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.]

## 339 comments

Comments feed for this article

18 June, 2007 at 3:19 am

beansSo 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. :)

26 May, 2010 at 8:08 am

BryantBeans,

One thing that I discovered as a math major is that as you gain more and more knowledge, things that you once would have considered “ingenious” become very easy for you. To any observer the ease at which you come to a difficult proof is inspiration when in fact it is the result of years and years accumulating knowledge and techniques.

I think one of the biggest mistakes that I made as an undergrad was accrediting more to “genius” than hard work.

9 April, 2014 at 1:35 am

HamzaBraynt is absolutely true. The things we once thought were quite difficult and we didn’t had the answer to them, seem very easy once we study them and strive to find its answer. All the inventions made today such as the mobile may seem to us a very common device just beacause its in the hands of every individual, but just imagine making a “mobile” when there was no name of telecommunication!

The ones who did such works which had no sign are called genius.

Don’t think that you’re a genius or not, think whether you can become one. The great people you think of being genius in the past were themselves common people but their talent led them to become history!

Study your subject of passion in the deepest details and you’ll find yourself a genius.

2 October, 2014 at 3:58 am

AshwinThis will help the beginners:

http://kissmyblackads.blogspot.in/2011/05/ira-glass-quote.html

19 June, 2007 at 10:28 pm

Pacha NambiI 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 GasarchAGREE 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 ArmstrongIs 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?

26 January, 2013 at 3:04 pm

StephenJust… what are you doing? To say that mathematical ability could use some hard work (with, of course, the ingredient of reasonable intelligence) doesn’t suffice to say he is an emergent in entire principle. Using your logic, you would denigrate feeding children healthy food, because they evolve. And theology, – as ridiculous as it is – cannot be a good device in – lets say – universal argument: reason ultimately is. I love the quote, though. Though, On the darker-side, the rest of the comment seems to be used to make you sound “smart.” Sorry, If I offended you with the bit about religion. I guess it’s more likely that you are a christian, than I bargained for: based on the person, whom you quoted. Never mind. If comments were written for their utility then it wouldn’t be here. “I-like-to-write” is the reason why its here. ^ here

7 March, 2013 at 12:51 am

KamalaykaYou have too, many, commas

Short of breath, brah?

7 March, 2013 at 2:57 pm

Anonymoushaha… simply cannot stand it when people do that, either. I started off well and then I got tired, I guess. Isn’t it very interesting when you read your writing from days ago. Sometimes I honestly can’t remember writing what I wrote – like it was a dream. The capricious writer.

21 June, 2007 at 2:23 am

reginaI 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 TaoDear 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

perceptionof 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.)2 October, 2014 at 4:14 am

AshwinDear Terence Tao,

It happens that NOBODY KNOWS why there is something in this Universe rather than Nothing.How much does this astonishing fact about the existence of matter and the very presence of consciousness bother you?

I encourage you to study these things which definitely will be a great help to humankind!

Thank You.

15 July, 2015 at 7:47 pm

Martin CohenAs far as I am concerned, this does not matter. I am, you are, the universe is. We have to deal with it. I do not see any advantage in trying to know your “why” and fell that this would be a waste of time far better spent on actions that will positively affect us.

21 June, 2007 at 11:52 am

John ArmstrongIn 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 katzI 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

AnonymousI 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

AnonymousN.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 TaoDear 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

analogousto 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.]

10 April, 2012 at 10:37 pm

hcaiVery interesing view.

I am a biologist, I think mathematics has been modeling the physical world very successfully, but it still helps very little in biology. Though biology is also part of the physical world, but at the scale that biologists are studying it is not easy to make consistant abstractions. For example, if we study the behavior of the mice, you will never see two mice behave identically. So the only mathematical tool widely used in biology is statistics. But, you know, statistics has a lot of limitations. It is also abused in biology, in many cases it is wrong. So I am wondering if mathematician could develop new mathematics for biology.

29 November, 2012 at 6:21 pm

Alpha&OmegaMay biologists don’t use math, but they surely use logical reasoning, which is the true foundation of Nature and Reality.

22 July, 2014 at 4:34 pm

ChrisI can’t disagree more. This type of romantic view of the world ended a few decades ago. Logical reasoning is certainly one of the most important tool for exploring Nature, but it doesn’t mean that Nature and Reality can all be explained solely by that.

28 June, 2007 at 8:30 am

ShibiDear 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 TaoDear 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.

31 October, 2014 at 12:57 pm

AnonymousWonder at what age Terrance Teo learned about Ramanujan ?

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.

23 October, 2015 at 4:12 am

Alex KierkegaardThere is no such thing as talent. It’s how much you practice or immerse yourself into something. Given the right resources and education, one can develop interest and abilities over time. I have found this to be true in music which is a field I’m quite fond of- playing piano. As a kid, I used to think people who play piano are so smart or geniuses, or that they are born with some innate special quality. This is not true. Because I’m self taught and dedicated myself to something I truly enjoy. It’s not talent but sheer desire and practice that makes one good or great at any fields.

29 June, 2007 at 8:00 pm

Jonathan Vos PostOf 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

averageJOEI 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.

26 January, 2013 at 2:39 pm

StephenCreativity as it is in its effective and regular nomenclature happens to revolve around great storytelling, humor, artists and the like. There are actually vague ways of testing that too. The creativity you meant was the new ideas – like general relativity. Well, those ideas do tend to expose themselves uniquely when they are delivered by unique people right? Here – you used the word defy(in the example defy all tests), which used is already assuming that creative genius – what? – wasn’t just hard work; it’s just incredibly special. Or else maybe someone is naturally inclined over others in a smaller dimension/field/category, and one number couldn’t define their intellectual success in this lifetime. I believe g has meaning, and believe you can put a number on someone’s ability to reason, but that doesn’t mean someone who comes a little shorter in that regard cannot succeed in intellectual activity – in the sense that they may have something different to bring to the table. At least there isn’t a rigid and proportional correlation between intelligence and intellectual success. Take the resounding and ultimate Albert Einstein for instance. Psychologists estimate his IQ to be between 150 and 160, yet he is still considered the greatest mind to ever live. The IQ of 150 seems rather high, until I say that it is just over the 99.9th percentile. Not all that rare, is it? On a related topic, we all understand the indignation we feel when someone else eases through an incredibly hard concept, but it is a mere anomaly you are experiencing! It seems to you as if they come about their ideas magically. Well, that isn’t the case and you may possess some similar power, but that’s only probability that it happens as so; it’s because people are different and bring different and quasi-exceptional ideas. This doesn’t obliterate people’s ideas of the “genius gene.” There is of course is strong genetics, and since life is – like that moron said; emergent – everything in a way can rest upon genetics. That’s going off on a tangent. He doesn’t say everyone should rest their lives on mathematics – he’s not insisting you become a mathematician – he is disparaging modern idolatry, in a way. What should be glorified instead, is hard work. I’ll end on the audacious note of how I believe these ideas are tied into a fascist country like the USA. But I’ll end off even more audaciously by leaving myself exposed, not extricating myself with explanations and ultimately… not spamming everyone. However, I do think it’s funny you spelled innane instead of innate, because it’s the opposite of the idea you are trying to convey.

8 July, 2007 at 4:24 am

franckDear 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 ParkDear 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

kaimingI 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 TaoDear 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

dodNo 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

KristySeveral 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

26 January, 2013 at 1:44 pm

StephenJust… what are you doing!?

2 May, 2015 at 11:25 pm

KittyI hope that I am not the only one to have a giggle fit over this.

13 October, 2007 at 8:35 am

AnonymousHi 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

anonymousDEAR 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

mightone 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

AdamDear 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

gohI 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

pierrefor 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

pierreand 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 Armstrongpierre, 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 SandersI 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 PostI 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.”

16 September, 2009 at 2:41 am

AnonymousIf you really had an iq of 180 you wouldn’t be calling it rather high. If you got a score of 180 off the internet then don’t trust it, IQ is defined with a mean of 100 and a standard deviation of 15. That means you would have to be in the most intelligent 300 or so people in the world to have an IQ that high. Also IQ tests are silly anyway.

6 August, 2010 at 4:21 pm

...http://www.jimwestergren.com/greatest-nerd-of-all-times-jonathan-vos-post/

26 December, 2012 at 10:43 am

BlackacreSorry, I couldn’t hear anything you said over the sound of your ego barfing and the clatter of all the names being dropped.

30 December, 2007 at 11:47 am

Todd TrimbleAs 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 NambiToo 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

pierreAs 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 TrimblePierre: 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

moralof 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

pierreWhatever 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 ArmstrongCite? 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 TaoIt 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.

21 December, 2010 at 3:30 pm

Matt C.I would say that there is too much left-hemispheric criminality in mathematics from the existence of too many mediocre thinkers polluting it. Tao shows an honorable humility, but confuses the fact that g is very important. Obviously, smart people have the least ability to acknowledge that fact.

26 August, 2011 at 9:59 am

human mathematicsCan you give some examples of “left-hemispheric criminality” ?

31 December, 2007 at 11:29 am

pierreAnd 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 ArmstrongPierre: 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

increasethe score to compensate for the fact that the scaling has gotten harder.31 December, 2007 at 12:56 pm

DavidI 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

pierreI 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

RichardI 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 Armstrongpierre: 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 numbersonthat pagewere 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

RichardHere 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

pierrepierre: 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 ArmstrongAnd how do these tests reduce “creativity” to the same linear scale as the WISC or the WAIS?

1 January, 2008 at 7:10 am

pierreAnd 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 ArmstrongBut 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

claimto work.I return to my original question: just what is it that IQ measures?

22 October, 2015 at 3:16 pm

Alex KierkegaardIQ doesn’t measure true intelliegence, but academic gobbledygook.

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 intelligence. The term “IQ,” a translation of the German Intelligenz-Quotient, was coined by the German psychologist William Stern in 1912 as a proposed method of scoring early modern children’s intelligence tests such as those developed by Alfred Binet and Theodore Simon in the early 20th Century. Although the term “IQ” is still in common use, the scoring of modern IQ tests such as the Wechsler Adult Intelligence Scale is now based on a projection of the subject’s measured rank on the Gaussian bell curve with a center value (average IQ) of 100, and a standard deviation of 15 (different tests have various standard deviations, the Stanford-Binet IQ test has a standard deviation of 16).

[wikipedia]

What does an IQ test measure? The ability to take an IQ test. Not much more than that.

22 October, 2015 at 3:18 pm

Alex KierkegaardIQ doesn’t measure true intelligence but academic gobbledygook.

1 January, 2008 at 2:42 pm

t8m8rI tried several tests and the score was always improving. So my “intelligence” was improving with IQ test practice. If IQ test is supposed to measure some intrinsic quantity how can I improve my score? Or should I suppose the scores of the tests will converge to some value?

1 January, 2008 at 5:15 pm

Terence TaoDear David,

Mathematics may have been a particularly competitive activity in the days when Nash was a graduate student, but I believe the situation is quite different today, for a number of reasons. Firstly, mathematics is much larger in diameter now; even the best graduate student cannot hope to master all subfields of mathematics at once during his or her study, and so there is more room for everyone. Secondly, mathematics is a more collaborative activity now, in part due to improvements in communication technology, but also, I believe, because many problems now require expertise from multiple fields in order to make progress. Finally, mathematics nowadays tends to place more emphasis on cumulative activities (e.g. deepening the understanding of some mathematical structure or phenomenon) than on singular ones, such as racing to be the first to solve some famous mathematical problem (though this is of course still a significant motivator). In particular, a breakthrough to one problem tends to open up several more interesting questions to address, and so progress by a “competitor” on one’s research problems (or thesis, for that matter) often creates more opportunities for further development than it closes off; certainly this has been my own experience when involved in a simultaneous discovery. [In fact, I feel it preferable to work in an area that many other people are also active in, than to work in an isolated area that almost nobody else cares about.]

As I said in in my main article above, professional mathematics is not a sport. In particular, whether one is “better” than one’s peers is not really the right thing to focus on; the more important thing is to ensure that one can do good mathematics in one’s chosen research area.

.

1 January, 2008 at 10:57 pm

DavidTerry,

Thank you for the response. It was very good for me to get out of mathematics and into a career where I was not afraid of being valueless. It was also a great disappointment to give up and move to more trivial things, but hey, at least I didn’t kill myself :-) If I had seen math as a field where a few people make great contributions and many people make small but meaningful ones, I think I would have continued. I hope discussions like these affect how math is presented to students in the future.

2 January, 2008 at 2:17 am

pierreWhat does an IQ test measure? The ability to take an IQ test. Not much more than that.

It’s a pitty that I have not enough data with me to disapprove this.

I return to my original question: just what is it that IQ measures?

>>I think I have ansewerd the question. Now If you want me to give you a definition of intelligence, I can’t because I consider it as taken for granted.

It’s like when Terrence uses the words “genius”. He does not define this word.

It’s my last post on this article.

2 January, 2008 at 7:19 am

yasiru89A counter-example, however unelegant, helps- savantes display ‘genius-like’ traites(the word is intentionally used loosely so as to correspond with widely-accepted notions) but don’t fall in the high Intelligence Quotient category. As far as outlier scenarios are concerned this augments perfectly for the case of exceptional mathematicians and certain other innovators with average IQ(we’ll say 95-140 to encompass a reasonable majority).

The gist of it, in so many words is that attributes such as creativity cannot be measured(atleast practicably) owing to psychological as well as phisiological differences(the wiring of the brain, etc.). Nothing need be said for motivation and thought spared since a test is inherently a test and one’s life’s work is – one’s life’s work!

15 January, 2008 at 12:16 pm

he jianDear Terence,

I am refering to your comment of 27 June last year regarding the nature of mathematics. My view is similar to yours but there is one thing I can’t answer: Math indeed acts like a language in describing the physical world, but we have freedom in creating a language as long as it doesn’t generate an effective amount of inconsistency. English, Chinese or Hindi can all desribe the same object (for example, Chinese is very loose in logic), but we don’t have this freedom with mathematics. There seems to be just one math with its own rules which we can only work hard to discover yet without the freedom to alter.

Also, language does not have the predictive capability as mathematics does. For example, I can not tell ahead the outcome of an event because I can describe it in English, Hindi or Chinese.

What is your view regarding this phenonmenon? Thank you for answering.

15 January, 2008 at 12:49 pm

Hu JianDear He Jian:

I think the problem comes from the worlds the common language and the math as a language try to represent. The world the common language represents is a mind world which is highly free to human mind where the common language births from, basically the mind itself; The physical world is much less free to human mind where the math as a language births from. Common lanuage describes an unlimitedly free mind world which basically is itself; while math has to describe a world of another world. When common language describe one object, basically it describes the perception of this object in mind, — perception is part of the mind world itself whether acually it is consistent to the object or not. However, if you want to describe an object in math, you have to make sure it is consitent to the object in another world to the mind.

Would like to hear from Terence.

Thanks.

JH

17 January, 2008 at 7:59 am

SteveSounds right to me. I have always kind of thought that.

I can compare it with Masters swimming. While many Masters swimmers enjoy competing and trying to attain best in their class,

at the same time most of them feel, win or lose, they are already ahead in what they have accomplished for their health.

Desire, focus, and good influences.

17 January, 2008 at 9:08 am

gT. Tao, thank you for this post. It is interesting that your comments on “reasonable intelligence” – “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” – are very nearly identical to problem solving strategies G. Polya describes in his book “how to solve it”, and things Feynman talks about in “surely you’re joking…”.

Sometimes I look at the crazy formulae by Ramanujan (1/pi equals WHAT?!?), or a “magic rabbit” proof in anaylsis, or similar, and think “the guy who thought of this has some mystical powers of insight, I could never think of this”. I am very relieved that these simple strategies described by you and Polya and Feynman are ways that “great mathematicians” solve problems, since this is basically what I try to do.

22 January, 2008 at 11:00 pm

AnonymousI have a simple answer:

If one’s intelligence can be measured by a number at all, he must not be a genius.

23 January, 2008 at 7:37 am

Jonathan Vos PostSo “anonymous” has an intelligence measured by what– a Dirac matrix? A nilpotent grassmanian? A large cardinal? I think that this is semantic evasion of the word “measure” — which has at least a statistical meaning for any given test instrument. There are legitimate arguments about the dimensionality of the data gathered by intelligence testing. To give “anonymous” the benefit of the doubt, one might plausibly reason that there is no a priori reason to expect intelligence to be a scalar. See the discussions involving keyword “g” as this eikipedia page begins:

The general intelligence factor (abbreviated g) is a controversial construct used in the field of psychology (see also psychometrics) to quantify what is common to the scores of all intelligence tests.

An illustration of Spearman’s two-factor intelligence theory. Each small oval is a hypothetical mental test. The blue areas show the variance attributed to s, and the purple areas the variance attributed to g.

An illustration of Spearman’s two-factor intelligence theory. Each small oval is a hypothetical mental test. The blue areas show the variance attributed to s, and the purple areas the variance attributed to g.

Charles Spearman, early psychometrician, found that schoolchildren’s grades across seemingly unrelated subjects were positively correlated, and proposed that these correlations reflected the influence of a dominant factor, which he termed g for “general” intelligence. He developed a model where all variation in intelligence test scores can be explained by two factors. The first is the factor specific to an individual mental task: the individual abilities that would make a person more skilled at one cognitive task than another. The second is g, a general factor that governs performance on all cognitive tasks. Spearman’s theory proved too simple, however, as it ignored group factors in test scores (corresponding to broad abilities such as spatial visualization, memory and verbal ability) that may also be found through factor analysis….

23 January, 2008 at 10:33 am

AnonymousQuoting from Terence’s “quote” page:

“When the only tool one owns is a hammer, everything begins to resemble a nail.”

