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.
See also the article “How to be a genius“, by David Dobbs, New Scientist, 15 September 2006. [Thanks to Samir Chomsky for this link.]
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18 June, 2007 at 3:19 am
beans
So there’s hope yet for me then! ;p
I think because it’s human nature to look at and ‘compare’ yourself to people who are ‘cleverer’ than you, we automatically assume that they’re geniuses. (However I have to say that most of my lecturers are geniuses!)
It seems that all maths undergraduates go through the same notions! Nice post. :)
19 June, 2007 at 10:28 pm
Pacha Nambi
I think doing any creative work in mathematics by anyone (genius or not) should be encouraged, recognized and appreciated. Also, one must enjoy doing creative work - without expecting any reward or recognition for it. Recognition and reward will come automatically to those who do good work in due course of time. Even publishing just one very good paper per year in a highly respected journal should be recognized. Quality more than quantity is what matters ultimately.
Thanks for giving me an opportunity to express my opinion on this subject.
20 June, 2007 at 12:29 pm
William Gasarch
AGREE with post. Also - there are so many different types of abilities in
math that people are good at in different combination, so a linear ordering
is not only a bad idea for getting work done, but perhaps impossible.
A few contrasts: problem solver/problem maker, algebraic/geometric,
clever/knowledgable, getting idea for proof/pinning down proof rigorously,
good reader of math/good listener to math. A clever problems solver who
can listen to math and get it quickly may LOOK more impressive,
but a knowledtable problem maker who is better off reading stuff
is just as valuable.
bill gasarch
20 June, 2007 at 1:13 pm
John Armstrong
Is math better done in quiet solitude or screaming across the alkali flats in a jet-powered, monkey-navigated…
You’re right as usual in your description, but surely you must recognize that some sides of these contrasts are selected for and some are selected against in the marketplace. To get a job it helps to be more of a problem solver than a theory builder, and woe betide those who hitch their wagons to the star of categories instead of analysis.
20 June, 2007 at 10:11 pm
Anonymous
“Do not disbelieve the prophecies simply because you had a hand in carrying them out.”
J.R.R. Tolkein
Isn’t what you are really saying just that genius is an emergent phenomenon?
(See: http://en.wikipedia.org/wiki/Emergence)
Then is it not the case that whether one worships it or finds it romantic
is a simple matter of theology?
21 June, 2007 at 2:23 am
regina
I am teacher of math and , actually I am writing books that help me to share my teachings: math + art + Science and history
It is a way to demystify those subjects
I would like to present to you the series of books entitled, “Caius Zip – The Time Traveller,”
The main idea behind the “CAIUS ZIP – The Time Traveller” series is to show the history made by great men and how mathematics and other subjects were important in their decisions. Caius Zip is a young man that participates in these discoveries and in the great battles. In each adventure, he acquires maturity and learns that to get out of trouble he must use his most important ability that he unknowingly uses very well: the power of deduction
The first book, ” Einstein, Picasso, Agatha and Chaplin:, How to explain the theory of relativity, cubism, travelling in time and unmask a murderer ” has been published
Description
Caius Zip, the young time traveller, arrives at Paris in 1905. The turn of the 20th century is a period that sizzles with ideas and realizations and the Universe is about to be contemplated as it never was before.
In this fiction, Einstein was resting in Paris before his innovating Theory of Relativity enlightened him. At that same time, Picasso was just starting on his idea of breaking with conventional perspective.
Both characters seek the same concept: space-time relation. The encounter between art and science is finally possible by means of a limitless imagination.
There are the descriptions of interesting places of the belle époque in Paris and the memorable dialogue between Caius, Einstein, Picasso, Agatha, André Salmon, the poet and Getrude Stein, the sponsor of the novice Picasso, at the Spanish painter’s atelier on how art, literature, science, travelling in time and mystery are intertwined.
Caius penetrates the birth of the theory of relativity and cubism and also manages to solve a murder mystery with the help of his two teenage friends, Agatha Christie, with her investigative mind and Charlie Chaplin, who provides a touch of magic to this surprising work of fiction.
After all and as Einstein once said: “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed”.
See a passage from the book in: http://www.caiuszip.com/relativiting.htm
best regard
Regina
21 June, 2007 at 10:34 am
Terence Tao
Dear John,
It’s interesting that you perceive analysis and problem solving to be so fashionable these days; this seems to me to be a relatively recent phenomenon. In the height of the Bourbaki era, for instance, I would have said that the emphasis in mathematics was more in algebra and in theory-building (and for quite a long time, this focus was spectacularly productive). And, of course, much of modern mathematics also involves a lot of geometry, topology, combinatorics, physics, etc. If anything, I think the marketplace these days favours mathematics which has some interdisciplinary component - need one hitch one’s wagon to merely one star?
