love with their field, and that they have formed this passion in a consistent manner.

So what’s going to happen if the student then decides that they’re “insufficiently intelligent” to provide a significant contribution to their chosen subject? I personally think that nothing good, even in the case that they’re correct. Even if the given person can only produce work that is 50-75% in terms of quality compared to a work of a “great” mathematician, this is still progress. Unfortunately there isn’t a technology out there yet that gives everyone the same clock speed at every single task. The caveat is, though, that the levels of achievement are also highly correlated with the type of motivation one receives (including from themselves), and thinking that one is not “intelligent enough” would quite likely trash any stamina and perseverance that the person had previously built up. If the student then decides, further to being demotivated to pursue their chosen field, to do something where they feel they have a stronger “comparative advantage”, they will still quite likely end up in a group of people where there would be someone with IQ higher than theirs. To me it looks like a vicious cycle which saps motivation until its victim loses all will to be creative and productive, or ends up doing something trivial really well relative to their immediate surroundings.

Obviously, if the student also happens to be wrong in their hypothesis, the outcome would be even more tragic.

Therefore, I firmly believe that framing achievement as being conditional on intelligence (especially for oneself! especially in a field where the person is already known to be talented!) is a very slippery slope, and should be avoided because all of the dangerous pitfalls it introduces. This approach would only work if one could a) reliably compare their intelligence to that of their peers; b) successfully avoid all of the psychological and motivational traps that are triggered by someone thinking “I am not clever enough for this” with respect to _anything_. I don’t think the vast majority of human beings can be self aware enough for this to reliably work – and we’re most likely talking about uncorrelated traits here (intelligence vs self awareness). Therefore, if ones goal is to maximise their productivity and happiness, this would be a very dangerous belief to accommodate.

Even more so considering that the validity of the claim being discussed is extremely dubious. The research paper that Artiom is referring to above was written in the fifties. Accounting for the Flynn effect, in today’s terms the median average IQ scores (average of three tests Roe conducted) of a prominent scientist would only be 136, with low at 108 and high at 162 (see http://www.iqcomparisonsite.com/roe.aspx ).

One could now exclaim “But the IQs of scientists would be increasing in lockstep with the average IQ of the population!”

Well, not necessarily. The rate of IQ increase due to Flynn effect is way too fast to be explained in terms of evolution (i.e we’re not becoming smarter due to better hardware). A much better explanation in terms of fit to the data would be the changing social environment, specifically proliferation of rational thinking patterns. You see, before the most talented scientists would achieve the required level of rigour in their thinking _despite_ the social environment. Now more people have the necessary resources, so they get closer to the level of a great scientist. Which doesn’t mean, however, that the thinking of the great scientists would improve at the same rate – I would say it is more likely to be a lot _more_ difficult to improve, kind of like breaking a world record in a sport repeatedly (advances in medicine vs average life span also come to mind).

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