IQ is such a hammer, it is irelavent if it is a number or a Dirac matrix…

25 January, 2008 at 10:10 pm

Allen KnutsonI think the marketplace these days favours mathematics which has some interdisciplinary componentBert Kostant’s dictum on this is “People who build bridges are pushed off them.” He refers to it as the (Peter) Lax problem.

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.One of the many standard realizations that every math grad student achieves on their way to a PhD is just how easy, nay automatic, it is that one become the world expert on some subsubsubsubfield of mathematics. The

realdanger is that that subsubsubsubfield can’t be made to sound interesting to other mathematicians; help in avoiding this danger is one of the key reasons to have a thesis advisor.29 January, 2008 at 12:51 pm

Bala AturHow much does age matter in this context? Is there a general decline that happens with age or is it possible to be as creative and successful (however you define both attributes) at 65 as at 25? Jordan Ellenberg wrote an article in Slate some years back that Mathematics was no longer a young man’s game i.e. that it took a long time to master all of the mathematics required to make an original contribution – at least that’s my understanding of what he was saying.

Say I want to enter a Ph.D. program in Math or a related field at the age of 50. Is that a good idea ? There are life-stage issues that probably would weigh a lot, but, from a sheer intellectual horse-power perspective – does it matter how old you are? Are there any statistics about folks starting in Math at a relatively older age?

29 January, 2008 at 2:51 pm

RobertaI don’t think there’s an age limitation on doing what you’d like to.

29 January, 2008 at 6:55 pm

AnonymousSay I want to enter a Ph.D. program in Math or a related field at theage of 50. Is that a good idea? There are life-stage issues that probably would weigh a lot, but, from a sheer intellectual horse-power perspective – does it matter how old you are?

No, it doesn’t matter, because you aren’t competing with your younger self. Hypothetically, maybe you would have done even better work at age 25. That’s unknowable now and therefore irrelevant. The relevant question is whether, at age 50, you can accomplish enough to feel satisfied with yourself and to get a job you are happy with. Age is not itself a big obstacle to this (plenty of mathematicians do excellent work in their 50’s), and individual variance between people is overwhelmingly more important than any systematic age effects.

Aside from the life-stage issues you mention, the main thing I would wonder about is age discrimination. Would a 55-year-old Ph.D. recipient be judged fairly in comparison with younger job candidates? I hope so, but I’ve never seen such an applicant in any job search I’ve been involved with, so I have no direct experience with how it would be perceived.

Are there any statistics about folks starting in Math at a relatively older age?

Probably not, because it’s not very frequent. In any case, selection effects would make the results almost meaningless. If people who take up mathematics at age 50 do amazingly well, maybe that’s just because only exceptional people dare to try. If they don’t do well, maybe it’s because only desperate people try. Either way, it’s not clear how much you will learn about your own situation.

A randomized trial could be illuminating, but I can’t think of a good natural experiment (and of course there’s no ethical alternative). War is the best experiment I can think of offhand, but it doesn’t delay anyone’s career for more than five years.

8 June, 2011 at 11:52 am

CrocAmazingly clearheaded view on age and its relation to research productivity. Thanks for posting.

30 January, 2008 at 10:10 am

AnonymousImagine that Guass or Tao never came across math until 50, they would still do better than average 50 year-old mathematicians who started at 18, don’t you think!!

It is all relative.

30 January, 2008 at 3:18 pm

t8m8rSchwartzschildt found his solution to the einstein equation when he was around 50 at a war front.

31 January, 2008 at 8:49 pm

jisungThis comments make me hopeful.

Thanks tao.

I’ll have to try.

1 February, 2008 at 9:14 am

Jonathan Vos PostMath over 50? Yes!

I was almost ruined twice by the destructive Romantic myth that Math is for the young geniuses. That, if you haven’t made a major contribution as a teenager, or your early 20s, you never would.

That hurt me as a child semi-prodigy, when I taught myself Calculus through differential equations in the summer after 7th grade (i.e. by age 13) from a college textbook, without any actual mentor (my parents being summa cum laude from Harvard and NorthWestern, but in English Lit, and knowing no Math beyond, technically, a little spherical trig my Dad used in teaching navigation while a flight instructor in World War II).

It hurt me again when I left for Caltech at age 16, hoping to do Mathematical Physics with Feynman. I did test out of some Freshman Math courses at Caltech and start right off taking some graduate courses. Why? Because I had relied on insight and raw intelligence, and had no discipline in Math, and no idea how ignorant I really was. Caltech’s Math department tolerated my uneven work, with flashes of quality (and some A and A+ grades) combines with stupid inability to grasp what other students could get quickly.

I then went into Computer Science specializing in Cybernetics and Artificial Intelligence (M.S. 1975, for a datastructrure and algorithm for parallel automated theorem proving on massively parallel processors which did not yet exist). PhD research in what I called “Molecular Cybernetics” — arguably the world’s first PhD dissertation in what was later called Nanotechnology and Artificial Life. My PhD is an “All But Degree” — for political reasons neither accepted nor rejected, but shown on my transcript as an “incomplete.”

Without the “union card”: I had little choice but to go into industry, where is did essentially Applied Math and CS for 20 years for Boeing, Burroughs, European Space Agency, Federal Aviation Administration, Ford, General Motors, Hughes, JPL, Lear Astronics, NASA, Systems Development Corporation, U.S. Army, U.S. Navy, U.S. Air Force, Venture Technologies, Yamaha.

I assumed (wrongly!) that I was too old for Math, and had blown my chances, and should focus on making money and/or doing fun work and/or mastering Management and/or publishing Science Fiction.

Then, suddenly, spontaneously, at age 52, I began to dream equations, wake up and write them down, see that they were true, and start writing papers. I applied this discipline. An average of once per day, set yourself a problem from a math journal, arXiv, online source, or wacky playfulness. Solve it completely or up to partial results + conjecture. Link it to paper and online literature. Polish it, annotate it, and submit it to an edited online venue such as the Online Encyclopedia of Integer Sequences (hosted by AT&T Research Labs, and edited by Dr. Neil J. A. Sloane and a couple of dozen distinguished associate editors), as I’ve done 1,864 times. Or the Prime Curios web site, as I’ve done 225 times. Or write a definition for MathWorld. Or coauthor a paper for a conference or arXiv. I repeat: EVERY day.

Now I am, again, a Mathematician, going farther than I could as a pseeduo-prodigy, and with the benefits of maturity and discipline and collegiality.

My bias is thus that I am both older (a career in aerospace, start-up high-tech company executive management, and other areas), and young again. As a revitalized student of Mathematics, with the cunning of middle age, and the discipline that I lacked when young, I learn joyfully.

I also have enjotyed teaching over 2,000 students as an adjunct professor of elementary Math (algebra, geometry, the Mathematical Physics in an Astronomy Lab), and in Elderhostel classes, and as a hideously underpaid overworked substitute Math teacher to inner city teenagers facing drugs, gangs, and failure.

I am not advocating what others should do, merely saying why I mildly disagree with a “math is for the young” philosophy. This also comes from my 42 years of working with computer software. I think it better to copiously comment one’s programming source code. It is agony (albeit sometime highly paid agony) to figure out what a programmer meant in legacy code, after the programmer has retired decades ago. I’ve had to reverse-engineer the “Day-Of-Launch” code that Rockwell used for NASA in deciding whether or not it was safe to launch the Space Shuttle. Then had to testify to Columbia Accident Investigation Board.

Math is not usually life or death; but it can be in mission-critical engineering. Math is somehow simultaneously about nothing but itself [Formalism], about the universe [Realism], about a deeper and more fundamental universe [Platonic Idealism], and about our own personal quest for self-understanding and wisdom.

When interviewed several years ago, Paul R. Halmos (1916-2006) was asked: What

is mathematics to you? He responded: “It is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing — one great, glorious thing.” A few years later, he was asked about the best part of being a mathematician. He said: “I’m not a religious man, but it’s almost like being in touch with God when you’re thinking about mathematics.”

8 June, 2011 at 2:31 pm

Gandhi ViswanathanThank you for this fantastic post.

3 February, 2008 at 7:14 am

Pacha NambiI liked reading Jonathan Vos Post’s blog about math over 50. I don’t think good mathematics can be done only by someone who is in the teens or early 20’s. Take the example of Paul Erdos. He did very good work till he died in his 70’s. He also collaborated with amazing number of mathematicians from all over the world. He can be a great source of inspiration to anyone thinking about becoming a mathematician.

Doing good work is all that matters in the long run. Age is irrelevant.

3 February, 2008 at 8:36 am

Thomas RiepeHere an article on aging and the brain:

http://tinyurl.com/yuf2ah

3 February, 2008 at 9:04 am

Thomas Rieperelated:

http://bldgblog.blogspot.com/2008/02/growing-old-in-age-of-lead.html

26 February, 2008 at 1:43 am

Thomas Riepean other related link:

“Hard graft, not genes, creates musical genius”

http://tinyurl.com/33e9l9

(exist similar studies on math skills?)

1 April, 2008 at 6:03 pm

做数学一定要是天才吗？ （译自 陶哲轩 博客） « Liuxiaochuan’s Weblog[…] 做数学一定要是天才吗？ （译自 陶哲轩 博客） March 30, 2008 — liuxiaochuan （原文：Does one have to be a genius to do maths?） […]

16 May, 2008 at 8:39 pm

Jacob FreezeThe idea that mathematics is a field where many people make small but

significantcontributions is a myth maintained by assistant professors who make small butinsignificantcontributions.If this myth disappeared, 10,000 tenure-track professorships would disappear along with it, so there’s always a thread somewhere on the net where

le menu peupleof the universities argue that their obscure research outweighs the role of genius, and it really isn’t an argument at all…It’s just the sound of

petit bourgeoisacademics pleading for their jobs.16 May, 2008 at 10:23 pm

Terence TaoDear Jacob,

In my experience, every significant advance in modern mathematical research is in fact built upon dozens of previous results, insights, folklore, or experiences, a good fraction of which would have only been deemed of minor interest in their own right, and/or originate from junior faculty who are mostly only known to other experts in their field. The situation is perhaps analogous to that of a punctuated equilibrium; the period between dramatic breakthroughs is often marked by steady but mundane progress, which slowly digests and clarifies the previous breakthroughs and isolates the key obstructions to further progress that researchers then focus on to produce the next breakthrough. Both phases of this process are important for advancing the field.

This is not to say that every minor result is of equal importance, of course; the mathematical taste of the authors involved makes a fair amount of difference. For instance, generalising a previous result for its own sake may not be terribly productive, but generalising a result in order to test how the proof technique of that result interacts with an interesting phenomenon, example, or obstruction that it has not had to deal with before, or to reduce the reliance on some overly restrictive hypothesis, can be ultimately be rather valuable. Similarly, compressing the proof of a known result into a smaller space for the sake of brevity alone may not be of too much value, but finding new proofs that deliberately avoid a specific class of techniques (with the future intent of extending the result to cases in which those techniques break down), or which are more consistent with some newly discovered paradigm in that field (or a related field), can be very instructive. Even failures – the inability to extend proof method X to problem Y, for instance – can point out some previously under-appreciated obstructions or limitations in current theory.

In short, there are many useful contributions to mathematics beyond the most dramatic ones. One does not need to be some sort of genius to be able to provide these contributions; but it does help to have certain amount of experience and awareness as to what real mathematical progress is.

4 November, 2014 at 5:59 am

AnonymousJohn Nash is a genius unlike those work on number theories as John was not prepared to follow others and set about finding out for himself from ground and climbed to Mt Everest of mathematics in his twenties where as in number theory say prime numbers it was already explored ,that means whoever worked on had the base camp already in place say at 25000 ft and only last leg to be climbed risking his health in pursuit of further advancement. Gre. Perolman refused field medal as he made only a minor contributions compared to those who worked on Poincare conjecture.He is a true genius for not seeking publicity and not to wastes time on this topic and get on with research instead.

17 May, 2008 at 2:04 am

DmouthI’ve always found it frustrating and bewildering when prodigies like you or Norbert Weiner downplay the importance of profound and powerful native ability.

As someone with just enough ability to realize his insuperable shortcomings, I wish I could believe them. Really…I have since childhood. But an honest if rueful look at reality compels me to think otherwise.

I will never be a Ramanujan, Von Neumann, or Tao; and they will always be able to do things that I could never do, no matter how diligently I work. Period. And that goes for anyone with appreciably higher Gf than me.

To say that a high IQ (or hard work, for that matter) is not a sufficient condition for professional mathematical success, is to state something trivial. And it’s just as obvious that a hard working genius will achieve more than his lazy counterpart.

Psychometrically speaking, genius occurs approx. around an IQ level where 145<IQ, which is 3 standard deviations above the mean of 100. With this threshold in mind, I’m confident in saying that to achieve significant success in modern day mathematics, genius is sine qua non. It’s not PC and it might not be the thing to say that gets the most people involved in math, but it’s the truth.

Prof. Tao, you mentioned your extensive work with graduate students informing your opinions in this discussion. Might I suggest that your sample set is decidedly atypical?

Surely amongst the supremely talented math grad students working with a Fields Medal winner like yourself, hard work or experience might account for a fair amount of the variance in the success of any single paper or problem.

But as I see it, you’re taking a set of high-IQ individuals and remarking that amongst them you see hard work and experience being better explanatory factors for success. And I agree with you. But does this conclusion hold for a more heterogenous population?

I’d be willing to bet you a tidy sum that they possess, as a group, a level of fluid g far greater than the average population. Seriously, if you were to give your students the RAPM (that you yourself took and did extremely well on as a child), I would be *astounded* if they all did not achieve genius or ceiling level scores.

That said, IQ, like much else in life, seems to suffer from diminishing returns. There’s a certain threshold that is required to perform copacetically in a certain field. (Math seems to set the bar the highest). And beyond that, the importance of other variables (persistence, expertise, providence, creativity, etc) becomes increasingly significant.

For many of us–considering the circles we move in, the schools we were blessed enough to attend, the books we read, etc–I think it’s very hard to fully appreciate the stupidity of many members in any significantly large population. Truly, I don’t mean to sound pompous. Only to merely point out that half are below average by definition.

Modern education is suffused with sieves–standardized tests, proxies for intelligence tests. The vast winnowing process that takes place at modern elite universities is incredible, if you sit back and think about it. If this is the milieu you move in, I think it’s all-to-easy to fail to appreciate the intellectual handicaps most people labor under everyday.

With all due respect, I think it analgous when one of the most exceptional mathematical prodigies of all time tells us genius is not required to do mathematics. If you could roll back the clock to when you were 4 years old you might have a notion of the average level of mathematical sophistication out there.

To be sure, you’ve worked hard to hone and refine your skills; I don’t want to deprive you of that. But, cognitively speaking, you were born with a Ferrari, have taken good care of it, suped it up, and added some post-production performance parts.

The rest of us our riding in governored U-Hauls.

9 November, 2014 at 8:42 pm

Ngoc NguyenI agree with Dmouth. It’s easy to be humble and gracious when you are so enormously brilliant. But would you continue to be of the same viewpoint if suddenly–due to stroke or some other freak trauma to your higher cognitive processes as the result of an accident (i.e., automobile collision?)–you suffer severe brain damage which, after weeks or months of recovery, you no longer have access to your 230 ratio IQ and your doctors and experts now estimate that you effectively function at an average adult IQ of 115 (the mean of all college graduates in this country). So what now? Will you be able to perform as a mathematician as you have before this tragedy? I really doubt it. But most of all, would you still believe and stand by the false hope that you are feeding most of your readers here? Don’t get us wrong. We don’t doubt your well-meaning message that just about anyone who can graduate from a four-year college can become a mathematician if they worked hard at it. But statistically almost none of them will be able to perform at the level you have as a mathematician, and I get the feeling that your audience here all secretly dream of becoming another “Terence Tao, prodigy” because–honestly–who wants to get into any field or calling (like mathematics) just to be mediocre at it, or, in other words, unremarkable and undistinguished at it? It seems to me that most students who enter into a highly creative and potentially rewarding field like mathematics do so in order to make a significant contribution, that is, to distinguish themselves from their peers and also make a name for themselves as a professional mathematician–as you have so outstandingly done. Unless I am patently wrong, that is the exception…and not the rule. Not everyone can be you, or, for example, a Grigori Perelman, the guy who solved the Poincare Conjecture. Whatever that elusive quality we call “genius” is or however it’s defined you either have it or you don’t. And people like you–and Grigori Perelman–have it. The rest of us do not, so we have to struggle diligently and apply ourselves and compensate for it in other ways–but no matter what no amount of hard work or genuine passion can make up for the difference and allow us to catch up to your level of super-profound, mathematical ability which we average human beings call “genius” for lack of a better word, as we don’t share your rarefied cognitive “frame-of-reference,” as most of us are statistically closer to the mean of the Bell Curve in terms of native ability. When you have outlier ability it is much easier to do outlier work (as a mathematician).

This is starting to get longer than I intended, so I will end on this point. Take, for example, Carl Gauss, “The Prince of Mathematicians,” the greatest mathematician since antiquity. As a professor at the University of Göttingen he rarely, if ever, taught classes to the students because his gargantuan brilliance could not tolerate the extreme boredom and ennui that overtook him every time he had to endure teaching students of almost no math ability or perceptible promise at all. Consequently, he avoided teaching maths to students altogether. However, there were some exceptions for those who showed mathematical talent and/or promise. One of these was Bernhard Riemann, a student who fulfilled his doctoral thesis under Gauss. It is noteworthy that Gauss was only too happy to give his student Bernhard Riemann the time of day because he immediately recognized this young man’s genius for mathematics and his exceptional promise as a mathematician. (Gauss was not disappointed by his likewise brilliant student. As a result, we today owe Riemann a debt for the legacy of the Riemann Hypothesis, arguably the most famous unsolved problem in number theory.)