Dear anonymous,
I would say that the perception of genius is an emergent phenomenon, and has non-trivial impact. But the concept of genius itself appears to play a minor role at best in any accurate model of mathematical ability and achievement. (The same, incidentally, appears to also be true for prophecy and models of prediction.)
21 June, 2007 at 11:52 am
John Armstrong
In truth I have only a limited data set, but I see my colleagues making calculations of known invariants getting more and better job offers than those who extend the domains of invariants or push for new ones. At the very start of one’s career, being a problem-solver means not having to convince hiring committees that the problems you work on are interesting. Being a theory-builder means you have to do exactly that.
I have not seen an analyst of my acquaintance struggle like the algebraic topologists often do. Fully half of the job offers I saw this last cycle said “analysis”, “functional analysis”, “applied analysis”, “dynamical systems”, and so on. They weren’t all exclusive, but they were strongly preferred. A scant handful claimed to want a topologist.
Interdisciplinarity is all well and good in theory, but in practice people want to put you in a nice, tidy box, and “category” is an easy box for many hiring committee members to put on the shelf and forget, even if you’re using it as a tool for studying something much more down-to-Earth than many older mathematicians think categories are used for.
21 June, 2007 at 12:42 pm
steven katz
I do not know whether you have already explained yourself on the subject and I am afraid my post is not at its proper place here, but I wish you could one time express your views concerning the nature of mathematics. I mean many people who could be described as great genius (es ? English is not my native language..) like Gödel seem to be on the platonic side as if their experience lead them there. Do you feel yoursel in this case, I mean do you really feel like evolving in a real mathematical universe independ of your existence, and does “mathematical intuition’ seem to you a kind of objective perception, like vision for physical objects, which is not equally distributed among people or is it just a romantic and naive point of view ?
21 June, 2007 at 3:42 pm
Anonymous
I don’t believe that the concept of genius is intended as a model to describe
how mathematical ability and achievement come about, a microscopic model, if you will. Rather, it is intended as a kind of macroscopic model for predicting what sort of mathematical achievements one can expect and from whom.
I believe it works out pretty well for this purpose as certain foundations which provide you with support might be willing to attest.
21 June, 2007 at 3:46 pm
Anonymous
N.B. Tolkein’s point about the prophecies was the same. They
were not intended to say how things would come about, only that they
would. Knowing more details about how they came about did not invalidate them.
21 June, 2007 at 8:32 pm
Sitting Duc » What it takes
[...] Terry Tao wrote in his article that mathematics is not a sport and it does not take a genius in order to do math. One needs plenty of patience and hard work. Reading his article, I feel a lot better in myself. I truly believe that if I don’t land on the moon, I would still land on another moon. Here I come, UFL!!!! [...]
27 June, 2007 at 8:38 am
Terence Tao
Dear Steven,
My view is that mathematics is primarily a language for modeling the physical world, or various abstractions of the physical world(*). So in one sense it is purely formal, in much the same way that English is a formal combination of letters of the alphabet. On the other hand, as our understanding of mathematics improves, our models fit the physical world better (both in terms of predictive power, and in terms of agreement with physical intuition) and so the mathematical objects we study begin to more closely resemble physical objects, though of course they are never actually physical in nature. It is certainly helpful though, when trying to create new mathematics, to think of mathematical objects as being analogous to physical objects; for instance, a mathematical object may “obstruct” another mathematical operation from taking place, and thinking about obstructions is a very useful way to make progress in mathematics.
(*) The physical world generally refers to tangible objects, but one can also consider abstractions of these objects, abstractions of abstractions, and so forth. For instance, a children’s ball is a physical object; it might be red. The property of “redness” is then an intangible abstraction, but still physical. The phenomenon of “colour” is then an abstraction of an abstraction, but again still physical; the concept of a “sense” is a yet further abstraction; and so forth. Somewhat analogously, mathematics tends to start with “primitive” objects such as numbers or points, then moves up to sets, spaces, operations and relations, then functions, then operators, (and then functors and natural transformations, in category theory), etc. [It's true that in set theory, all of these mathematical objects can be described as sets - much as all parts of speech in English can be described as strings of letters - but this is only one of many equally valid interpretations of these objects, and not one which corresponds perfectly to physical intuition.]