22 October, 2015 at 4:41 pm

Alex KierkegaardDear Ngoc,

I dont think you don’t have a be a genius or be very intelligent to be good or great at something that matters to you. The truth is that no one is born a genius or having some supernatural innate quality that distinguishes them from others. Imagine if you had no access to books or any musical instruments as a kid. Do you think you can be a genius at anything even if you think you are super smart or have some special innate quality then? Thus, the environment and resources plays a significant role in the development of one’s ability. One can have mediocre scores and be brilliant at something later on if one tries hard enough. Comparisons are insidious. For instance, Amy Tan got mediocre Verbal SAT scores, I believe it’s like in the 400s. But look at where she is now. I would be careful not to compare with anyone else’s achievements and focus on what you can do with your abilities.

30 April, 2014 at 3:22 am

AnonymousDmouth – I could not agree with you more (except for the operator < usage). Thank you for this reality check :)

17 May, 2008 at 3:10 am

John R RamsdenThere are so many angles and offshoots to this fascinating discussion. For example, bearing in mind Dmouth’s lament (previous reply), there’s no denying that raw talent gets you through Thomas Edison’s “99% perspiration” stage *quicker*.

With scholarships and top degrees, and what not, it also gets you into academic environments optimized for nurturing your talent, with mentors and others of comparable abilities. So in that sense “talent builds on talent”, just as “money makes money”.

But if you’re single-minded and patient, and learn your subject thoroughly and in good order, there’s no reason why you can’t reach the coal face in due course, even if it takes rather longer, and the Internet is a huge help with this (as Tao’s blog and others like it attest!)

The great historian Lord Macaulay wrote something like (from memory, as I can’t find the perishing quote online!) “No one ever wrote any great and noble work except in a language they had forgotten when or how they learned”, meaning that to have a stroke of genius one must have first made the underlying knowledge or prowess part of oneself.

Again, as Darwin wrote, modestly using a word other than “genius” for his own achievements, “Luck favours the prepared mind”.

On a related subject, again harking back to Dmouth’s reply, I sometimes wonder if people always find their real metier, any more than they always meet a true soul mate.

They say “genius will out”, and a compelling interest and talent will find expression. But nature and society can be wasteful – I mean just on the basis of probability there must be hordes of Ramanujans toiling in rice fields or working as station clerks all their days, and there must be far more people nearer their vocation but still for one reason or another knowingly or otherwise never making the connection. (Think of how many famous mathematicians almost took another fork in life, and in that event would probably never have contributed as they did or at all.)

17 May, 2008 at 3:16 am

AnonymousOh Dmouth, that’s just sad, I don’t understand any reason for being so bitter. You might read Paul Halmos’s biography “I Want To Be A Mathematician” sometime. Halmos did lots of good significant math, wrote important textbooks, taught lots of students who became important researchers or teachers themselves, and served a stint as math dept. chairman of a huge state university where he did the hiring and administrative stuff to reorganize their calculus program and improved the calculus instruction for 1000’s of non-math (i.e. science and engineering) students every year (resulting in safer airplanes for everyone, or whatever), etc. He had a very rewarding and worthwhile mathematical career and was a highly respected figure. But with no disrespect intended, he wasn’t a genius, he wasn’t going to be the guy to solve the Poincare conjecture, but rather, it’s clear from the book that he was like most of the rest of us who had some reasonable amount of math aptitude and liked the subject and worked hard at it. I think most of us could have pursued a life like that if we felt math was important and urgent enough that we could devote a lot of mental energy to it constantly, even if our raw talent level was not that great. (In my case, I felt that way myself for a while, but eventually that undergraduate idealism wore off and my interest flagged, so I didn’t stay with it).

It’s the same thing with musicians–I know lots of respectably successful ones who aren’t going to become mega-stars, but they do what they do because they like it and it keeps them excited and they are able to put energy into it, which leads to doing worthwhile stuff automatically. That’s the main thing. If it’s just a boring assembly-line job, why pursue it in the first place? There are other opportunities that are easier and pay better, I’m sure.

You might also look over on Scott Aaronson’s blog where he and his readers talk about this sort of topic sometimes.

17 May, 2008 at 5:10 am

Jacob FreezeTerry: Thanks for replying.

Without getting involved in silly questions about tests mainly designed to make distinctions among the “lower orders,” when I talk about “genius” in mathematics I’m usually thinking about Riemann, and if I have to think about more than one of those monsters at a time, the concept doesn’t get too blurry by extension to Grothendieck, Poincaré, and even that most freakish of all one-hit mathematical wonders, Galois.

Why them? Why not Gauss, Archimedes and Euler, or the rest of anyone’s top ten list of great mathematicians? Let’s just assume that you see a common factor in my peculiar little set, and notice that for three of the four of them life went horribly wrong.

God only knows if Grothendieck is still alive after a couple of decades as a wandering saint or lunatic, Riemann more or less starved, Galois got shot, and only Poincaré had the good fortune of being born into such a rich, famous, and brilliant family that he could have

gibberedhis way through life without suffering much harm. Other plausible members of this micro-fraternity also paid a crazy price for their glorious originality and depth of vision: Newton turned into a paranoid civil servant with alchemy for a hobby, and as for André Weil… His sister paid for both of them.If it were in my power, I would constantly multiply the number of endowed professorships so that every newly minted Ph.D would get one, and even if an economic catastrophe forces us to relocate a fraction of the bourgeoisie from their current home on

easy streetinto the hard world of physical labor, I hope that every MBA on Wall Street has already dug a few ditches before the first assistant professor of mathematics gets his or her hands dirty.So the animus of my previous post wasn’t really directed at the industrious tribe that creeps or even occasionally leaps toward a settlement of Erdos’ conjecture about arithmetic progressions or some other trendy mathematical question, in the relative security of universities from Los Angeles to Australia.

But however pleasant it may be to pretend that a smooth continuum connects all those comfortable academic bureaucrats to Riemann and Galois and Grothendieck, that pretension also insults the incomprehensible affliction of those great individuals, and like every other insult to

suffering,it deserves a harsh reply.17 May, 2008 at 8:32 am

NirmanDear Prof. Tao,

suppose I were to generalize this question to “Does one have to be a genius to do X” what would your views be? If I would put X to be music or math or physics most people would answer YES. If I would put X to be school math, elementary geometry or high school physics most people would say NO.

Let me digress in a slightly different direction for a while. I will generalize in a different dimension, what if we were to replace “genius”, which is a state of mental strength to say physical strength and X to be some task which strong men have done. Say : Does one have to be like XYZ to be able to beat Mohammad Ali at Boxing?

I think, for most of these questions there is a tipping point. If X is harder or beyond that tipping point, hard work cannot possibly get one there. No matter how hard I try, I am not going to be able to beat Mohammad Ali at boxing. Its easy .. he probably weights a 100 pounds more, he is taller and so on. I could work hard to make myself stronger but could I make myself taller? If not, then I possibly could not increase my weight beyond a point still keeping healthy. Long story short, there are some things entirely beyond my capacity. This is not a proof, its empirical proof. I have tried to become stronger, run faster. It has brought me fruit and improvement in both directions. But I do not see myself becoming Ali by any amount of hard work. In fact if I were to work too hard, I feel the toll on it making me weaker.

So why does not the same apply to mental efforts?

Gauss was known to have computed the sum of a series, by using a clever trick, at 3 years of age. That shows the spark of genius he had. I am pretty sure all the mathematical geniuses referred to in earlier comments have had some such story associated with them, which I am not aware of.

If I had to go from California to New york, I could go running or drive or fly. Either of them works but the amount of hard work and time I would need in the three cases would be different. Most people, if they were to keep working diligently at math, keep themselves inspired somehow despite years of failure would probably discover as much as Gauss, but it could take up to 100s of years doing that, which is more than their lifetime.

The above rant I think merely proves that in order to do math comparable to what Gauss, Riemann, Ramanujan or Tao have done, one needs to have a spark of genius. However, I think the main question being discussed here is – “Does one have to be a genius to do _some_ math”. For that I agree with Professor Tao and several other commenters. With enough hard work one can get to a point where he/she can start doing some research in Math. By a stroke of luck, or working with a genius they may even get credit for a major result.

But I think, a simple change to the question – “Does one have to be a genius to do GREAT Math” easily has “Yes” as the answer.

17 May, 2008 at 6:25 pm

He JianJust some random reflections to share:

1.) I have noticed that nearly all of the greatest mathematicians are humble and their humility seems honest. I think there must be some truth in their opinions of themselves. The particular quality of “genius” mentioned here exists more on a physilogical level, which is significant but isn’t the core of genius. True, Ali is tall and big and fast, but Bruce Lee is only fast, but Lee has unusaul mental energy. I think if genius truly exists, it exists more on a spiritual level: the will. Gauss does math like Van Gogh paints a canvas. It isn’t that Gauss WANTS to do math, but he WILLS it out, like a woman giving birth to life, same with Mozart writes a piece. Quality difference between works by “genius” and by the lesser lies in the difference between Works willed out and works thought out. The former is truthful (thus universal), the later factual (thus trivial), the former spiritual (thus original), the later intellectual (thus consequential). Therefore genius is more of the quality of heart. I know amazing former artistic prodigies (one was 5 years old, the other a Downs-syndrome. They were true extraordinary artists, not celebrated fakes like Warhol or Picasso, by the way ), they are now about 30, are ordinary artists now. To my observation, it is a consequence of the loss of innocence but the decline of physical skills.

In fact, most of the above comments show “I WANT to do great math” attitude, yet unfortunately, there is precisely where genius escapes.

2. Forgive me, I think math and arts and their geniuses are overrated, because human culture is in general overrrated. If we see them as usual, then much of the puzzle fades away. I think the mystery of genius is more about people’s mentality towards it (admiration, envy, puzzlement, frustration… etc.), somewhat like the mystery of the beauty of a woman is more about people’s reaction to her: men’s desire, women’s envy. etc. Beauty itself is no mystery. In fact. in nature, beauty is everywhere. I am of the opinion that the ROLE of genius is the product of culture though most people believe the reverse is true. Lots of folk song singers in China are as good as Mozart, since they don’t ORGANIZE their abilities, they are just beauties in nature but “geniuses” in society. Civilizing only means organizing.

3. Hope western readers forgive me here since I think the oddity of the role and concept of genius is more typical in western culture, which in my humble opinion is unhealthy since it has misled a lot of young people into pretentsion of Van Gogh or Einstein.

18 May, 2008 at 3:09 pm

DmouthHaha, oh, I’m certainly not bitter! I’m a quite buoyant realist, content with his abilities ;) Thanks for the book and blog recommendations. I’ve enjoyed what I’ve read so far!

I think a fair amount of the disagreement is a result of the ambiguity of the term genius. Intelligence is defined relative to a certain population. And it seems genius has been relativized here to a significant extent. So that only the genius among geniuses of geniuses is recognized as such. Part of my point was that when you spend your time at elite universities or cognitively demanding jobs, surrounded by super-high performers, it can distort your perspective of the larger landscape.

So if by “genius” we exclude all individuals so as to leave only the Von Neumann’s, Erdos’, or Sidis’ of the world, then I’d agree and think it uncontroversial to say genius is not required to do good math.

20 May, 2008 at 11:48 pm

Jacob FreezeThe

petit bourgeoisstyle is so pervasive in academia that it’s like the air everyone breathes… you can’t see it, you can’t taste it, you can’t smell it… and it’s evenmoreintangible than the air, because most of the creatures immersed in it don’t even know it’s there.So when I claim that the significance of minor-league math research is primarily an economic question, because thousands of jobs depend on that particular myth, and the wrong answer would turn the normally mild-mannered

petite bourgeoisieof academia into an angry mob, just like any other mob whose livelihood is threatened…The economic aspect of the question is so far out of sight and out of mind that even a bright guy like Terry Tao simply passes over the possibility that when he defends the significance of “junior faculty” research, he’s simply reciting the credo of his class, an article of faith essential to its very existence, on a par with the

Shmah:“Hear O Israel, the Lord is our God, the Lord is One.”The fundamental principle of academic employment isn’t any more susceptible to free debate in a university than the existence of God is susceptible to free debate in a Temple.

This isn’t to say that I didn’t appreciate Terry Tao’s reply to my original post, not so much for the relatively obvious apologetics as for the first-rate English prose. It reminded me that verbal scores are a better predictor of future mathematical eminence than math scores on the GRE, with the appropriate restriction to applicants at top-tier graduate schools. It would be easy enough to recognize the best mathematician on this thread, even if you had never heard of the Fields Medal.

This same distinction also applies to unusual personalities like Richard Borcherds. His range of expression is expectably restricted, but within that range it’s word-perfect, and I doubt that even a brilliantly exacting editor like the late Barbara Epstein at the New York Review of Books could have found much room for improvement in the recent discussion of higher-order arithmetics between Borcherds and Terry Tao.

27 May, 2008 at 7:28 am

Genius, Sustained Effort, and Passion « Apperceptual[…] seems to be a growing consensus that genius is all about hard work; really hard work, for at least a decade. What can motivate that […]

1 June, 2008 at 9:41 am

Jacob FreezeTerry Tao begins this discussion with a quote from José Ortega y Gasset, and it’s hard to imagine a more ridiculously inappropriate headliner for an essay defending the dignity of “junior faculty” in mathematics. This is not to say that you can’t find a few passages in

La rebelión de las masaswhich reflect Professor Tao’s argument like the hallucinatory images of passers-by in a fun-house mirror:“Science has progressed thanks in great part to the work of men astoundingly mediocre, and even less than mediocre. That is to say, modern science, the root and symbol of our actual civilisation, finds a place for the intellectually commonplace man and allows him to work therein with success.”

In spite of his superficial agreement with Professor Tao, something has obviously gone horribly wrong with Ortega y Gasset’s

attitudeabout the common soldiers in the army of science.“Previously, men could be divided simply into the learned and the ignorant, those more or less the one, and those more or less the other. But your specialist cannot be brought in under either of these two categories. He is not learned, for he is formally ignorant of all that does not enter into his specialty; but neither is he ignorant, because he is ‘a scientist’, and ‘knows’ very well his own tiny portion of the universe. We shall have to say that he is a learned ignoramus…”

Terry Tao’s junior faculty are exactly the

señoritos satisfechosthat Ortega y Gasset despised more than any other class… These are the “self-satisfied little gentlemen” whose narrow technical education supposedly endows theirex nihilodiscussion of concepts like “genius” with a mysterious respectability, and Professor Tao’s ludicrously inappropriate citation of Ortega y Gasset unintentionally confirms the cultural nullity of the scientific “mass men” who are pilloried inLa rebelión de las masas.28 December, 2010 at 2:36 am

MJacob thanks for the comments here. I really enjoyed reading them. It was worth reading the platitudes of the rest to find yours. However, by this I don’t mean that what you said was completely agreeable to me; You make a very good point, but like all people who have a real point you emphasize it a bit much that helps for the exposition but ate the expense of deviating a little far away from the truth. (I hate using the word truth.)

As you know the junior faculty also contribute a lot to mathematics but in a different way. Since I am an undergraduate heading toward graduate school who is statistically very likely to end up becoming one of these junior faculties, I’d feel obliged to give my best shot to defend the institution of junior faculty and the like.

The junior faculties, they teach and spread the knowledge, they set up seminars, restate the results, polish proofs and publish it as new and etc. They find slight variations, make slight improvements. All of these activities are quite necessary, and hence I think worth doing. More importantly, the institution of Junior faculty positions and not so good mathematicianship is necessary to produce Terry Tao’s. When graduate schools are admitting students there is no precise way of knowing who among the admitted student is going to be the one who contributes most significantly to mathematics. You need to admit many in order to find the Terry Tao’s. And to be able to admit many, there must be some job, some position available for these people for after when they graduate. So in short, the system in order to produce the great, must be able to accommodate the mediocre as well due to the inherent uncertainty for detecting the great.

14 June, 2008 at 11:39 am

这等牛人也在wordpress上写blog！ « Just For Fun[…] 里面有他的research，lecture notes，math related advice等等，内容很丰富，他和读者的交流也是挺充分的。这里转一篇他的文章”Does on have to be a genius to do maths?” [原文链接]，有个人翻译了这篇文章在[这里]. […]

15 June, 2008 at 12:26 pm

kyI have a question in general that I would like to hear others’ comments.

I read an article by John Nash that he believed following the progress of the research in trend and immersing oneself in the current too much destroys his/her creative and innovative sense as a mathematician (or any artist in general). It was the excuse he made for not going to any of the class in his Princeton days. So on one hand, if one wishes to maintain creative thought and unthinkable imaginations from the normal standard, he/she may wish to refrain from following others’ research too much.

On the other hand, of course, for anyone to start to investigate a new research topic, one has to understand what has been studied so far thoroughly by voluminous reading and writing perhaps a survey of the topic and then trying to solve some conjectures that have been raised by others already.

Any thought on this regard? I suppose John Nash was one of the most peculiar and original kind but I have read many other accounts on which the so-called “genius” was always daydreaming in his own original thoughts rather than reading others’ works day and night, say Einstein. It also depends on whether a researcher wishes to become a “genius” type with one unique finding that is absolutely unique or a typical scholar who produces hundreds of “little” but certainly admirable work, I guess. Or should most of us realize that we are not “geniuses” and simply follow the path of the latter, the typical scholar, and keep ourselves busy by reading others’ works most of the time?