28 June, 2007 at 8:30 am
Shibi
Dear Terry,
Pardon me if i am wrong in asking this question(or asking it the wrong way). But do you mean to say that there is no such thing as genius? I mean i always thought that someone like Ramanujan was a genius. Also, i would like to ask you about the relative levels of understanding. I mean some people have to read several books and solve endless number of problems before they understand a topic, which might take months or years, while someone like Riemann was purported to have read and understood a great classic book in just a week’s time. Do you think that the “time” it takes to understand things(i guess this significantly varies amongst various people) affects their research. Also about understanding a concept the “correct” way. i mean a few (brilliant?) people just learn stuff in a clear crisp manner. the fundamental concepts just stick in their head . while for others they tie themselves up in hopeless logical knots and endless circles of misunderstanding and understanding things the wrong way. this obviusly affects them and their research;
Also about people who seem to have “physical insight” like Geoffrey Taylor in Fluid mechanics. He was probably not as great a Mathematician but he had the uncanny knack of separating the “essence” of a problem and reducing it to a manageable portion so as to solve it. He himself claimed that he was no good at things like Chess or Crossword puzzles but he seems to have this amazing amazing ability to solve Fluid Dynamic problems. What do you have to say about those kinds of people;
As a humdrum workaday(and defintiely not brilliant) graduate student in Aerospace Engineering this essay has given me immense solace and comfort. i really want to thank you for posting this essay.
29 June, 2007 at 1:58 pm
Terence Tao
Dear Shibi,
It is true that some mathematicians can be vastly more efficient than others at learning material, but I feel this is more due to experience and an efficient means of study than to any innate genius ability, though of course innate talent is still a contributing factor. For instance, among the graduate students I have advised, the first paper they read in a subject often takes a month or so to read (and they have a question on almost every page on the paper); but after a few years, they can get the gist of a new paper in the subject within a day, skimming past all the “standard” (or at least “plausible”) portions of the argument and focusing on the key new ideas. The key, I think, is to find one or more efficient ways to internalise the subject - either by using formalism, or geometric intuition, or physical intuition, or some other analogy or heuristic. Each mathematician has their own different way of doing this. Ramanujan, for instance, apparently performed a tremendous number of numerical computations, and derived much of his intuition from the patterns he observed from those computations. The intuition wasn’t always correct (for instance, he famously gave an incorrect formula for the n^th prime), but he did discover a number of amazing results this way, some of which took a long time to prove rigorously.
29 June, 2007 at 4:53 pm
Jonathan Vos Post
“need one hitch one’s wagon to merely one star?”
No. One may hitch one’s wagon to a galaxy of stars. The Online Encyclopedia of Integer Sequences, the arXiv, these are part of the future of mathematics. The killer app of the Web is collaborationware.
In that, I believe that I agree with Pacha Nambi.
As to prophecy, the Foundation Trilogy and sequelae by (and authorized by) the late great Isaac Asimov had a lot of deep cogitations on “psychohistory” as a mathematics of emergent predictability of sufficiently large numbers of human beings. He analogized to Kinetic gas theory. Isaac Asimov, very late in the process, undercut his own metamodel with a short story about the Chaos Theory aspects of prophecy.
Terry Tao’s answer to Shibi is interesting. He is parallel to a business partner of mine who used to say: “repetition gives the illusion of intelligence.”
19th century psychophysics quantifies this by indicating that, up to some inherent limit of performance, a person’s speed of performance of a task declines as the log of the number of repetitions.
Archimedes, Newton, Gauss, Ramanujan, Erdos, Feynman, et al. did a large number of repetitions of calculations and higher-order mental tasks, and got very fast and very good and very smart.
Is there genius? I think so, having met Feynman and Gell-Mann and John Forbes Nash, Jr., and Stanislaw Ulam and others. But they worked very hard and very long to give the illusion of it being easy for them.
29 June, 2007 at 8:00 pm
Jonathan Vos Post
Of course, I mean “a person’s speed of performance of a task INCREASES as the RECIPROCAL of the log of the number of repetitions” or “a person’s length of time to complete the performance of a task declines as the log of the number of repetitions.”
*sigh* Sorry.
6 July, 2007 at 10:33 am
averageJOE
I would tend to agree with those who believe in genius. Certainly , a person of my limited talents, given a large finite amount of time could not hope to come close to understanding the works created by brilliant mathematicians ( yourself included). And then there is creativity which seems to defy all measurable tests ( IQ exams included) … perhaps this is what is meant by intuition and insight … the benefit of due dilligence and repetition with opens eyes to see paterns and to generate insight into otherwise seemingly unrelated problems… anyways… I found this discussion interesting and thought I should drop my 2 cents worth… To Terrance Tao … your son, does he possess the innane gifts that you have or is he like the rest of us .. ( or maybe somewhere in between?) … just curious … the question hints of inuendo of what percentage of genius is environmental vs. genetic.
8 July, 2007 at 4:24 am
franck
Dear Professor Tao
Even from an outside point of view (I am not a mathematician) I agree with most of your point. The word “genius” is not precise enough, and even if there are some real genius in each generation - and in many ways you seem to be the nearest of a mathematical genius humanity has those days - the mathematical field is far too great to be explored only by genius.
But in the other hand, I don’t see why, nowaday, a medium gifted mathematical student would choose a mathematical career. In my case, for exemple, I was a schoolboy with some gifts in mathematics - nothing like competing in International Olympiads or anything, but easy with the concepts in high school and university levels. Being a frenchman, I “naturally” went to one of the best “grande école” and got an engineering degree with a lot of mathematical insights. There I met many people more gifted than I was, and undestood (maybe late !) that even if I could make a living in mathematical research and teaching, I probably would not become a well known figure in that area, and would earn much less money (with a factor of 3 or 4) than I would in industry.
So I got a master in Telecommunications and IT and now, at 40, I am deputy CIO of a medium french company. The job is usually interesting, very involving and I make good money - not much according to US Standards, but we are in France :).
But in the back in my mind there is always the idea that what I do has some futility, and that in the big picture scientific research, especially mathematics, is, probably with art, the only activity we human must me proud.
15 July, 2007 at 10:11 pm
Johm Park
Dear prof Tao,
I thank you for posting this article. But I guess I should not use this objective article as false encouragement.
Would you please define what is the ‘reasonable amount of talent’?
I was born outside of America, am a 25 year-old economics graduate enrolled, think I love math, though I think I am not much brilliant or especially knowledgeable, and am considering the options of either studing econ in America or pursuing Law career in my own country(I didnt major in math as undergraduate).
Since this ‘natural’ talent question’s the main psychological blockage to my choosing the first option, your help would make my decision a lot more objective and lead me to better chance of offering something good of my own.
17 July, 2007 at 2:20 am
kaiming
I believe it is true that one does not have to be a genius to do maths.
But if you are smart as a genius, you have a greater chance to make discoveries or solve mathematical problems as important as those by Albert Einstein or Isaac Newton. The less capable persons will only make the less important achievements.
17 July, 2007 at 8:50 am
Terence Tao
Dear Johm,
It is true that graduate economics these days does require a certain amount of mathematics (notably several-variable calculus, probability, and game theory). See this post:
http://gregmankiw.blogspot.com/2006/09/why-aspiring-economists-need-math.html
As for “reasonable intelligence”, it is hard to define precisely, but roughly speaking I refer to the ability to analyse situations (particularly hypothetical or other abstract situations) at a level deeper than a superficial level, e.g. by drawing analogies, asking questions, formulating and then testing hypotheses, reasoning logically, finding ways to double-check one’s conclusions, identifying any limitations to one’s analysis, and so forth. It’s not a skill that is restricted to scientific or mathematical situations - the same type of skills are also needed to a diagnose a mysterious computer bug, solve a crossword clue that lies outside your direct knowledge base, or dissect the complicated plot of a novel or movie. But one does need some level of higher mental faculties in order to do mathematics; it’s not a subject that can really be learned by, say, memorising flash cards.
17 July, 2007 at 11:28 am
dod
No one is perfect!
19 July, 2007 at 3:56 am
Carrière-advies van Terence Tao at QED
[...] een postdoctorale carrière. Tao legt onder andere uit dat wiskunde meer is dan bewijzen, dat je geen genie moet zijn om een goede wiskundige te worden, dat je naar conferenties moet gaan, ook buiten je eigen [...]
21 July, 2007 at 10:56 pm
Kristy
Several physics issues wants to ask. What is the quantum network? Relativistic how the formula is derived, which Riemannian geometry of the importance of understanding how? Earth around the sun operation, in the classical Newtonian mechanics to explain gravity, and Einstein appears, This is because the bending of space around the sun. Ordinary people do not have an advanced knowledge of mathematics can understand? Assumptions in a vacuum is a very long stick, which was one end of the rotation center, as the angular velocity unchanged, Then there will be bar a linear speed to meet or surpass the speed of light, which may be? Does this campaign need a reference system? Mathematics in the physical phenomena of the study is idealistic? Relativity applies to the macro world, and quantum mechanics is applied to the microscopic world. Physicist this thinking, as a mathematician, you will be involved in this field of physics? I used the automatic translation function of the translation, there may be statements not understand. Please forgive me! Thank you for your time
13 October, 2007 at 8:35 am
Anonymous
Hi everyone here!!How many hours do you sleep every day??how to make yourself forget zout the sleep but concentrate to the math?
15 November, 2007 at 5:02 pm
anonymous
DEAR DR TAO,
THERE IS SO MUCH INTERESTING STUFF ON YOUR BLOG THAT I HOPE YOU WILL LEAVE IT ON THE INTERNET, ALONG WITH THE COMMENTS, PERMANENTLY. THIS IS ESPECIALLY IMPORTANT BECAUSE THERE IS SO MUCH INTERESTING STUFF TO LEARN IN MATHEMATICS AND PHYSICS AND OTHER SUCH INTERESTING THINGS THAT ITS HARD TO FIT THE TIME IN TO KEEP UP WITH YOUR WONDERFUL BLOG TOO, ESPECIALLY FOR ME BECAUSE I SPEND MOST OF MY TIME JUST STRUGGLING TO MAKE A LIVING NOW AND I BARELY HAVE THE TIME TO READ YOUR BLOG LET ALONE TO STUDY.