22 June, 2008 at 1:47 pm

Some related thoughts on intelligence « The Mendicant Bug[…] Does one have to be a genius to do maths? […]

27 June, 2008 at 9:53 am

Passer-byThe question is rather empty. If all one means by genius is an operational definition comprising three-digit values, then by no means is it necessary. And another problem: what is maths, anyway? There are many varieties and levels in which mathematics and mathematical thinking consist. If you take math to be mere arithmetic (and this does occur in some quarters), then clearly the answer is even a monkey can mechanically add the values of 13 and 24 to reach the sum of 37. A problem arises when one subscribes to different criteria to localize a possible meaning of these terms, a fact which rests in the couch of convention for the most part. Frankly, I find IQ to be only a small indication of relative (with reference to a large population sample) capability and not a qualifying stamp of absolute determinacy regarding that individual’s eventual output. Even in spite of that, — and Prof. Tao must admit that whispers exist surrounding his contributions to thought, which reductively describe his works in the sole terms of his alarmingly high 189 IQ (i.e., look at some of the comments above) — many pledge their minds to this blind analytic determinism. The question, therefore, hinges on the ambiguity, even vagueness, of the term genius and maths and plays on the presumptions others carry in their acquiescent heads regarding these two elements.

However, my definition of genius is not open to the psychologist’s fallacy nor is my definition of maths limited to limited arithmetical processes. In point of fact, many individuals with these high IQ values (as one meaning of “genius” would have it) hardly approach the historical importance of others and there have been hundreds of mathematicians without high IQs or great historical significance on par with other individuals.

As such, my short answer is: no.

What then is left of the term “genius”? There have been many individuals in mathematics and in other (even related) fields who had to overcome almost insuperable difficulties, difficulties which before did look absolutely insurmountable. I hold that a genius is one who not only accomplishes this, but desires to share this insight with fellow human beings. There are even geniuses who had no native talents except an unquenchable thirst to explore new, alternate paths along the road of life.

Our values determine our standards. Our values can even determine what we aspire to become in but one instance.

29 June, 2008 at 6:10 pm

AlexDear ky,

I guess my answer to your question would be “some of both”. One of the difficulties in managing a research career is that there is conflicting good advice. For instance, we’ve all heard of the advice “look before you leap”, as well as “the early bird gets the worm”. Independently both statements are very reasonable, but together they contradict one another.

Similarly, at some point (usually after finishing the doctorate), one is often advised to work on a broader range of problems. Meanwhile, one should also take the time to work on areas of already obtained expertise, to look for deeper results (or at least more publications).

The “ignore other’s work” vs “read other’s work” is a member of a more general framework of common problems one faces, roughly under “risk tolerance”. I’d say that given time, you’ll converge to your own risk tolerance level.

P.S. I don’t think a “typical scholar” in math doesn’t usually have hundreds of contributions. That’s a pretty above average career!

30 June, 2008 at 8:37 am

Jonathan Vos PostThe argument about the “contributions” of Mathematicians is indicated by these quotations in this blog thread:

(0) Thesis: ” 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’.”

[Terence Tao]

(1) Retroactivity: “If I had seen math as a field where a few people make great contributions and many people make small but meaningful ones, I think I would have continued.”

[David, 1 Jan 2008]

(2) Antithesis: “The idea that mathematics is a field where many people make small but significant contributions is a myth maintained by assistant professors who make small but insignificant contributions.”

[Jacob Freeze, 16 May 2008]

(3) Thesis Expansion: “In my experience, every significant advance in modern mathematical research is in fact built upon dozens of previous results, insights, folklore, or experiences, a good fraction of which would have only been deemed of minor interest in their own right, and/or originate from junior faculty who are mostly only known to other experts in their field…. In short, there are many useful contributions to mathematics beyond the most dramatic ones. One does not need to be some sort of genius to be able to provide these contributions; but it does help to have certain amount of experience and awareness as to what real mathematical progress is.”

[Terence Tao, 16 May 2008]

(4) Ad Hominem Lemma: “whispers exist surrounding his contributions to thought, which reductively describe his works in the sole terms of his alarmingly high 189 IQ (i.e., look at some of the comments above) — many pledge their minds to this blind analytic determinism.”

[Passer-by, 27 June, 2008]

(5) Quantification: “I don’t think a ‘typical scholar’ in math doesn’t usually have hundreds of contributions. That’s a pretty above average career!”

[Alex, 29 June 2008]

In my humble opinion, there is more going on under the surface, in unstated assumptions, than in the explicit discussion. For example:

(A) There is a purpose for (0) and (3) related to the purpose of the blog which contains them. Terry Tao is trying to recruit and retain Mathematicians from a pool of potential practitioners (pre-Mathematicians both of the student type and those from other fields of thought) and frustrated practitioners. Retention is dealt with somewhat in (1).

(B) “good and useful contributions” as in (0) may be overlapping sets.

(C) Terry Tao has elsewhere written a wonderful classification, with annotations, of what constitutes “good” contributions in Mathematics.

(D) He likewise indicates that “useful” and “fruitful” are among the bases of “good” in this context. But the full list compresses an amazing amount of analysis about a stupendous amount of literature, and should be read carefully.

(E) Thesis (0), and follow-up (3), suggest decoupling between IQ (or other measure of genius) and potential to make contributions in Mathematics.

(F) Antithesis (2) suggest strong coupling between IQ and contributions.

(G) Antithesis (2) gives fuzzy use of “small” and “significant” to suggest the need for quantification. See also: “Least Publishable Unit.”

(H) In reply, (3) gives fuzzy “many” and “dozens” as numbers of contributions that are subsumed by major contributions, and contrasted with “dramatic.”

(I) The distinction between “small” and “dramatic” roughly corresponds to Thomas Kuhn’s distinction between “normal science” and “revolutionary science.” This is a sociological and psychological and philosophical distinction, not obvious from the content of any specific contribution.

(J) There is an implicit assumption that “small” is done by junior faculty and “dramatic” by senior faculty.

(K) Quantification of the number of contributions is traditionally done by a count of publications and citations and various measures that weight by the significance of the venue of the publication.

(L) Quantification is itself undergoing a phase change in this age of wiki-science, blogs, online databases, preprint archives, and the like. Can one make a significant contribution through blogs? Terence Tao uses both blogs and traditional papers as exemplars of the new paradigm. So does, for instance, John Baez, to pick one of many fine and highly visible examples. The use of IQ has been rejected on various grounds as a useful quantification, both in blogs and in, for instance, “The Mismeasure of Man.”

(M) The OEIS (online encyclopedia of integer sequences) has over 140,000 contributions as measured by web pages, some of which have dozens of formulae, comments, and citations to traditional and electronic literature embedded.

I’ve made, as of today, 1,966 of those, of which 157 cite (and link) to the arXiv preprints in Math, Physics, and Biology. My contributions in that venue are “many” and each is “small” and yet these cite to both other “small” contributions and to “good” and “useful” and “dramatic” contributions. I do some work on problems which are not “small” and a subset of these those slowly and painfully become refereed papers. But I am agreeing with Terence Tao in practice, by intentionally submitting to trusted on-line venues (OEIS, MathWorld, Prime Curios, and the like) roughly once per day over a period of roughly 7 years.

(N) Given the vast amount of data available, both traditionally indexed as in SCI, and more directly data-mined as by Google Scholar, one can find support for Thesis and Antithesis alike.

(O) The fact that almost all senior faculty began as junior faculty makes these kinds of analyses important for issues of promotion and tenure.

(P) This blog thread does not so far appear to demand nor contain “proof” either metamathematically nor statistically, for Thesis nor Antithesis.

(Q) The discussion, on a plane of Philosophy, Psychology, and Sociology of Mathematics is for various reasons, above and not yet mentioned, an important one.

(R) This blog comment is a small contribution. I did not have the time to compress it to fewer alphanumeric characters.

Thank you for your contributions.

4 July, 2008 at 6:59 pm

Joe ShipmanOne certainly does not have to be a “genius” in the IQ sense (IQ>=150 on a variety of tests; 150 is about the limit beyond which multidimensionality of intelligence makes scalars inappropriate) to make permanent and valuable contributions to mathematics. However, those whom history regards as “great mathematicians” would almost all be capable of achieving that IQ score.

Very few people should avoid going into a field because they can’t be “great” in it (really, only those who have a shot at “greatness” in some other field should be discouraged), so I don’t think there’s really all that much disagreement on this thread.

Especially since even someone who is not a “great mathematician” still has a chance to discover a great result. If your name is attached to a theorem in the books, you’ve achieved immortality even if you won’t be remembered as a “great mathematician”. There are lots of such names.

I’ve never worked as an academic mathematician, having always held industrial jobs (finance, software, statistics, etc.), but I’ve published two papers with some theorems of permanent importance.

One last point: it is NOT necessary to learn a huge amount of technical math to make valuable contributions to mathematics. For professional reasons, graduate students like to slog their way out to a frontier where very few others have gone, so that they will be very likely to discover something new, but there is plenty of buried treasure in well-plowed fields. My recent paper “Improving the Fundamental Theorem Of Algebra” contains results that could have been proven by tens of thousands of mathematicians since Gauss because it depends only on undergraduate-level Algebra and Group Theory. (Gauss’s 1816 proof, the first truly rigorous one, showed, essentially, that any field of characteristic 0 in which odd degree polynomials had roots and every element was a square or the negative of a square could be made algebraically closed by adjoining i. I showed that “odd degree” could be replaced by “odd prime degree” and that the result was true in any characteristic, and found a necessary and sufficient condition for all implications between sets of axioms of the form “all polynomials of degree N have roots”.)

17 August, 2008 at 7:41 pm

anonymousI’m also a maybe-aspiring mathematician, seeking vague guidance.

This is a strange discussion. But it’s also maybe useful. I like the post because it’s hopeful, and I think in many cases that’s best, however true.

Anyway, it’s hard to say what someone will or won’t be able to do.

16 September, 2008 at 7:24 pm

TheTruthA lot of fools here, where does truth come from and how is it derived? Did truth precede us? Answer those questions.

As for great mathematicians being humble, many are anything but… Read up on George Cantor and transfinite numbers, his ideas were bitterly resisted by Poincare and the others. Also remember George boole was also criticized and outright ignored by the mathematical community during most of his life as well.

From wikipedia:

Cantor’s theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature of God,[4] on one occasion equating the theory of transfinite numbers with pantheism.[5] The objections to his work were occasionally fierce: Poincaré referred to Cantor’s ideas as a “grave disease” infecting the discipline of mathematics,[6] and Kronecker’s public opposition and personal attacks included describing Cantor as a “scientific charlatan”, a “renegade” and a “corrupter of youth.”[7] Writing decades after Cantor’s death, Wittgenstein lamented that mathematics is “ridden through and through with the pernicious idioms of set theory,” which he dismissed as “utter nonsense” that is “laughable” and “wrong”.[8] Cantor’s recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries,[9] but these episodes can now be seen as probable manifestations of a bipolar disorder.

The only genius is truth, just because prodigous men are able to see it more clearly means little, all truth already is, it is merely being derived from what already was.

A true genius knows that the truth itself is genius, and submits himself to it humbly, he knows that character is worth more then ability and he is a student of socratic way of living. That is true genius, the rest may or may not be on the path.

Socrates (wikipedia)

Socrates often said his wisdom was limited to an awareness of his own ignorance. Socrates believed wrongdoing was a consequence of ignorance and those who did wrong knew no better. The one thing Socrates consistently claimed to have knowledge of was “the art of love” which he connected with the concept of “the love of wisdom”, i.e., philosophy. He never actually claimed to be wise, only to understand the path a lover of wisdom must take in pursuing it. It is debatable whether Socrates believed humans (as opposed to gods like Apollo) could actually become wise. On the one hand, he drew a clear line between human ignorance and ideal knowledge; on the other, Plato’s Symposium (Diotima’s Speech) and Republic (Allegory of the Cave) describe a method for ascending to wisdom.

In Plato’s Theaetetus (150a) Socrates compares himself to a true matchmaker (προμνηστικός promnestikós), as distinguished from a panderer (προᾰγωγός proagogos). This distinction is echoed in Xenophon’s Symposium (3.20), when Socrates jokes about his certainty of being able to make a fortune, if he chose to practice the art of pandering. For his part as a philosophical interlocutor, he leads his respondent to a clearer conception of wisdom, although he claims he is not himself a teacher (Apology). His role, he claims, is more properly to be understood as analogous to a midwife (μαῖα maia). Socrates explains that he is himself barren of theories, but knows how to bring the theories of others to birth and determine whether they are worthy or mere “wind eggs” (ἀνεμιαῖον anemiaion). Perhaps significantly, he points out that midwives are barren due to age, and women who have never given birth are unable to become midwives; a truly barren woman would have no experience or knowledge of birth and would be unable to separate the worthy infants from those that should be left on the hillside to be exposed. To judge this, the midwife must have experience and knowledge of what she is judging.

26 September, 2008 at 1:35 am

I blue eyeThe perception of genius is the most important part of a genius. Without it you could have a set of skils and do good job but not at a genius livel.1% is mor important that all that 99%

28 September, 2008 at 11:41 pm

TosinI like this “new kind of science” It’s like, so democratic, anybody can do it.

How do you make practicing scientists more welcoming of others?

How do you make practicing scientists more content with their own contribution and not constantly unhappy that their neighbour is better?

How do you make practising scientists happy – in other words I see the sticks (I’m not doing enough, I’m going to fail to make tenure, blah blah blah), where are the carrots (I’m doing well, this fits with my picture of a good life, yada yada) ?

In my experience of “science” it had formed as a community inimical to the pursuit of happiness. We know that this is unnecessary. I think that this is changing.

1 October, 2008 at 8:31 am

SushilDear Dr. Tao,

Your post has truly inspired me to work harder and the right way. But I hope I am asking this question correctly.

Do you really think there is no genius in this world? You mention that working hard and efficiently is the key, but I find people who are able to find answers quickly are generally more confident in approaching newer problems.

I am an undergraduate student in electrical engineering, and I know for certain that I am not brilliant; but sometimes I do wish that hard work gives me some amount of satisfaction in the form of results or discoveries, even if it is after a month or a year. But many times, when I see things failing, it saddens me. It makes me wish that I had an oracle that will just give me the right answers, and make happy all over again. Sometimes I wish that I shouldn’t be doing electrical engineering after all.

Don’t you think that this oracle can be genius?

2 October, 2008 at 2:00 am

I blue eyeIt is a great diference beetwen the “amor intelectualis” and the passion for scince. The first make you good , the second make you a genius. You understand ?

It isn’t a new kind of science it is just the perception and point of your viwe about the science. I have more repsect for the Perelman because he isn’t so covnentional neither crazy.

11 October, 2008 at 1:48 pm

percy liThank you, it seems my dream can still be realized!

23 October, 2008 at 9:07 am

rcourantDr. Terence Tao has an IQ of 220-230.

17 November, 2008 at 10:08 pm

joeWhat you say about how talent can hurt one’s ability to work hard is so true. I am by no means a math genius, but in my high school I was on the top as far as mathematical abilities goes. Because of this I thought math was… “easy” and not really worth much effort. I have realized how wrong I was. Now I am in college and working very hard. I wish I had put more effort into mathematics at an earlier age, but at least I still have tons of fun math to learn now!

8 December, 2008 at 12:54 pm

i blue eyewhen we want a emancipation of mind we need pasion not 1000 of hours of train. If we do not have pasion we just want to be apreciate and this is an egoistical point of view of our world. To view just the social part of thinks is idiocracy.

8 December, 2008 at 2:05 pm

AnonymousThe original post (as well as much of the ensuing discussion) suffers drastically from the fact that “genius” has not been defined. In my view, the definition of a genius is someone who works work, is directed by intuition, literature, and enjoys good luck. In other words, posses the attributes that Terry describes as being characteristic of a successful mathematician. In which case the answer to the question becomes “yes”.

8 December, 2008 at 11:20 pm

i blue eyeResearchers need to be in close touch with truths which provides that, if we allow, is called intuition. What does that mean? That without a strong relationship between researcher and a problem that has to solve it is almost impossible for him to have the chance to be a person recognized in their field. Sure, a lot of talent and a high IQ is very easy to work hard.

In order to make more concrete thoughts will bring up the fact that the entire universe is a forest. Now, in some cases can only think of the forest at the trees and not only understand a part of the beauty starting as a researcher to analyze the trees to understand the universe or we can look at the forest, which is connected total of truth and intuition. Great truths are feeling! That the most profound understanding is intuitive.

9 December, 2008 at 3:51 am

IndianDear Prof Terence Tao, If my query has annoyed you ( I would like to apologize), and my query is:

Is ‘genius’ nothing but “efficient thinking”? You said ( in reply to a query by someone how some people are able to comprehend complex material, you said it— by an efficient means of study than to any inherent genius ability)

So Prof Tao, can you explain (either tacitly, or elaborate) what efficient thinking/efficient means of study is? (Either with respect to problem solving or etc).

Your blog is really engrossing, and I really appreciate what you have done.

I request you to answer my query, I thank you in advance.

9 December, 2008 at 9:07 am

i blue eyenot all eficient thinkers are geniuses. This is absurdity.

9 December, 2008 at 10:53 am

IndianIs it so? Then whom do you consider as a genius?

10 December, 2008 at 12:25 pm

i blue eyeYou think that a hard trained person is a genius ? You are wrong.

A genius is a profound thinker who do something by pasion and power of will. Your example of social engineering is an absurd idea. Most great concern in the world nonsistematic and therefore the conditions in which genius is natural it seems absurd to call the concept of “effective thinking” which is something like a social resource in making geniuses. As you can see there is no such thing!

10 December, 2008 at 6:17 pm

IndianI guess your reasoning is fallacious ( in some sense at least); Geniuses are the product of “effective thinking” , “common sense” , “passion”, “hard work”, I feel you are digressing from the topic, and your mode of communication obfuscates me; I ask you to be more clear in your mode of communication ( use words correctly, convey your thoughts effectively).

I feel Dr Tao can resolve this query, because he himself is a genius/prodigy.

11 December, 2008 at 1:16 am

i blue eyeDc tao is just verry verry good at what he do.