I HAD TO DROP OUT OF COLLEGE AND HOPE AT SOME POINT TO DO RESEARCH IN MATHEMATICS AND PHYSICS BUT I WAS AN UNDERGRADUATE IN PHYSICS WHEN BECAUSE OF DIFFICULTIES IN THINKING THAT I’M NOW SOLVING THAT I HAD TO DROP OUT. I’M 51 BUT REFUSE TO GIVE UP ALTHOUGH I DEFAULTED ON STUDENT LOANS SO THAT PAYING FOR LIVING EXPENSES AND TUITION WHILE STUDYING WOULD BE A POSSIBLY INSURMOUNTABLE PROBLEM, ESPECIALLY SINCE I GOT TERRIBLE GRADES ALTHOUGH I’M APPROACHING THE POINT THAT I COULD GET STRAIGHT A’S ALMOST EFFORTLESSLY.
THANKS FOR YOUR TIME.
18 December, 2007 at 10:14 am
Beans
“reasonable amount of talent”
That is the catch - talent!
I am being encouraged to consider doing post-graduate studies, but do not think myself capable of such a task. It is easy to look at the negatives when one evaluates such possibilities, and they can be rather numbing and painful! (I am 19 and in my second year of undergraduate studies, so still have about a year to make my mind up).
What advice would you give to someone like me, who isn’t particularly clever, but might one day want to do post-graduate studies? I mean is there anything I can do now “in preparation”? I don’t really know much maths at this moment in time, but have an idea on two people who I wouldn’t mind being my supervisors. (I have been thinking about this for way too long!) Do I look into their field and make my module choices dependent on that, or find a field and then choose a topic? [One lecturer is group theorist and the other is a number theorist - both are extremely cool! Does there exist a topic which means that both can be my supervisor?!]
I can’t seem to remove this idea that to do post-graduate studies you have to be a genius! Ah well, I best shut up now.
18 December, 2007 at 11:49 am
Adam
Dear Beans,
Group and number theorists surely have a high school level open
source open problem (e.g., smart integration by parts). If you can solve
it, they might be impressed and even mention your work on a webpage.
22 December, 2007 at 3:29 am
goh
I am a high school student that loves Mathematics very much. I wish to major in math in university and even be a math professor. However, I do not think that I am brilliant enough in math. If I do all the things as you have mentioned in the paragraph 2, can I achieve my ambition?
Besides, can you recommend some good universities for me to major in math?
29 December, 2007 at 2:25 pm
pierre
for some definition of a genius, go here:
http://www.lygeros.org/0116-M-classification_English.htm
http://www.lygeros.org/0134-class2.html
By the way, the author of this article is a universal genuis, with an IQ of 189.
29 December, 2007 at 2:57 pm
pierre
and here for a global view:
http://www.lygeros.org/mclassi.html
After talent, patience, and works, Tao could have also mention the educational environment, which is, I think, very important. Look at for example Erdos, one of the most prominent mathematician of the all times, both of his parents where maths teachers. And I could give hundred of other examples.
I don’t want to depreciate Tao’s view since he is a great mathematician, and I am not, but I think his view is very “consensual”. It is like saying: “high intelligence is not necessary for doing research, but it helps”.
Personally, I wanted to become a mathematician, but I realized that my IQ of 130 was not high enough.
29 December, 2007 at 3:52 pm
John Armstrong
pierre, I’m not going to name any names, but I know more than a few mathematicians whose IQ (whatever that horrible linear scale actually means) is far below 130. I’ve met some who are downright morons. But they do good math despite that fact, and doing good math is all that matters.
29 December, 2007 at 10:12 pm
Andy Sanders
I don’t think there’s any precise way, beyond someone being a genius, to measure what their success will be in mathematics. Apart from a few people who are truly gifted in a unique way: it seems every other working mathematician is a combination of incredible work ethic, significant cleverness and a healthy dose of good luck (although I think this probably comes from working until the luck comes around.) I am current a graduate student in Math, and while I am certain many of the professors I walk the halls with are brilliant men, they are likely not geniuses in the sense that I think people are using that word.
I don’t know if I’ll ever do anything great, but I am going to give it a shot. From there, we’ll see if I get that dose of luck and moments of cleverness.
Nevertheless I enjoy articles like this from folks like Professor Tao. It shows everyone has to work for it, even those who seem to have more talent than just about everyone else.
30 December, 2007 at 9:59 am
Jonathan Vos Post
I strongly agree with pierre and John Armstrong on the Erdos, IQ, Tao, on genius dynamics.
I have a rather high IQ (circa 180) but am clearly not as smart as my wife, our son, or many of the undergrads I knew while at Caltech. Let alone the supergeniuses I spent extensive time with, such as Feyman, Gell-Man, Hawking, and Heman Kahn. Despite their press coverage as miraculously brainy, they were all tremendously hard workers.
More important, my ostensible intelligence and rapid grasp of simple Math (i.e. taught myself calculus at age 12, tested out of some required courses at Caltech by exams I took age 16) worked very much against me getting my B.S. and beyond. The quick and effortless learning and occasional insight led to me to seriously neglect acquiring good work habits and probem solving discipline.
Again and again, wherver I go, I see people of limited intelligence who achieve astonishing results, while people of enrmous giftedness go nowhere.
What matters is not what cards you are dealt by genetic lottery, and parental support, and early mentoring, but how well one learns to play the hand that one is dealt.
Wherever you go, there is someone smarter, prettier, richer, and/or better connected. Get used to it, and move beyond it.
“Chance favors the prepared mind.”
30 December, 2007 at 11:47 am
Todd Trimble
As an amusing side note on IQ scores: in the book No Ordinary Genius, it’s reported that Feynman had an IQ of something like 127 (from when he took an IQ test in school). His sister, with tongue firmly in cheek, would gloat that she was smarter: hers was 128!
Please let’s not obsess about IQ scores. It’s just ridiculous. Let’s not worry about ‘genius’ either — an awful lot of romanticism tied up with that word, most of it completely wrong-headed IMO.
I’d say that the sine qua non for success in mathematics is sheer fascination with the subject and the drive for greater understanding. From that a great deal follows, including the willingness to do the necessary hard work.
30 December, 2007 at 6:42 pm
Pacha Nambi
Too much importance is given to IQ. I think it is about time the idea of IQ is discarded the way the idea of ether was thrown out.
One should simply enjoy doing what he/she finds fascinating in mathematics, physics, chemistry, biology etc. The reward and recognition will come to those who do very good work in due course of time.
31 December, 2007 at 3:33 am
pierre
As an amusing side note on IQ scores: in the book No Ordinary Genius, it’s reported that Feynman had an IQ of something like 127
>>>No, as Jonathan means it in a post I read on internet, its an urban legend, like the one that Einstein failed a maths exam.
I am not a mathematician, but I know what a genuis is. I definitly think that you cannot expect to go far without an high IQ.
For those who speak french, go here, for an view on maths and intelligence.
http://www.lygeros.org/0218-Mathematiques_Cognitives_et_Intelligence_Extreme.htm
>>Now, does someone have an idea of Erdos’s IQ? Iam looking for it.
Thx.
31 December, 2007 at 8:08 am
Todd Trimble
Pierre: Feynman was of course a notorious embellisher of legends about himself, and this may be one of them for all I know. But, the story is told in James Gleick’s otherwise well-documented biography on Feynman (on p. 30 according to Wikipedia); I don’t have the book to hand, but it’s apparently it’s part of the Feynman lore as told by himself and his sister. (And, there may well be documentation available to prove the story correct.)
Whatever the actual truth, the moral of that story is that Feynman himself didn’t take IQ scores seriously (far from it!). If a super-genius like Feynman doesn’t take it seriously, why should you?
There are more kinds of intelligence in heaven and on earth than are dreamt of in IQ testing. And that’s the last I’ll say on this topic here.
31 December, 2007 at 9:06 am
pierre
Whatever the actual truth, the moral of that story is that Feynman himself didn’t take IQ scores seriously (far from it!). If a super-genius like Feynman doesn’t take it seriously, why should you?
Ok. Lets put aside the story about Feyman . The theory of IQ is a statistic theory, with its “outliers” like every statistic theory. Feyman was one of them
I agree that intelligence is a complex structure that cannot be characterized by a number.
But on a large scale, it is consistent with reality.
31 December, 2007 at 10:27 am
John Armstrong
Cite? And what objective quantity is it that IQ measures? And why is it that (what with constant rescaling) everyone from the late 19th century would score as mentally retarded on modern IQ tests?
31 December, 2007 at 10:41 am
Terence Tao
It is strange that IQ has such a hold over the popular imagination, because as far as I can tell it plays no role in academia whatsoever. In professional mathematics, at least, we don’t brag about our IQs, put them in our cv’s, or try to find out other mathematician’s IQ when trying to evaluate them; it has about as much relevance in our profession as the Meyers-Briggs Type Indicator.
More generally, the skills and traits that are popularly associated with “intelligence” or “genius” become largely decoupled, after a certain point, to those that are needed to do good mathematics. For instance, a very creative person may have a hundred innovative ways to attack a mathematical problem, but what one really needs is the rigorous thinking, comparison with existing literature, intuition and experience, and knowledge of heuristics in order to winnow these hundred ways down to the two that actually have a non-zero chance of working. Indeed, being overly creative at the expense of true mathematical skill may in fact impede one’s progress on a mathematical research problem, due to all the time wasted on the ninety-eight hopeless avenues.