Yes , “effective thinking” , “common sense” , “passion”, “hard work” make geniuses but a genius will be so creative and so inspired that he always live in another Universe. What it is a genius who do a job ? It is just an “effcient thinker”. It is fair to call him a “genius” ? Look at Beethoven , Einstein , Tesla , Pitagora , Hypatia , Gautama , Mozart. They are geniuses and they think like a genius. IN a profund way and powerfull. The genius isn’t a social product but a very instictual person who do somethink according his own being.

Sorry for my mode of communication.

22 January, 2009 at 10:59 pm

Bryan ArnoldI really enjoyed this post. I agree very much with you, but I was wondering where you think Ramanujan fits into this idea. Some of his insights did seem to be “more” ex nihilo -mainly due to his lack of formal training and unfamiliarity with the idea of proof. Then again, he was sometimes wrong as a result.

Again, I agree with you whole heartedly, but I was just wondering about your opinion on the subject.

23 January, 2009 at 7:36 pm

Terence TaoDear Bryan,

I of course only know of Ramanujan through his work and through secondary sources, but I understand that he performed a prodigious amount of numerical computation and experimentation in the course of his research, which may have pointed him the way to some of the amazing identities and other mathematical results that he discovered, and would have given him a rather different intuition and “box of tools” than what a more mainstream, theory-based approach to mathematics would give. Still, his talent was exceptionally unusual, and he is one of the few successful mathematicians of the modern era that I would see as a plausible contender for the title of “genius” as the term is popularly used. But the bulk of mathematical progress nowadays comes from more prosaic individual and collaborative mathematical effort – much more Hardy than Ramanujan, so to speak.

6 February, 2009 at 11:43 am

NKHHi all,

I guess if Einstein, as a young student at the ETH Zurich, knew much about those IQs tests and some of the posts in this discussion, he’d give up his study and earned his money with his violin (prob. as a free musician) in Bonn or Bern for his life.

It were maybe the judges of (almost) all physics professors in Europa that time about Einstein as they refused to accept him as a doctorant.

Best regards.

21 February, 2009 at 5:43 pm

RJHi Terry,

I’m currently a high school student. I’m really good with maths but I don’t think I’m one of those “naturally gifted” students. My dad (who is my primary mentor) told me that I wasn’t really good in math when I was little but I just worked my way to what I am now. Even though I find it a bit discouraging to hear (especially from my parents) that “I wasn’t that good” back then (as opposed to people like you who were born with natural math instincts), I am really interested in pursuing a mathematics degree and going to graduate studies in math.

Do you think I have a chance to be successful pursuing this dream? I really enjoy doing math and it’s my “first love”, but I am still not quite sure if I can handle it.

A response would be really helpful. Thanks and more power to you.

23 February, 2009 at 8:20 pm

Jonathan Vos PostIs Genius Born or Can It Be Learned?

By John Cloud Friday, Feb. 13, 2009

Is it possible to cultivate genius? Could we somehow structure our educational and social life to produce more Einsteins and Mozarts — or, more urgently these days, another Adam Smith or John Maynard Keynes? … The latest, and possibly most comprehensive, entry into this genre is Dean Keith Simonton’s new book Genius 101: Creators, Leaders, and Prodigies (Springer Publishing Co., 227 pages)

[truncated]

24 February, 2009 at 1:58 am

tmrDear RJ,

It is in some sense good that you weren’t really good in math when you were little but you just worked your way to what you are now, because that means you have the control over where you will be in, say ten years from now. Whereas if you have to be a genius to do math, there is only the blind chance of genes and chromosomes and what not.

26 February, 2009 at 8:29 am

There’s hope for us yet « Christian Pinawin[…] 26, 2009 in Uncategorized By following a post onTerry Tao’s blog, here’s an article in New Scientist that’s both sobering and […]

3 March, 2009 at 10:26 am

NUR SYAZWANISometimes, i thought the one who are genius in mathematics are they whose have the ability to create so much idea as we know there are many ways on how to solve any mathematical problem.As we know,there are same understanding concept on mathematics for people but there might be some differences on the development of idea to solve the problem

18 March, 2009 at 6:32 pm

Fractal Art: Waiting for Your Love « zaytuun | Fractal & Islamic Art[…] Does one have to be a genius to do maths? […]

25 March, 2009 at 12:07 pm

TomDear Terry,

Would you agree that most work in math today can be done through email if you are collaborating with someone? Or do you have to be physically in the same room as him?

Thanks

27 March, 2009 at 8:44 am

Does One Have to Be a Genius to Do Good Science? « Academic Career Links[…] Terence Tao: Does one have to be a genius to do maths? […]

21 April, 2009 at 1:40 pm

AnonimousI’ve read this article a few months ago. I come here now to tell you that I believe that I am a counter-example. I think I’m reasonably intelligent (and other people think so too), I work hard every fuckin’ day and it doesn’t make any fuckin’ difference. Doomed to mediocrity. I respect Mr. Tao very much, but please, belive me, not everyone can do mathematics. Tao is too much of genius to understand the average mind (I think).

22 April, 2009 at 9:12 am

CorwinHonestly people, why are you debating the “true” meaning of the word genius?

Like all other English words, it has been given a formal definition and has an informal definition formulated through aggregated personal beliefs and understandings of English speaking people. Your own definition of the word, while valid, really doesn’t matter.

The commonly held definition ties genius to IQ. You don’t have to like it, nor do you have to agree with it. Unfortunately for you, however, it is reality.

On the other hand, those of you that are arguing about the validity of IQ, I applaud your skepticism. It is true that there are no studies linking IQ to academic success, or even “success in life” (whatever you may take that to be). From personal experience and as a person whom has been haled as a genius for quite some time (due to multiple tests as a child and as an adult revealing an IQ of well over 190), I can tell you why it has no bearings on success.

I received horrible grades for the simple fact that I found school a waste of my time. That, naturally, lead to the inability to get into a decent college. In my particular case, I managed to become well appreciated in my chosen field and have a very high paying and respected position sans collegiate graduation. Many others, in similar situations, do not do as well for various reasons. Perhaps a minor degree of luck played a part in my story along with desire and motivation.

Genius, or rather, “High IQ”, does not guarantee any form of success. It is more akin to being given the task of building your own structure, along with the rest of the world, but you are gifted with more materials and better tools. You can spend all of your time watching others build. You can have some Scotch and look at the sunset. If these are your choices, you will most likely fail. Others will build up around you – people with lesser advantage. However, if you have the same dedication, you can create your structure in much less time with far less effort. It can be better built, more elegant, and more efficient.

Being “gifted” is exactly that. You have a “gift”. It is only the application of that gift, the exploitation of your full potential, that ultimately matters.

So, in a sense, I agree with Mr. Tao’s belief that genius is not a prerequisite for mathematical success. It can be, however, a great advantage.

25 April, 2009 at 5:24 pm

Tom O.Could we stop all the crap about IQ?

First of all, the most fundamental idea that needs to be understood is that IQ tests are primarily pseudoscience. In order for a true intelligence test to exist, it would have to utilize a pathology for what is being measured. Namely, the result of an intelligence test shouldn’t be so sociologically determined. If intelligence as we know it is such a biologically deterministic ability, then there has to be corresponding genes or brain patterns that we can study to determine intelligence. Likewise, since what we’re studying is biologically mapped, there shouldn’t be “improvements” on IQ tests. As far as I know, the IQ test fails to meet these requirements, and until it does, the burden of proof lies on them.

Likewise, I don’t understand why people would get upset over not being intelligent. I’m a high school student of very average intelligence. Does this mean that I may *not* be god’s gift to the rest of the human race and that my existence will be judged as unnecessary? Yes. Deal with it. Unlike Lake Wobegon, not all children are above average.

More importantly, if you enjoy mathematics as a study, then your ability at it really shouldn’t matter. If your current competence level is something that you view as distressing, then you can choose to either try to improve the factors that you do have in control, or do something else. There’s no point in studying something that’s only going to cause further agony.

25 April, 2009 at 5:37 pm

Tom O.Oh, and to further add to my previous comment, it seems extremely narcissistic to only care about the faith of the faceless masses that run and control your life when you fear that you are a member of them. To have such a delusional sense of entitlement that I expect as an axiom to be a genius of great talent, and to view other people as an unwashed mass of sheep strikes me as slightly more than disturbing…

5 May, 2009 at 3:35 am

AnonymousI want to say *thank you* for allowing us the opportunity to explore the very interesting topic of genius vs. hard work in the field of mathematics. I am a home educator who incidentally has a high IQ child, and I am involved in many online forums with parents of high IQ children. It appalls me to see an all-to-prevalent attitude among some of the parents of these children that hard work can never hope to compete with the so-called intrinsic power of IQ. I have always been a strong advocate for hard work, independent thinking, and the power of passion. I appreciate hearing you burst a few bubbles by correctly pointing out the vast superiority of hard work.

5 May, 2009 at 6:55 am

barterfortyoneAnonymous,

How to measure autistic IQ in a light way?

18 May, 2009 at 11:41 am

DrrachelHello there! First of all, congratulations to your achievements! You made us proud!! I am currently a doctor working in England, and your achievements have certainly inspired us, chinese, to work harder. Nothing is impossible!

It is a little strange that you, yourself who were thought to be a genius do not believe in the ‘genius’ idea at all! I certainly believe that besides the gift that the god has given to each of us, hard work, perseverence and self believe is essential to make us achieve great things in life!!

Keep going!:)

26 August, 2009 at 5:31 pm

AnonymousI personally subscribe to this theory.

http://www.paulcooijmans.com/genius/genius.html

In my opinion this theory is simply great, it of course includes intelligence and associative horizon which is considered to be innate and decided at birth, but also includes another important facet considered indispensable for genius. That is conscientiousness. In a very crude manner conscientiousness can be summed up as ‘working hard’.

I have found this to best model of genius so far, of course one not need to take his hard ‘nature’ approach, but even with that approach it still leaves much room open for hard work to play its role, but perhaps we can temper this with something else Cooijman writes.

“The general under-valuation of intelligence in modern society may be more due to the intellectuals’ failure to recognize the value of intelligence than to the attitude of the greater public. It’s like with money; it’s always the rich who say “money doesn’t bring happiness”. Never the poor. With intelligence, it’s the intelligent who often say “intelligence isn’t everything, we are really all equal”. But the dumb know better.”

1 September, 2009 at 10:40 am

abhishekparabI have read this post more than 20 times and yet, every time I read it, I am inspired to do more Maths!

15 September, 2009 at 9:50 am

Enseñar cálculo con base en probabilidad y otras cositas « Probabilidad y ciencia[…] Es más, el mismísimo Terry Tao, con toda su innegable genialidad, así lo sostiene (véanse esta, esta, esta y esta entradas de su blog, por ejemplo). Tao no es solo su genialidad, de hecho él es […]

8 November, 2009 at 7:25 am

Jess PorterExcellent Article:

I have been reading [mostly applied],math books for 20+ years.

I am a member of SIAM and a local Mathematica users group.

I have 5 or 6 years of college, but never got my degree.

Math Rocks!

Jess

29 November, 2009 at 3:05 am

amlan chakrabotyBrother! I am a lot interested in mathematics especially im geometry and trigonometry.I am an undergraduate (class-9) and I wish to win medals at Imos’and become a mathematician. . But when I think about it as a career I feel it queer because in India very less relevance is given to post graduate and research mathematics.What do I do??

9 January, 2010 at 10:43 pm

cnPhil » 做数学与做学问[…] 老师在他的博客上写过一篇文章, “Does one have to be a genius to do maths”, Zhiqiang […]

10 January, 2010 at 7:04 pm

转载：做数学家必须是天才吗？ « CodeWay:我的博客[…] 原文地址:Does one have to be a genius to do maths? 中文翻译:做数学一定要是天才吗？ […]

11 January, 2010 at 8:34 am

vadimAs other commenters I also agree with the post. But one cannot deny that there are differences between equally educated people, even if they put the same amount of effort. Usually these differences are marginal, but they still exist and sometimes are noticable. This happens of course not only in math.

12 January, 2010 at 4:01 pm

PierreI put a post 2 years ago, and I am happy to see that it is still active.

I totally agree with Terence Tao, but too me, this question is like:

is it necessary to be tall to play basket ball. Well, look at the size of the international players…

25 March, 2010 at 8:44 pm

AnonymousI am only a college student, pursuing a degree in mathematics.

But what i find very interesting about the previous discussions, is that despite the fact that prof. Tao and many others are supporting the idea that being a ‘genius’ is not the key to being successful in mathematics, i am getting the impression more and more that one needs to bo such, indeed, in order to be able to do mathematics.

26 March, 2010 at 9:38 am

Zeeshan MahmudI think part of the problem of teaching mathematics in academia has been the strong logical-mathematical approach to it. With full respect to Polya that mathematics is not a “spectator sport”, despite having no calculating talents, I find it more satisfying to browse wikipedia and metamath because of my strong penchant for linguistic flair, ie fascinated with exotic names like de Rham cohomology or Axiom of Choice. In that way, mathematics can be thought of as learning a foreign language.

And again despite my anathema for mathematical calculations, my love for maths stem from philosophical foundations or implications of it.

So my point being each and everyone has her own form of genius and it’s best to capitalize on them and approach mathematics via that corridor.

Just my two pence. :-)

9 April, 2010 at 2:07 am

做数学一定要是天才吗？ （译自 陶哲轩 博客） « Yalearts's Blog[…] （原文：Does one have to be a genius to do maths?） […]

14 May, 2010 at 1:29 pm

JosephTo me, genius is a person who persibe light where others see only darkness, and Euler in Basel problem or Godel in the foundations of mathematics, while talent is who does quality work that others could do.

For instance, Gauss in Differential Geometry was a genius, but he was only a talent in Astronomy, because other mathematicians could maked the same work.

It is common that some mathematical geniuses take out bad grades and math tests suspended at some point in their lives, for example, Fermat, Newton, Galois, Hermite, Poincaré, Einstein, Hadamard, Ramanujan.

Perseverance is what separates the geniuses of the mediocre, but still not enough. There are many people who do not have persistent that magic touch of inspiration that lets you see the solution to a problem in which one thinks very much.

Many geniuses have died young, like Abel, Galois, Ramanujan … Tom Wolff, while others have lived a lot, like Newton, Euler, Gauss. Well Tao, take care, goodbye.

20 May, 2010 at 5:09 am

YU JI ZEDear Prof Tao:

Thank you very much for reading my comments! It is the first time that I write to a great mathematician like you and I feel really nervous.

I come from China and love mathematics very much.I was awarded the third place in my province in mathematical Olympid when I was still in senior high school.Now I am having bridging course in National University of Singapore(NUS).I have decided to learn pure maths when I entre NUS

I have read the advice that you write for undergraduate students.I wonder could you please give me some more advice.Such as what books I should read for advance study because I found the textbools I am using now cannot meet my needs of learning. I am now learning calculus and liner algebra. I am sure that the advice you gave me will surely help and encourage me a lot.

Again thank you for reading my comments and all the best wishes!

yours

JI ZE

7 August, 2010 at 8:31 am

...http://www.chessbase.com/newsdetail.asp?newsid=6187

9 November, 2010 at 11:40 pm

SergeyI think that the question whether intelligence or hard work are sufficient to be successful as a mathematician or a theoretical physicist or other scientist/professional is oversimplified.

The more one’s intelligence is, other things being equal, the more productive a person would be and the more the probability is that he/she would contribute important results.

Likewise, the harder one works, the more that probability is. Other things are likely to increase this probability too, such as good working skills, stimulating environment, some personality traits, good mental health, good physical health etc.

Looking at it in another way, if we could select the most successful mathematicians and scientists, I would expect that people who have the most of the mentioned traits would be the most often represented ones.

Concerning the significance of IQ, mainstream scientists seem to disagree with those who claim that IQ has no significance at all. It is sufficient to look at http://en.wikipedia.org/wiki/Mainstream_Science_on_Intelligence and at correlates of IQ at http://en.wikipedia.org/wiki/Intelligence_quotient#Validity_and_social_significance. I think to downplay importance of IQ may be as dangerous as to exaggerate it.

11 November, 2010 at 11:22 am

raminDear prof Tao:

if i was to run 100 meters and there were those that could do it in less than 10( or even 1) seconds, and i could –with a lot of practice– do that in 15 sec, then the fact that it takes me 15 times as it takes them to do the same task wouldn’t bother me. but when it comes to mathematics and problem solving, it is a totally different story. what you do in 1 month i can’t do in my entire life. i wish with hard work i could do the same thing in 10 years. but i just can’t.

this seems so strange to me. and i always wonder, what causes such huge difference? i mean even computers aren’t that different in speed.

brains of fairly same size. same shape.(i have read that guass’s brain is kinda like every other brain) but when it comes to problem solving….

so i would be glad to know what you think the source of such huge difference is?

(((it appears to me that it is like there is a wall full of buttons on it. the taller you are the higher the buttons you can press. so if you are not tall enough, then you start jumping. but there is a limit for jumping. do you agree with such image? )))

2 December, 2010 at 2:42 am

amirsDear Prof Tao:

Does brain-age have a rule in doing math?

is your mine as powerful as your 20th , in don’t mean knowledge , i mean inteligence ,power , …

Regards

amir saeidy

13 December, 2010 at 10:43 am

raminHi again

to everyone:

above i asked a question which no one showed any interest in. ((the one about the nature of the huge difference between problem solvers))

let me be more explicit. i am wondering if it is possible to make the difference between problem solvers, linear. i mean suppose i study the way Prof Tao does, will that make the difference between him and me linear? so that i can do whatever he does in lets say twice the time it takes him do that exact same thing.

and to amir:

salam amir khooobi?

19 December, 2010 at 4:25 am

earthNo. one does not have to be a genius to do mathematics.in fact, we are all born with a sense of numbers,logic,quantity and space.A normal healthy brain can always think spatially.Special education suited for everybody may help to bring out the best of mathematical ability from everyone.

but the problem is , how can we be successful mathematicians?

for that ,the answer is best given by the great mathematician Jules Henry Poincare, who once said “Mathematicians are born, not made”

but only mathematics.i think its everybody’s game. http://plus.maths.org/issue19/features/butterworth/index.html

19 December, 2010 at 8:55 pm

quantum probabilityThank you for making this point from your renowned position. Too many people hurt themselves or make bad decisions because of their belief in genius.

23 December, 2010 at 12:18 am

VijayDear Prof. Tao, I believe that while you mean well, your statement is not correct and even harmful. If the Wikipedia entry about you in correct, you were a child prodigy and were teaching elder kids mathematics *at the age of two*! While you may work very hard, so does every dumbo like me who cannot understand even the abstract of one of your papers without many months of effort.

I suppose you do not realize exactly how dumb the rest of us are. :-) Let me repeat : I cannot even *understand* your papers, forget about contributing to mathematical research.

It seems to me that the “cult of genius” has more credibility than the “cult of hard work” being promoted these days.

Just take a look at Tiger Woods putting away at age two : http://www.youtube.com/watch?v=MLL541C9wBs. (There is another one of him at age 4, but I could not find that). Clearly, the kid was born with it.

Yet, I find statements to the effect that, “It was through focused and deliberate practice that Tiger Woods became the great golfer he is today.”, and also that “Tiger Woods was born with nothing special that you and I were not born with”.

I believe that a more honest statement is: “You can enjoy and even contribute to mathematics without being a genius, but don’t expect to scale the heights that I (Terence Tao) did without a huge overdose of God-give/Nature-given genius. And the same applies to every field”.

That professor who is peddling the concept of “deliberate practice” needs to be jailed for brainwashing so many people into sticking at something they were not born to excel at.

Prof. Tao, not knowing one’s (genetic) weaknesses is a HUGE meta-weakness, and it has taken me many years to learn that. This goes against most people’s conditioning (“you can be whatever yo want to be”, “Just do it”, etc;), but it’s reality.

23 December, 2010 at 5:57 pm

Terence TaoThat particular story about myself seems to have grown in the retelling. I personally don’t remember it, but as I understand it, I was trying to explain to some slightly older kids how to count, using number blocks. I suppose this could be called “teaching mathematics”, but not of a particularly advanced nature.

Also, one should not underestimate the cumulative effect of many years of dedicated study and focus on a subject through one’s primary, secondary, undergraduate, graduate, and postdoctoral education. I remember distinctly as an undergraduate going to the library to browse through some recent mathematics journals, and finding that I could not understand even the abstract of most of the articles there. In particular, I doubt my undergraduate self would be able to read most of the papers that I and my collaborators write nowadays. Nevertheless, after several years of PhD study and a few more years as a postdoc, one can reach the cutting edge of research in a given subfield of mathematics and start contributing to it; I have seen this dozens of times now with the graduate students here at UCLA, who generally start just with a decent undergraduate maths education, a reasonable (but not “genius-level”, whatever that means) amount of intelligence and mathematical ability, and a mature work ethic. The relationship between initial talent and final performance is quite a complicated one, and it is not even clear that it is completely monotone; certainly one needs to have a certain minimal level of mathematical ability in order to be able to make good contributions to mathematics, but exceptional mathematical talent does not necessarily translate to exceptional contributions to mathematics, and in some cases an innate talent in one area of mathematical ability may even be counterproductive by reducing the incentive to improve upon the other areas. So I do not believe it is particularly accurate or desirable to focus excessively on exceptional talent as being the main predictor of mathematical achievement. (Nor do I believe that exceptional work ethic is the main predictor either; as mentioned in the main post, there are many factors involved here, one of which is knowledge of one’s own limitations as mentioned in your comment.)

Finally, as I said in the main post, professional mathematics is not a sport, and as such, sporting analogies (such as those invoking Tiger Woods) are not really a good guide to the subject. In most competitive sports today, the bulk of the attention and career advancement is given only to a tiny sliver of practitioners, with one’s position in various rankings being of extreme importance. While there are certainly (for better or for worse) some competitive aspects to professional mathematics, there is not nearly as much of an emphasis on rankings, or a domination of the subject by a very select few. If one looks at the progress in a given field, only a fraction of this progress (and not necessarily the “best” fraction) will come from exceptionally well-known and recognised mathematicians; the bulk comes from a significantly larger community of mathematicians who are very good and talented, with years of experience, knowledge, and developed intuition, and are known and respected within their field for their contributions, but who would not be readily thought of as being “geniuses”, “prodigies”, or having “exceptional innate talent”. The outliers who do display these sorts of traits are of course also important to mathematical progress, but not by as much as one might naively think.

2 January, 2011 at 9:19 pm

VijayThank you for taking time to write this reply!

> “I remember distinctly as an undergraduate going to the library to browse through some recent mathematics journals, and finding that I could not understand even the abstract of most of the articles there.”

+

> “I have seen this dozens of times now with the graduate students here at UCLA, who generally start just with a decent undergraduate maths education, a reasonable (but not “genius-level”, whatever that means) amount of intelligence and mathematical ability, and a mature work ethic.”

That is indeed an eye-opener to me! I suppose it will take time for me, but I will re-consider my views, revisiting this discussion and re-reading as necessary.

11 June, 2011 at 3:06 pm

anonVery nice reply ! Thanks for this nice info !

23 December, 2010 at 8:05 am

chanakyaI am a student of high school, probably a not worthy commentor, but I would like to express my opinion. Basically, we are born with empty mind, a new computer with 0 occupied hard disk space. We take ideas from the external world and we develop certain beliefs and interests. Pr. Tao, has been influenced by numbers and mathematics, which he enjoys and feels himself strong in, works for him well. Even in case of great mathematicians or scientists, their fascination towards something in childhood made them what they are. Basically, there is great work or genius work but no genius individual. For other people, their external world is certainly contrary. So depending on ones influences and beliefs, we develop our skills. Even when in the mother’s womb, the infant is exposed to different things and even to his/her mother. My parents are normal educated people, who influence me with their beliefs and ideas, to the extent they can imagine and go. I am exposed to certain facts of life, which makes me different to them. Even the concept of hardwork which Pr.Tao beliefs in, is not from him directly, his mentors or teachers influenced him with their belief of working hard. Its as though our minds are interconnected with invisible forces called ideas, which constantly flow from one medium to other. Even a creative work or new thing is an extention to previous belief or idea. The origin is unknown, but idea is basically an experience.

23 December, 2010 at 11:22 pm

Gábor Petechanakya, you are expressing this very nicely, but I think it’s only part of the truth. We come to this world not at all empty. There’s all of the human and prehuman evolution, your genes, your personal body, your parents’ histories, and so on. (Look at C. G. Jung’s great book “Man and his symbols”, for instance.) The way you put together all the ideas going trough you depends a lot on all of this. You could say, of course, if you want, that there is still no individuality, there’s only the evolution of cells and genes and ideas, and you are just one form of expression of this. Fine, this is a nice image, I think, to explain WHY we are individually here in the world, but doesn’t say much about HOW we are. Should I do math or not? You can say that, on the input side, “talent + influence” has in fact only one term, “influence”, but does that help in deciding whether you are happy with your output/input ratio?

Well, maybe it does help: if you have a better idea where your ideas come from, that certainly helps decide if you want to carry on. For instance, from my own work so far, the one that required an idea about which I have the least clue where it came from, so in some sense required the most of a genius-like leap, that is probably the least important work of mine… So, at least I know I’m not earning my bread being a “genius”.

OK, sorry for this clutter, you probably wanted a reply from Prof Tao….

24 December, 2010 at 4:04 am

chanakyato Peter:

Sir, I agree with you that we are not all empty.

I strongly feel that influences affect me because I like to dedicate my whole time to mathematics and mathematical physics, irrespective of outcome or anything for that matter. But due to the external pressure of gaining a graduation degree from a prestigious college, I have to divert myself from my interests. I have to do work on things which I don’t care much, for that tag beside the name. Its an issue of respect and prestige to my parents, which is just something created due to influence again. I really don’t bother whether I will be known to the world for achieving something, but at least for my parents, I have to do something for that tag, which makes them happy. I don’t blame anybody for the situations which I have to face, but I always felt like breaking those strands to live my dream and vision. Even my short term aim of entering the international mathematical olympiad is to find some hope that I am capable, and to be free to live the way I want.

I always thought that I had a mentor who would understand me and guide me in the vision I dream, which is in turn based of luck factor. So what to do? Its simply my experience and view point, which need not be true.

24 December, 2010 at 4:14 am

chanakyato Pete:

frankly speaking, even my purpose of commenting on this is because it relates to myself more, and secondly, to find someone who can give few valuable guidelines. I am thankful that you did spend your valuable time on my opinion and sorry, if I have been too personal.

26 December, 2010 at 5:03 am

MarloDespite the subjective nature of the term intelligence, it’s pretty obvious to most people that it varies between individuals. No matter one’s level of intelligence, there will always be people who are smarter than you, as well as people who are dumber than you. Everyone is not born equal.

Nonetheless, barring some kind of learning disability, it is certainly possible for people of “average” intelligence to learn abstract mathematics. Average students will not learn as quickly as gifted students, but still they can learn. This can be achieved with a combination of good books (an extreme rarity in math), good professors (even rarer) and a certain degree of open-mindedness about learning things you thought you already knew.

Whether or not such people can become “rockstar” Mathematicians is a more difficult question. This requires something innate, and probably a borderline obsession with one’s work, which isn’t necessarily derogatory. This is just to say that the academics who make the most profound discoveries are the ones who find their research to be more deeply fulfilling than their peers do. I would guess that this type of fulfillment is the main quality that seperates famous Mathematicians from the rest. After all, the world is abundant in high IQ people who have aptitude but lack contributions in science and math.

26 December, 2010 at 9:25 am

raminto everyone and specially to earth.

“”about making the difference between prof tao and us less.””

I think your NO is an emotional, one. please read my arguments.

suppose we have two runners A and B in a village, one day a man who is looking for a talented runner comes to the village and wants to find the best runner in the city to train him for lets say Olympics. to choose the best runner he makes the volunteers run 100 meters and the winner is the one he chooses. A and B and some others take part in this little competition. A wins doing just a little bit better than B. so A gets trained for the next several years and B goes to lets say college to study medicine. after several years A and B are not comparable (as runners) anymore. although the difference between them was not that huge several years ago. this is just an example to show how chaotic human life can be.

look how Prof Tao emphasizes “tools”, “why this tool works here why that one doesn’t” etc. most my profs don’t think about these things. they do mention, but i can see that they spend very little time on these things.

i think this give us some clues that the difference is getting huge because most of us, most of the time are doing things that will never get to a solution.

i think this is knowing the tools that is making prof Tao faster and faster.

a closer model can be A and B being students in a school. A being twice as fast as B in solving problems. and suppose that all students are supposed to solve some weekly homeworks. every friday they go home with some homework they are supposed to do during the weekend. A is faster so he solves the problems in saturday and on sunday he spends some time thinking about the “tools”. B is not that fast he spends both days doing homework. so next week A is even faster than before, cause now he knows something about the “tools”. and he has even more time to work on the tools but poor B is still solving problems the same stupid slow methods he did before. and little by little the difference gets huge. then after some years B regards A as a magician, genius.

most of us have been “the brightest student”, the one that makes everyone else go wow, in some classes we have attended during our life time. were we really that different from the rest of the class? i don’t think so. in some cases we were not even the most talented. we just knew better than the rest of the class how that subject worked. we knew how to study that subject. correct me if i am wrong.

I am not claiming that we can be as good as prof tao, but i think the difference can get much less, if the time is spent on the right thing. instead of going on to the next page of our textbooks without improving our problem solving methods, thinking methods etc.

—and about Henry Poincare, who once said “Mathematicians are born, not made”:

Hilbert was very very intelligent and i think better than poincare but he was completely wrong about the foundations of mathematics, which is a much easier subject to study than human mind. so lets put this quote of poincare aside and don’t think that because he was good at math he has the right to tell us what we can’t do.

((the models i mention are just to give an idea of the perspective from which i am considering the subject. i know that they are very childish and stupid.))

2 January, 2011 at 9:51 pm

science and mathI think genius ness is required to do maths.

But anyone can start doing basic mathematics and doing basic mathematics makes us genius for advanced mathematics.

20 January, 2011 at 7:55 am

做数学一定要是天才吗？（陶哲轩）[转载自刘小川WordPress] | 宇宙海洋™[…] （原文：Does one have to be a genius to do maths?） […]

25 May, 2011 at 2:22 pm

tyroSays the “Mozart of math” lol…

29 May, 2011 at 1:10 am

science and mathHmm.

Do you know any mathematician who is not a genius?

To practice mathematics we don’t need genius , But to do mathematics we must be a genius.

26 July, 2013 at 9:30 am

varunramaprasadthere is no clear cut difference between practicing and doing math!

31 May, 2011 at 4:47 am

“the capacity to be alone” « overcome man[…] thought that Einstein’s famous quote (of genius being 90% perspiration) and more recently Terrence Tao’s comments were merely modesty on their behalf. Now I understand that there is, of course, no modesty […]

9 November, 2014 at 6:27 am

Mike RossThat quote is attributed to Edison, not Einstein!

8 June, 2011 at 5:32 pm

everybodyelseThe distinction between knowledge and inspiration is clear one when looked at from the perspective of the source of the material. From the perspective of final results the two are indistinguishable. Along these lines genius is favorable. From a third standpoint of practicing a discipline as a livelihood I believe knowledge is the more favorable quality since we are limited to but one life and, perhaps, insuring our contributions (to more than just math, but for the sake of this discussion…) are in favorable proportion to the time spent is our only duty.

9 June, 2011 at 3:20 am

NgocAccording to the researches of psychologist Lewis Terman, after an IQ of 120 the intelligence quotient becomes superfluous and even irrelevant in predicting academic and vocational success. More important things like hard work, motivation, perseverance, desire, and determination come into play at above this level of IQ. What’s remarkable is that an IQ of 120 is significantly below the popular notion of a “genius IQ” of 140 or above–or, better yet, the more rarefied genius IQ of 160 or above. In other words, a genius-level IQ or amount of ability is unnecessary–or not required–for success in academics and vocationally. Although a fair amount of natural ability always helps, this research does tend to support Dr. Tao’s emphatic statement at the beginning of this thread that hard work and application rather than some kind of magical, inspired “genius” makes good mathematicians at the end of the day.

12 June, 2011 at 1:37 am

AnonymousTerman was not using his study to analyse the success of people in academia. You can’t extrapolate his results to academic professions, where intelligence (or IQ if you will) plays a greater role (sorry, I’m not trying to ‘bash’ you or anything). A person who has a reliable IQ (I use IQ loosely, but I really mean IQ based on tests that better correlate with ‘g’ or ‘fluid intelligence’, such as the raven’s advanced progressive matrices test. Whenever I refer to intelligence, I mean it in terms of problem solving talent) estimate of ~160 will, on average, contribute to maths and other fields (e.g., physics) more substantially than a person who has a reliable IQ estimate of ~120. I would argue that Feynman’s low estimated IQ was an unreliable estimate of his raw problem solving talent. Bottom line is, intelligence plays a crucial role whether one likes to admit it or not. The difference between a person of 150 IQ and 160 IQ probably means little in terms of determining who would contribute more to maths or ‘hard sciences’, but in my opinion, the difference between 120 and 160, or even 140 and 160 is very noticable. The 120 IQ person would probably contribute very little.

Maybe you don’t have to be a genius to do maths (or any of the hard sciences, especially the ones of a more theoretical nature), but you better be at least close to one if you want to have a ‘good chance’ of contributing anything substantial. Of course hard work is always required. But if you don’t have the intelligence, hard work will not ALWAYS make up for it.

For fields like economics, finance, literature, etc, intelligence plays less of a crucial role as they aren’t usually involved with solving very difficult problems. Therefore, success in these fields depends less on how intelligent you are.

‘Geniusness’ or close to it doesn’t, by itself alone, predict ‘success’ in maths but it is probably a necessary condition. However, if you only go into fields like maths based on how well you think you can contribute to it, then you probably wouldn’t contribute much, since it would appear that you don’t have the intrinsic motivation to study it to begin with.

These are all, obviously, only opinions.

12 June, 2011 at 9:34 am

AnonymousI believe there have been some ‘Seers’ in the history of Math and other sciences, Newton, Riemann, Einstein, etc. But to get to that vantage point you must first be a hard working technician. And anyone who can carry on an intelligent mathematical discussion with Prof Tao and his colleagues are all geniuses to me.

12 June, 2011 at 10:46 am

NgocYes, I agree. Compared to us (mathematically, especially, that is) they are geniuses to you and me–as Prof. Tao and colleagues in his circle are some of the best mathemticians in the world today, no doubt.

Furthermore, it is not everyday that someone wins a Field’s Medal for mathematics.

25 August, 2011 at 11:59 pm

AnonymousWhile I agree that hard work increases your knowledge of any subject you may study, I also believe that if one’s mathematical abilities are not nurtured/stimulated at a young enough age, they may become stale and limited. While I was an undergrad, I never met anyone who had not been of the “math olympiad” type who was as advanced as those who had both the nurturing and the curiosity.

2 September, 2011 at 7:27 pm

Mathematics: What advice would you give to someone who is just beginning to learn mathematics? - Quora[…] , which is by and large is a subject not like Philosophy . .http://web.mit.edu/newsoffice/20…https://terrytao.wordpress.com/ca…Read Blogs Of Mathematicians too.Rahul Rai • 8:27pmView All 2 Comments Edit Link Text Show […]

12 September, 2011 at 12:47 pm

exogenistI found this post exciting and inspirational.