Similarly, a very intelligent person may be very comfortable with abstract concepts and abstruse reasoning, and a certain amount of this can indeed be an asset when learning some of the more theory-intensive portions of mathematics, but at some point one has to be able to digest this theory and connect it with more mundane, “common sense” concepts (e.g. geometry, motion, symmetry, information, etc.); there is a risk of an excessively intelligent student getting overly enchanted with the formalism and esotericism of a subject, and neglecting to keep his or her knowledge grounded in reality (and to communicate it effectively with others).
In a third direction, a very quick thinker may be able to pick up new ideas rapidly, to find snappy rejoinders to any question, and to complete tests and examinations in a remarkably short amount of time, but these attributes may in fact lead to excessive frustration when such a student encounters a genuine research problem for the first time - one that requires months of patient and systematic effort, starting with existing literature and model problems, identifying and then investigating promising avenues of attack, and so forth. In athletics, the best sprinters can often be lousy marathon runners, and the same is largely true in mathematics.
To summarise: as I said in the main article, a reasonable amount of intelligence is certainly a necessary (though not sufficient) condition to be a reasonable mathematician. But an exceptional amount of intelligence has almost no bearing on whether one is an exceptional mathematician.
31 December, 2007 at 11:29 am
pierre
And what objective quantity is it that IQ measures?
>>Intelligence quotient measures intelligence in the following fields : >>mathematics, sciences, philosophy and literature.
And why is it that (what with constant rescaling) everyone from the late 19th century would score as mentally retarded on modern IQ tests?
>>I doubt it, altought there is that kind of “Flynn effect”.
Go here:
http://www.iqcomparisonsite.com/Cox300.aspx
31 December, 2007 at 12:48 pm
John Armstrong
Pierre: what is this “intelligence” in these fields that an IQ test purports to measure? You’ve just pushed around words like peas on your plate, but you haven’t eaten anything.
As for the “Flynn effect”, that table is all but meaningless. Besides the fact that there’s no mention of how these numbers were derived or how they were “corrected”, a correction should act to increase the score to compensate for the fact that the scaling has gotten harder.
31 December, 2007 at 12:56 pm
David
I was admitted to several top-10 (US) math PhD programs and ended up going to a top-5 school, but once I realized I was not head and shoulders above my fellow students, I couldn’t bring myself to continue. A major factor was the fear that I would spend a year working on a problem, and then some brilliant guy who urgently needed the result would turn his attention to it and solve it in a week. Then I’d be left with nothing to show for my year of hard work. Not to mention the humiliation that an entire year of my work was worth less than a week’s worth of work for a better mathematician. I don’t think I could bear to explain that to anyone.
Maybe I had read too many biographies of famous mathematicians — like John Nash, who (in cahoots with another bright guy) used to get a kick out of solving other grad students’ thesis problems in a weekend. Anyway, I always had the impression that the vast bulk of professional mathematicians were toilers whose work went unread and was ultimately superseded by work done by the top guys. I tried to tell myself it still wouldn’t be such a bad life to be a mediocre mathematician, but unfortunately, it takes more motivation than that to get through a PhD program.
31 December, 2007 at 1:05 pm
pierre
I am not a specialist of intelligence, I am only interested in it. I dont have the time nor the space to explain everything, but here is a definition of intelligence: the creativeness,dicovery or inventiveness capacity in the fields.
Besides the fact that there’s no mention of how these numbers were derived or how they were “corrected
>>see: historiometry in wikipedia
31 December, 2007 at 2:12 pm
High IQ and Mathematics « Vishal Lama’s blog
[...] 31, 2007 in Uncategorized Tags: IQ, mathematics, Terry Tao Here is an interesting article (by Terry Tao) and the accompanying discussion on the relation between [...]
31 December, 2007 at 2:12 pm
Richard
I completely agree with Terence in his last comment here, and attempted to post something similar late last night until fatigue took over. I might add that this view generalizes to other fields as well. My medical specialist with an MD PhD and many papers under his belt put it this way in regard to the medical field: “the whiz kids don’t necessarily make the best researchers.”
31 December, 2007 at 3:12 pm
John Armstrong
pierre: I fail to see the creativity, discovery, or inventiveness in a standardized test such as the WISC or the WAIS. Maybe the stumbling block is lacking your 130 IQ (no, I don’t know what mine is, though it’s in my parents’ records somewhere).
Now, as to your reference to Wikipedia, that doesn’t answer my question at all. I know how IQ scores can be generated, but I don’t know how those numbers on that page were generated. Maybe people with 130 IQs can think perfectly without citing their sources, but I’m just not that smart.
31 December, 2007 at 7:34 pm
Richard
Here are a few random thoughts.