Exciting because Tao, with his main post and various commentaries, has given an image of modern mathematics analogous to the image of mastering an art, such as playing the piano or mastering kung fu.

This bodes well with me because I cannot be certain of my abilities through “Natural talent”, however I can be certain when hard work and experience has internalized, as second nature, the “essence” of thing for which I want to master.

This brings me to why I found this post inspirational. Perhaps the best motivation I have found for learning mathematics (currently at an undergraduate level) is its similarities to learning an instrument. For me to be able to play the piano well, I had to figure out a way to internalize the piano. That is I had to figure out a way to hear clearly what it is in my mind to play and how to actually play it. To do so I had to understand scales, notation, rhythmic patterns, and harmonies. After I understood it I found that I could see these rhythms and patterns on the keys of the piano and play them. The beauty I found in that process is that my knowledge of the piano became more general and as it did I found that my skill increased. If mathematics is anything similar then the only real limit is one’s inability to internalize its various parts.

To me it seems that IQ is a distraction with the danger of emotionally inhibiting one from internalizing a subject in the attempts of mastery.

22 September, 2011 at 10:41 pm

Joshua Lee HarwoodI admire your article and what you included in it. However, allow yourself to consider the possibilities of the unknown (as of yet). Specifically, I mean genius is not a certain type of person but rather a cultivated manner of approaching information and processing it, to get results without focus on the means hindering clarity of the focus on the goals one sets up for their thinking. To learn like a genius, one must learn to learn everyday more and more effectively so that the quality of thought can improve quality of life, or what comes from thoughts. So, I ask as a friend, please do not yet be dismissive of the idea that people can “produce these (very rare) inspirations on…a regular basis…with reliably consistent correctness”. The axiom “If there is a will, there is a way” consists not of empty words, I believe. I ask that you look up “e-prime” (a.k.a. “English-prime) and try to use it. It is difficult, but with practice it is a very useful tool for expression of thought. I admit that clarity fails me in moments I wish it does not, but I admit my limitations when they manifest. However, I will not surrender to limiting my thinking. I think a couple traits of genius are optimism tailored with determination. Results always require hard work, but that doesnt mean hard work has to be unpleasant or unrewarding. In that way, hard work is not hard, but very rewarding indeed and depending on one’s perception can seem easy to some who choose that paradigm. Paradigm shifting, in my opinion, perhaps can be viewed as the most difficult task for the human brain to perform with mastery. I intend mastery not as an actual ends, but a means towarding attaining continual improvement. Some individuals could generate inspired thoughts regularly that turns out correct more so than others, but perhaps the education system could improve towards a goal of leading all learners towards ways of thinking that allow them to communicate, process, and express thought best. Planning, hard work, etc. are secondary tools people with genius traits (which all people can cultivate–I view all people as geniuses, but regard those with greater governance over their learning and paradigm shifting capabilities as more applicative of the genius traits), as I believe the primary tools for any genius or person, really, are how they think and what they do with what they know, learn, experience, etc. All tools are up for grabs for any genius/person who simply are driven towards a goal. It is a mindset, rather than a specific modality really, but to impose limits on a person or to make any claims of what works best for all would be to miss the big picture. Essentially, geniuses learn to become big-picture thinkers (they will ask “how can I find the general in the specific?” and I will repeat the saying “As it is above, it is below.”). They also learn to think in analogy, learn from allegory, think symbolically, hone their visual-spatial reasoning skills, hone other skills, and essentially not just think with language or one modality of thought. Their thinking becomes all-encompassing, integrative, holistic, and expansive yet well-managed. For an area of weakenness in a certain area of skill or thought, a genius will find ways of strengthening themselves (self-reliance) and that allows executive functioning to improve as the person sees fit. Interpersonal effectiveness is another important tool for many reasons. Actually, the 4 skills learned in any Dialectical Behavioral Therapy group (interpersonal effectiveness, mindfulness, emotion regulation, and distress tolerance) can help many people clear any blocks preventing them from holistic, overall growth for their mind-soul-body.

I hope that my words do not fall on deaf ears, but on those who can benefit from any grains of salt they may find.

Thank you for your post Mr. Terence Tao!!

23 September, 2011 at 7:41 am

rumpydogThe fear of math is so wide-spread. It’s difficult to convince people in early child care that when they ask children simple questions such as “how many do you have there?” or encourage cooking that you’re instilling a love of math.

5 October, 2011 at 3:49 am

MiroDear Prof. Tao,

I would like to ask You one question. I was wondering if someone’s math talents had to be discovered in early stages of his/her life in order to be a great mathematician? I mean, if we take a look at all great mathematicians, they all were gifted at younger stages. But what if someone did not have conditions to spot and develop his talent and started to be fascinated by mathematics and ideas at the age of 20? I would like You to give me some counter-example and to throw some light on this topic. Thank You

4 November, 2011 at 1:01 pm

AshwinHi Miro!

I think a suitable counter-example to this would be Harish-Chandra, who actually shifted from Physics to Mathematics during his PhD

Source: http://en.wikipedia.org/wiki/Harish-Chandra

6 October, 2011 at 2:13 pm

AnonymousThat genius race are High Functioning Autistics

6 October, 2011 at 2:33 pm

SungwookIQ test generates too much illusion that it actually measures one’s natural intelligence or smartness. As someone said, it only measures how good of an IQ tester you are.

1) Believe or not, you can improve IQ test score by PRACTICE. See for yourself. It is no more than a computer game which some people may show great playing ability at first but eventually most of people can excel with lots of playing time. There was a time when some people believed Tetris can be a good measure of how good one’s brain is.

2) IQ test was NOT created to evaluate adult’s brain capability. Researchers devised the test to see if they can discover children with learning disabilities. That means low score is the only meaningful criteria. The test was designed to estimate if the children are behind or advanced to their own peers. In other words, it only shows one has a brain as good as normal adults with perfect score.

3) Despite these facts, IQ test can sometimes distinguish smart/less smart people in certain areas. However, so can other measurements (e.g. education, family background, living environment). The issue here is how accurate the measurement is in evaluation of brain functionality. Basically, you will have to measure how fast and efficiently your neurons are working along with density of neurons. Even then, this measurement may have nothing to do in judging one is a genius or not.

4) Nobody’s brain is infinite and so is genius’ brain. One can maximize the productivity in a certain area by eliminating the need of wasting the brain in other areas and solely focus on one area, say mathematics. You can brag about how much you know and fast you can solve a problem; however, developing a new idea is limited to still to the areas you are familiar with. Unlike 1600s, there is too much to know and understand to master the entire area of mathematics or a piece of area in mathematics.

5) Speaking of understanding, how do you accelerate the speed of understanding? As prof. Tao pointed, one should have enough knowledge to process the reasoning. As you know more about the field, your brain can eliminate the things that are irrelevant to the subject and also relate to the ones you already know; thus, works more efficiently. For instance, study the strategies in chess, then, you would be able to beat normal people in most of times in chess game. No matter how genius(?) an individual is, s/he cannot beat out an expert in understanding the subject without proper training or learning depending on the difficulty of subjects.

Still, the illusion of genius is fun to enjoy in a similar way to we want to admire an excellent individual in each area. I think it is no different than praising a sports star. To me, genius is a person who understands the most of social dynamics. Those insights are not something that can be taught, rather something that one should develop his own based on everyday life.

11 November, 2011 at 5:15 am

Matias HeikkiläDear Dr. Tao

I’m an undergrad in physics and I have been thinking about changing my subject to pure mathematics since I find it more interesting. However, it is a field I personally find slightly intimidating due to the notion of genius often attached to it.

Thank you for this inspiring and encouraging post.

Matias Heikkilä

20 November, 2011 at 4:33 pm

AnonymousHard work can only take you so far. This article gives some insight into the true picture – http://www.nytimes.com/2011/11/20/opinion/sunday/sorry-strivers-talent-matters.html?_r=1&ref=opinion

22 November, 2011 at 8:25 am

maryamThank u so much for ur post.I actually hold a phd in pure maths and after 6 years of teaching i am still unable to publish a single paper and that makes go mad and very unhappy. i become to think that i am a failure!

I think as u rightly put it, maths needs a very hard work, knowledge, and patience a lot of patience.

Thank you again

maryam

23 November, 2011 at 8:06 pm

why it actually matters « Math For Parents[…] earlier posts I’d mentioned interesting articles on mathematics by Bill Thurston, Terry Tao, and Paul Lockhart. Today I’m going to try and relate these articles to the topic of […]

28 November, 2011 at 7:50 pm

TimThanks for the post..How do you deal with being discouraged? Or getting bored or frustrated with a problem? How often would you say this happens to you?

11 December, 2011 at 4:36 pm

MaryTerry,

Since in your quote you seem to dismiss the notion of genius..Do you consider yourself a genius? Capable of understanding beyond else..or simply as gifted and skilled (through training and practice) as some of the other best mathematicians of our time. And perhaps then do you agree that referring to certain people such as Einstein as having superhuman “genius” overrates them and seems to make them seem smarter than anyone else when such a notion is false as well.

Thanks,

Mary

29 December, 2011 at 7:54 am

AADIL AMANDear prof. Tao,

You said that “professional mathematics is not a sport”. But we all say that (and I believe) that though its not a sport but it needs a great sporty spirit to play this game. Well, we all know about the universal definition of sports as well as the “game” MATHEMATICS. In sports we keep a very close eye on the opponent’s behavior, whereas here we don’t need it at all. But certainly its a “GAME” which we like to play. “Sporty spirit” is all what you described by the motivating words like “working hard” collaborating etc. and all that you described. So, I would be very happy if you raise your opinion on my topic here.

Also, THANKS a LOT for this post. Indeed a very nice post here that you posted. I simply LOVED it. This gave me a lot of confidence as a student of MATHEMATICS as well as a Sport person.

3 January, 2012 at 5:25 am

Để làm Toán có cần phải là một thiên tài? « 59CLC's Blog[…] Tổng hợp California, Los Angeles, Mỹ. Bài viết này có tiêu đề Tiếng Anh là “Does one have to be a genius to do maths?”. Dưới đây là bản dịch Tiếng Việt của TS Cung Thế Anh, đăng trên Nội san […]

13 January, 2012 at 6:49 am

RendellThank you for sharing your brilliant ideas. Now that I understand that you don’t need to be a genius to study advance mathematics, I am now more motivated to continue my study of mathematics. I have always wanted to have a masters degree and a phd but I can’t afford the expenses and the time needed to achieve these degrees. I am thinking of exploring applied math over pure math. I wish I won’t be labelled as a crackpot if I try to publish papers.

21 January, 2012 at 10:47 pm

QuoraWhat is it like to have an understanding of very advanced mathematics?…* You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a few powerful gene…

5 February, 2012 at 10:15 pm

Applying Polya’s principles to problem solving | Between numbers[…] do a couple of mistakes, it’s the big picture that counts. You can learn more about this at this awesome post by professor […]

4 March, 2012 at 1:36 am

Harimohan singhMathe is magic

31 March, 2012 at 3:50 pm

Boltzmann方程，H定理和非平衡态统计力学 « Fight with Infinity[…] 09年在清华的时候，卢老师是我分析课的任课老师。他极其敬业(令我印象很深的是他曾自费打印Tao谈数学的文章分发给全班)，对我个人也很关照。那时我隐约知道他是做统计物理学的。10年Fields奖颁出时我已南下赴港，没能听到他对Villina工作的评论。另一方面，港大的Mok恰好是这届Fields奖的评委之一，不过他也没有公开评论获奖者的工作。 […]

9 May, 2012 at 2:47 am

02.01.11 | Learning Jigsaw[…] mental maths teachingThe Benefits of a Business Checking AccountSmart Energy ConsultBUSINESSDoes one have to be a genius to do maths var analyticsFileTypes = ['']; var analyticsEventTracking = 'enabled'; var _gaq = _gaq || []; […]

25 May, 2012 at 3:45 am

Seng Hoi NgWhat’s the motivation behind the creation of a new branch of mathematics? Without this motivation coupled with a lack of insights, how is one able to carve out a new path in the darkness of the night?

29 May, 2012 at 6:24 pm

Some final life lessons from grad school « Life of Bai[…] of the things above were inspired by reading Some of these were inspired by Terence Tao’s blog. Like this:LikeBe the first to like this post. Leave a […]

31 May, 2012 at 9:30 am

AnonymousI fully agree with the views of author that one need not be genius to be a mathematician. Though some sort of intelligence coupled with common sense ,hungriness to face new challenges,dedication etc. In my opinion it is the environment created around you that determines you performane. Moreover one should not compare himself with other mathematicians . In fact he should keep on doing good to the subject and ultimately one will achieve better results .

8 June, 2012 at 8:11 am

[An] IQ test … only measures how good of an IQ... • see things differently[…] in each area. I think it is no different than praising a sports star. Sungwook Moon(Source: terrytao.wordpress.com) View the discussion […]

4 July, 2012 at 9:13 am

AnonymousDr. Tao,

the post is brilliant and very reassuring, especially this comes from a former child prodigy . A man’s passion is often worth more than some magical genes spurring innate talent. Of course, intelligence may also plays a role in such field, but without the fascination and the desire to do mathematics, genius is nothing but empty and purposeless.

4 July, 2012 at 11:08 pm

AnonymousThis is excellent advice – and can save your research career.

9 July, 2012 at 9:12 pm

MichaelThanks TT…very inspirational. I tend not to read math papers because I don’t want to take the time to learn all the rules. Maybe someday I’ll feel up to it.

16 July, 2012 at 7:59 am

AnonymousStricktly speaking, i think, the man who are inborn genius(of math) would perform well on math as likely as not. BUT, we must not forget the chances or fully probable possibilities that we can make a excellent geniuses (of effort). i think with confidence the very person who is really respectable is the latter. (relatively)

12 August, 2012 at 8:17 am

LaurieNo one in this thread seems to have entertained the idea that determination itself may be innate (or at least learned at a very young age). Consider the experiment with young children by Walter Mischel and the follow-up studies.

16 August, 2012 at 12:45 am

marcHow do you know if you have “a reasonable amount of intelligence”?

Im not peticulary good at math, as in, i have no innate talent for the subject (i did pass all my math courses with alot of study). However i want to take Bsc in physics. I am wondering, can i do it? can i become good at math through hard work? I have a natural curiosity for science, i quite enjoy math but like i said, im not really that good in it, however i would like to think im quite logical.

5 September, 2012 at 2:06 pm

dagdaki cobanI am stupid, slow and I am trying to learn abstract algebra. But I am very stupid I can barely read 2 3 pages in a few hours

6 September, 2012 at 10:31 am

NgocMaybe you just find the material boring and insipid. Try another source.

6 September, 2012 at 10:55 am

NgocEveryone here on this thread seems impressed by the notion that it takes genius to become a mathematician. Am I the only one here who is more impressed by the alternative notion that it takes possibly superhuman genius for a mathematician to solve Riemann’s Hypothesis, either individually or collectively.

13 December, 2012 at 3:49 am

KahbaAttacking the Riemann Hypothesis is dangerous .

Even if you’re a genius of the caliber of Riemann the price is mental sanity .

This peculiar problem made a lot of mathematicians totally crazy.

Real maths problems have a powerful power to restructure brain links to reality and once a step is taken it’s like going to the mouth of the devil .

I don’t talk about a 1000 page proof made of hundreds of cautious computations and orgical theorems (Person1-Person2-Person3… theorem)

I’m talking about the existence of some direct proof that would be all natural with no magical theorems that hide mathematical reality .

An acultural proof would make it’s genius owner mad .

4 October, 2012 at 5:52 am

AnonmathDear Dr Tao,

I often visit this blog whenever I am down(mentally) to gain encouragement.

A lot of us start learning a subject from books which are highly condensed and have become classics over the course of time. I was wondering if pedagogically, you consider this a sound approach i.e. learning from condensed texts.

As a high school kid, I am trying to battle with Dr Rudin’s Principles of Mathematical Analysis(rather unsuccessfully) and I have become discouraged by the book. (I do have a knowledge of calculus from a proof-based calculus text)Is there a way one can handle the situation?

Hopefully, you will notice this post.Thank you.

6 October, 2012 at 9:41 am

S.I am in a similar situation with the difference that I study in my spare time. I’ve been reading Rudin for at least two years now and I was, and is still, a little discouraged by the book. I didn’t let that discouragement stop me however and I have supplemented the book with others like, How to prove it by J. Velleman, which have been a source of inspiration.

While I am not Prof Tao I did discover a few things. Reading books just above my mathematical maturity level filled in the gaps that Rudin assumes in his principles of mathematical analysis (which, at the time, was far above my maturity level). For instance the book how to prove it goes into techniques for proving various theorems presented in Rudin’s book.

Secondly, the greatest joy I find in studying math is not the end goal of contributing something useful nor winning some prize. It is the act of exploring, discovering and creating mathematical objects. There is a thrill to it which is addictive even if not on a professional level. For instance, I was able to derive the derivative of a continuous function for myself on the assumption that “f(x)+d/h = f(x+1/h) where h>0” . This led me to discover that it is possible to treat 1/h as an “infinitesimal” number. Obviously “infinitesimal” needs definition and soon I found myself deriving the upper bound property of numbers, Axiom of completion in most books, to help me define 1/h as an infinitesimal.

The point here is that books can kill the inspiration of discovering things for one’s self if they are relied upon too heavily. If anything they are a source of techniques, guides, references, hints and examples to develop ones mathematical maturity. As ones mathematical maturity grows books such as Rudin naturally becomes more accessible over time. This is true for anyone who studies math and I’m sure it was true for Prof Tao too. The difference between him and the rest of us is that he, like a prodigious piano student, developed his faster than most.

This does not take away importance for the average practitioner of mathematics because there are more theorems of average difficulty to be discovered than there are Prof Tao’s to discover them. This does not imply that average difficulty theorems are in anyway less important than theorems that would take one’s life time to prove.

2 November, 2012 at 5:09 am

AlexMaybe progress in time is somehow represented by exp(c*g*w*t*x)

g=genius level, innate skills, etc.

w=work

t=time

x=int(random variable,0..t) (luck)

c=some-constant

and if this is the case… SO WHAT??, after all:

“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”

We need all levels of g to progress as a whole not just the one needed to solve poincare conjecture or kayeka conjecture.

You should try your best with your g to be happy, even if maximizing exp(g*w*t*x) is not neccesarily what makes you happy.

10 November, 2012 at 5:57 pm

AnonymousL would like to know how do you think about Srinivasa Ramanujan?

19 November, 2012 at 11:12 pm

AnonymousMaths is the hardest subject 4 me

How I overcome it

11 December, 2012 at 10:43 am

What is it like to have an understanding of very advanced mathematics? « Quantified[…] 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… ]. […]

13 December, 2012 at 2:51 am

Hamid zbibouWhen you say mathematics isn’t a sport ,actually this is contradictory as the only reason you are a mathematician is because you bypass the vast majority of other mathematicians in technical power ,you are a problem solver and not a creator of new mathematics .

You could have been much better if you took the time in your childhood to ponder mathematics more slowly .

You could have been a Grothendieck !

But sports-mathematics made you like the computer version of Ramanujan .

For me your fields medal is of non-importance ,if mathematics was a garden you would be more like a temporary scent in the air .

You are a genius for memorizing and mirroring copious amounts of mathematics but you aren’t a genius in the sense that you have not changed mathematics face .

For me you seem more like a “phénoméne de foire” ,simple question how many intuitions do you have per day ?

I knew Grothendieck indirectly and i can ensure you that hundreds of intuition poured from his mouth…

I know why you made this apology of non-genius it suffices to see your face in the fields medal ceremony and that smile from ear to ear to understand that mathematics is unconsciously for you just a matter of prizes .

11 January, 2013 at 11:26 am

Carl MuellerYou’ve made a terribly ungenerous comment about Terence, who is a terrific mathematician. His blog entry about genius is one of his best; almost every math student and young mathematician is afraid of not being a genius. Just working and enjoying math is the healthiest attitude.

30 January, 2013 at 8:29 am

Envyisabit..Oh oh, someone is green with envy …

26 December, 2012 at 1:30 am

jarronegro.ComDoes one have to be a genius to do maths? Whats new, ended up being a good post title to give this article.

Where could I read even more about this?

26 December, 2012 at 5:18 pm

Exo“(*) The physical world generally refers to tangible objects, but one can also consider abstractions of these objects, abstractions of abstractions, and so forth. ”

Yes! This is the method behind the madness! Abstractions upon abstractions. You see we live in the mind. Everything you observe is by default an abstraction. Our neurons abstract from the senses signals made from an independent source. Then it is abstracted upon by the conscious processes of our physical brain. One must understand and keep to heart that every thought that one believes is an abstraction has a real measurable value of energy in our tangible universe. That is, my thought has a pattern that relates directly with reality or not (Energy may exist in many forms, is it folly then that energy may exist in a form that is entangled with some other form such that a manipulation of one form corresponds to the energy of some other form? Yes people pour minds are entangled with the universe). If I have a thought that has a direct relation to reality and I abstract upon it another thought which also has a direct relation to reality then the abstraction must have a direct relation to reality. But for a thought to be abstracted upon another thought it must connect somehow. A connection is simply a relation between two propositions, Thus A successful mathematician is one who describes correctly the relation between two thoughts which are true descriptions about something which does or does not exist.

13 January, 2013 at 1:30 pm

raindrop11I’ve made/am making a Malay translation of this article here http://eraserboxtips.blogspot.com/2013/01/perlukah-seseorang-menjadi-genius-untuk.html Please tell me if you want it removed

27 January, 2013 at 11:08 pm

Thomas WatsonIf we want to do better in the formal kind of math, we should see how it arises from natural mathematics.

28 January, 2013 at 12:19 pm

S.what do you mean by “arises from natural math”? I understand that the inductive property is natural in that we gain an intuition for it from counting in our natural environment. It becomes formal when we attempt to fit it into some logical system. Thus are you saying that a better understanding of formal systems is directly related to our interpretation or definition or first assumptions of them.

If so I can agree that natural math is indeed dependent upon intuition which is cultivated through experience and observation. But at which point does our intuition fail so that intuition arises from a formal kind of math.

A beautiful example of this is analysis. The derivative indeed arises from various intuitions or natural math dealing with rates. The formality that arises in the use of the derivative forces one to develop ones intuition to deal with seemingly unrelated problems. The development in turn forces one to formalize the difference in their updated intuition and thus a greater understanding of a formal system, with updated techniques and practices, results in one doing better in the formal kind of math. Is this what you mean Mr Watson?

12 February, 2013 at 12:39 pm

mathformathsakeI have read a few of the comments in this blog and I wanted to say this:-

I think the primary motivation to pursue a career should not be success but “I want to do this because I enjoy doing it” attitude. In mathematics, I think the motto should be “I want to understand this!” rather than “I want

to win a Fields medal”.

Eventually someone will “succeed”. Congratulations to him. Note that this does not conflict with with the motto above.

As far as I understand no “genius” ever did something he/she hated doing. Prof. Tao, Ramanujan etc., all do (did) what they really love(d) and they are the “successful” ones.

12 February, 2013 at 2:57 pm

nabilI am with this philosophy :

I want to understand this!” rather than “I want

to win a Fields medal”.

We do some something because we love it!!

23 March, 2013 at 3:08 pm

sundar amarturThanks Prof. Tao. As an electrical engineer I always have felt a sense of inadequacy or an inferiority complex in the presence of colleagues who were trained more extensively in mathematics. Many a time I have felt that I could have taken more advanced courses in mathematics when I was at school (which was many years ago). Now I feel there is no need stop at any point in one’s life to learn. I am doing self-sudy on mathematical subjects I had missed all this time.

sundar

10 April, 2013 at 10:14 am

johnCome on prof Tao, you’re a genius and you know it ;)

26 April, 2013 at 2:18 am

Hyeongjin BaeI agree. I’m not genius, but I’m pleased.

29 April, 2013 at 12:45 am

SimonI thing that maths is the most specially subject, and i want to become a good mathematician, i thing africa and my country need mathematician

3 May, 2013 at 7:17 am

Fighting the “cult of genius” | The College Classroom[…] As both a student and teacher of mathematics, I think about fixed versus growth mindsets all the time. I never considered myself to be naturally adept at mathematics—in fact, in the 7th grade I can remember proudly declaring that I was “an English person” and I wanted to be a writer. Well, that didn’t happen, and it’s been a long, not always smooth journey to where I am today. I firmly believe that my success has been because of a growth mindset and a constant struggle against the “cult of genius” which pervades much of mathematics (a great blog post about this: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/). […]

17 May, 2013 at 6:26 am

The Coffee Stains in my Book | The Journeying Mind[…] might be different if I possessed the genius talents of say Terence Tao but alas I don’t. I have to downgrade my coffee manipulation […]

23 May, 2013 at 9:29 am

mathtuition88Thanks, Professor Tao. This article is really inspirational.

3 June, 2013 at 7:03 am

Bisogna essere un genio per fare matematica? - Maddmaths[…] Samir Chomsky per avermelo segnalato.] (Tradotto per gentile concessione dell'autore, qui il post originale in inglese) Terence Chi-Shen Tao è un matematico australiano nato nel 1975, vincitore nel 2006 della […]

17 June, 2013 at 8:54 am

George DeMarseYou don’t have to be a genius to do mathematics–but you do have to pass algebra 1 and geometry, which approximately half of students fail to do (more pass in well resourced schools–more fail in rural and urban schools).

If the average student does manage to pass these with a C–a 50% failure rate awaits them in college algebra.

This is with “hard work” for many students…but they’ll never get to pre-calc anyway. So much for being a mathematician.

The Sage of Wake Forest

28 June, 2013 at 8:12 am

Rouselle CorderoReading the title gave me hope…that I still have a shot in Math. It interests me until halfway through I got dumbfounded. Crap. Anyway! Kudos Mr. Tao keep doing what you’re doing while I just stick to what I do…cooking Chinese food. haha

19 July, 2013 at 8:57 am

Does one have to be a genius to do maths? | Singapore Maths Tuition[…] Source: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/ […]

11 September, 2013 at 12:13 pm

SamI agree with most of what this article says. You can learn Math if you are highly motivated to learn and you work on it very consistently. Desire and consistency are the two main keys.

I started liking Math years ago when I was in high school. Long story short, one of my teachers was the main reason I got super motivated about Math and started having that burning desire to understand it better.

First thing I did, I purchased a calculus text book and started reading and reading till my brain found a more structured way to approach problems. Once you get to that point, the love for Math is forever and it becomes part of your life.

I think one of the biggest responsibilities of Math teachers today is really to introduce that motivation and desire within each students to learn Math. The rest is a matter of consistency, repetition and hard work.

Thank you for sharing this very valuable article.

13 October, 2013 at 5:52 am

Does one have to be a genius to do maths? | Maths? No Fear! Blog[…] See on terrytao.wordpress.com […]

15 October, 2013 at 4:03 pm

MattHi Terry,

I love math so much, but I’m not really sure I have enough mathematical talent to pursue it as a career. I really didn’t have an interest in high school as I was interested in pursuing music. When I got to college I studied philosophy and logic lead me to look in to set theory which then opened the whole world of math to me. I did reasonably well in my applied math courses for Econ but I wasn’t stellar, but to be fair I really didn’t find them that interesting. I aced calc one, but haven’t taken any math courses after that. I have done a little self-study. I would literally spend hours just flipping through math books bewildered and wishing I could understand the language. I just don’t have the best track record with math, and I’m missing the quickness that everyone seems to value, at least it appears I am. I’m currently torn, because I love math but I don’t think I could be excellent at it and it would drive me crazy. What is your advice? How important is “quickness”(making calculations quickly) to mathematics? Should I pursue this career even if I’m not sure I’ll be great at it?

Oh and by the way, I’m a good friend of Ben’s(grad student at UCLA). Tell him I say hi.

Cheers,

Matt

29 October, 2013 at 8:14 pm

Where to begin? Maths and the idea of maths. | Behind the calculator[…] solves maths problems by plucking the answers from the air is a myth. It needs to be debunked. It has been debunked. What separates mathematicians from non-mathematicians, and good maths students from […]

1 November, 2013 at 1:23 am

Ivan MontesiFor me to think of a “natural talent” it sounds idiotic, not so much for the model with which u try to view certain subjects (genes), but because (obviously not only in mathematics) the concept of genius is something purely abstract that is acquired during the development of neural networks (hard work). According to my point of view we can only speak in favor of luck (statistics), from the series: when a particular subject has learned a certain pattern over the study of a topic, may serve him stuck “better” or during the day he happened a event that led him to bring him back to the problem. So, the point is that if a person is interested can cultivate genius (probably), otherwise no, there isn’t innate talent

Ivan

3 November, 2013 at 9:25 pm

zuchongzhiDear Professor Tao:

You link in the end of the post appear to be broken.

[Can you be specific? I was able to access both links listed at the end. -T.]4 November, 2013 at 9:29 am

Saurabh SharmaIf you want to do anything there is only one thing you need. That is mathematics. Right up from when you learn to how to walk

4 November, 2013 at 7:21 pm

[转载]做数学一定要是天才吗？ | 小小泪[…] 陶哲轩, Does one have to be a genius to do maths? 译文作者: 刘小川, 做数学一定要是天才吗？ 做数学一定要是天才吗？ […]

17 November, 2013 at 9:17 am

elijahdon’t worry about your difficulty in mathematics, i can assure you, mine are still greater – Albert Einstein

8 December, 2013 at 11:44 am

Creatividad | Unlock Knowledge[…] min 04:18] como muestra Donald MacKinnon [21] (a diferencia de como puede suceder en matemática o no?). Ackoffva más allá y […]

13 December, 2013 at 6:30 am

JOf the tens of billions of people who lived over the centuries, a few thousand have stood out in Mathematics. So you are telling me the other 10,000,000,000 – 10,000 have not worked hard?

13 December, 2013 at 8:31 am

Terence TaoNo.

13 December, 2013 at 9:12 am

SAs I encounter (and struggle) with rigorous math for the first time in my math degree, finding your blog was the inspiration I needed. It’s a pleasure to find someone so intelligent, yet humble, that offers sound advice to students of all ages. Truly a child prodigy who has gone on to achieve success and contribute significant value to the community. Thank you.

14 December, 2013 at 7:13 am

JWell may be I should have said “so you think the other $10^10-10^4$ missed one of the following – hard work(which includes learn one or more fields, communicate with other professionals), reasonable intuition(for instance thinking about big picture), literature, and a bit of luck”. Still your counter argument holds. I would think to have a shot at joining the $10^4$ (which is much more than 100 of the possible greatest ones), one needs more than a bit of luck and a bit of reasonable intuition. Of course having a superlative amount of all three but being born in Africa/India will make it tough for you to get into the 10^4. However having superlative gifts and being born in US middle/upper middle class/Europe/Australia makes it more than a bit easier than being born say even in Japan/Korea. Also your ideas on the topic do not explain some facts like why some cultures in some time periods (jewish for instance in the 19th and 20th century) have dominated and have added a lot more to the 10^4 than other cultures. One could argue Jewish/American/European middle class people have more of a good mathematics culture or had luck being part a better society (than say African/South American/Asian ones) that makes it easy for them to access hard resources(books, classes, scholarly people) and soft resources(seminars, being part of an inquisitive culture). However the success statistics in mathematics among the middle class in these societies is still racially biased which seems to lend credence to the good proteins and connections in the human brain from good genes theory for success.

14 December, 2013 at 11:22 am

iDear professor Tao.

Will it be possible in the future for the general public to work on collaborative math? I was involved in an experiment a few years ago where on a forum individuals would pose questions and the forum would offer opinions on them. What happened was it showed that people with differing perspectives and knowledge were able to answer even the most obscure of problems collaboratively. I believe that modern mathematics is ripe to move from the university halls to the screens of our mobiles or computers. I know you are involved in this in some limited or major way but I was hoping to get insight to whether a more organized purposeful approach will be conducted in the future. I have a very strong opinion that modern math will start seeing more problems investigated and researched by a public of people of differing sophistication on all levels and across the board. Math is too wide and too deep for someone of even your level to have touched all its parts. Plus it will provide a way for the lay man to be more interested in a subject that is still clothed in much obscurity.

3 January, 2014 at 8:13 pm

Bạn có cần có những khả năng thật đặc biệt để làm toán ? | Nhật ký Toán học[…] 2/Terence Tao’s Blog -https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/ […]

20 January, 2014 at 6:56 am

Sai Teja SomuDear Terry,

I have an idea,I am sorry if it is silly(I am an amateur mathematician)

Before 15th or 16th century the equation was thought to be having no solutions and rightly so.During that time nth degree polynomial was thought to be having less than or equal to n roots.But when was considered as a number we got nth roots of unity which are clocks modulo n and using those numbers we could prove many theorems which are very difficult to be proved using elementary methods.

Now from Lagrange’s theorem every nth degree polynomial equation modulo a prime has less than or equal to n roots modulo p.The equation

has no solutions.Is it possible to extend the set of natural numbers like real numbers to complex numbers such that every polynomial equation of nth degree has exactly n roots.May be if we do

that we may have generate something useful numbers like nth roots of unity.

20 January, 2014 at 11:23 am

Sai Teja SomuI meant the equation has no solutions

20 January, 2014 at 8:22 am

Yusuf Auwal IbraheeymI need to be mathematician

20 January, 2014 at 8:33 am

Yusuf Auwal IbraheeymI need to be mathematician, because i founded that many students were hesitating the matters related to mathematics. I have the courage of offering it, thats why i like it, although i am still in secondary school.

However in my first year at jenio secondary school i found it extremely difficult to comprehend due to i am considering the mathematics as difficult as running 200km without resting!.

6 February, 2014 at 9:31 am

GilYoung CheongThis is a great answer to one of the questions that “wastes my time” too much!

I think it is also relevant to ask how to distinguish between studying mathematics and taking various exams on mathematics. I am not sure if you wrote about this before (if so, I would appreciate a link), but it seems to me that you have thought about this.

14 February, 2014 at 3:41 am

Kai DuI agree with Terence totally.

The fact, I think, is that mathematicians are a huge team that constructs the huge building of Mathematics, consisting of rare designers and chief engineers who are definitely great genius, and countless builders and workers with varies of levels. Not everybody is talent, but everybody is doing his work in the process.

19 February, 2014 at 12:07 am

Hyeong-JinCan I scrap this post?

30 March, 2014 at 3:36 pm

Tiansu YooI’m very appreciate your opinions.

And I noticed you mentioned about “literature”, which contribute to one’s mathematical thinking. Out of my favorite in it, I’d like to ask: Do you like literature? What kind of book would you usually to read besides math?

7 April, 2014 at 2:40 pm

EliI am a freshman at MIT who is possibly (and maybe now probably) going to go into mathematics. I’ve always loved math, but I constantly feel like I’m the only “normal” person in a sea of geniuses here, and I have never even thought that I could make it in the math community. I literally started sobbing when I read the first line of the post, and it made me really think that maybe I could actually contribute something. Thank you, Terence.

30 April, 2014 at 3:32 am

AnonymousYou got into MIT …need I say more? :)

7 November, 2014 at 7:03 pm

Ngoc NguyenI agree with Anonymous. A freshman at MIT? You should be able to accomplish anything that you put your mind to, including maths. Congrats, Eli!

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”