People have different approaches to solving problems. Some will launch directly into a frontal attack, while others methodically circle and envelope the problem before going in for the kill. The later type of person is probably not ideally suited for excelling at IQ tests, or any kind of exam for that matter, yet their working method works for many kinds of mathematical problems. Remember the story of the tortoise and the hare?
I’m old enough that I remember when TV was all black and white, so my memory of that grade school IQ test is a little vague. However, I believe that it was just a long series of disconnected questions and problems. Mathematics is highly conceptual in nature, and involves understanding deep connections between these complex concepts. Moreover, a good researcher must have a good instinct for what is truly interesting and what really matters. I just don’t see how an IQ test can measure any of ability of this sort.
What is intelligence anyway? While hiking with my dog last summer in the woods in our favorite conservation area, we ran into a familiar volunteer there. Usually we discuss the complexity of nature, but that day she told me of a relative of hers who is borderline autistic, but who can multiply two long integers in his head with perfect accuracy. He just rolls his eyes up, and he soon has an answer. She said that he has some system working with “twelves”. When asked if he works in base 12 arithmetic, she said “something like that.” * Anyway, although I was one of those grade school kids who went through a period where I was factoring numbers almost compulsively in my head, and even earlier, thinking almost instantly 7 x 9 = 7^{2} + 7 x 2 = 49 + 14 = 63 rather than simply retrieving the result from a memorized multiplication table, that sort of feat is just way beyond and incomprehensible to me. Is this person intelligent? They have an extremely specialized skill to be sure, but I doubt that they will ever grasp the notions of measure or abstract topological spaces and algebras. I wonder if in fact the ability to grasp and manipulate highly abstract concepts is yet another highly specialized skill decoupled from almost everything else.
* I’d be interested to hear any comments from number theorists about the “twelves” thing. I couldn’t get much more detail from this person.
1 January, 2008 at 2:10 am
pierre
pierre: I fail to see the creativity, discovery, or inventiveness in a standardized test such as the WISC or the WAIS. Maybe the stumbling block is lacking your 130 IQ (no, I don’t know what mine is, though it’s in my parents’ records somewhere).
>>I can give you some tests, unlimited in time, that test creativity.
1 January, 2008 at 2:36 am
John Armstrong
And how do these tests reduce “creativity” to the same linear scale as the WISC or the WAIS?
1 January, 2008 at 7:10 am
pierre
And how do these tests reduce “creativity” to the same linear scale as the WISC or the WAIS?
>>I dont know; I am not the author of those tests.
1 January, 2008 at 9:59 am
John Armstrong
But you’re the one who is advocating these tests as examples of how IQ measures “creativity”. And now you’re saying you have absolutely no idea how they even claim to work.
I return to my original question: just what is it that IQ measures?
1 January, 2008 at 12:45 pm
Jonathan Vos Post
“Intelligence testing began in earnest in France, when in 1904 psychologist Alfred Binet was commissioned by the French government to find a method to differentiate between children who were intellectually normal and those who were inferior. The purpose was to put the latter into special schools. There they would receive more individual attention and the disruption they caused in the education of intellectually normal children could be avoided…. Binet himself cautioned against misuse of the scale or misunderstanding of its implications. According to Binet, the scale was designed with a single purpose in mind; it was to serve as a guide for identifying students who could benefit from extra help in school. His assumption was that a lower IQ indicated the need for more teaching, not an inability to learn. It was not intended to be used as ‘a general device for ranking all pupils according to mental worth.’ Binet also noted that ‘the scale, properly speaking, does not permit the measure of intelligence, because intellectual qualities are not superposable, and therefore cannot be measured as linear surfaces are measured’…. American educators and psychologists who championed and utilized the scale and its revisions failed to heed Binet’s caveats concerning its limitations. Soon intelligence testing assumed an importance and respectability out of proportion to its actual value…. According to Harvard professor Steven Jay Gould in his acclaimed book The Mismeasure of Man, these tests were also influential in legitimizing forced sterilization of allegedly ‘defective’ individuals in some states.
By the 1920s mass use of the Stanford-Binet Scale and other tests had created a multimillion-dollar testing industry. By 1974, according to the Mental Measurements Yearbook, 2,467 tests measuring some form of intellectual ability were in print, 76 of which were identified as strict intelligence tests. In one year in the 1980s, teachers gave over 500 million standardized tests to children and adults across the United States. In 1989 the American Academy for the Advancement of Science listed the IQ test among the twenty most significant scientific discoveries of the century along with nuclear fission, DNA, the transistor and flight. Patricia Broadfoot’s dictum that ‘assessment, far more than religion, has become the opiate of the people,” has come of age.’”
http://iq-test.learninginfo.org/iq01.htm
“An intelligence quotient or IQ is a score derived from one of several different standardized tests attempting to measure intellige