Does one have to be a genius to do maths?

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.

329 comments

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18 June, 2007 at 3:19 am

beans

So there’s hope yet for me then! ;p

I think because it’s human nature to look at and ‘compare’ yourself to people who are ‘cleverer’ than you, we automatically assume that they’re geniuses. (However I have to say that most of my lecturers are geniuses!)

It seems that all maths undergraduates go through the same notions! Nice post. :)

26 May, 2010 at 8:08 am

Bryant

Beans,

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

Hamza

Braynt 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

Ashwin

This will help the beginners:

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

19 June, 2007 at 10:28 pm

Pacha Nambi

I think doing any creative work in mathematics by anyone (genius or not) should be encouraged, recognized and appreciated. Also, one must enjoy doing creative work – without expecting any reward or recognition for it. Recognition and reward will come automatically to those who do good work in due course of time. Even publishing just one very good paper per year in a highly respected journal should be recognized. Quality more than quantity is what matters ultimately.

Thanks for giving me an opportunity to express my opinion on this subject.

20 June, 2007 at 12:29 pm

William Gasarch

AGREE with post. Also – there are so many different types of abilities in
math that people are good at in different combination, so a linear ordering
is not only a bad idea for getting work done, but perhaps impossible.
A few contrasts: problem solver/problem maker, algebraic/geometric,
clever/knowledgable, getting idea for proof/pinning down proof rigorously,
good reader of math/good listener to math. A clever problems solver who
can listen to math and get it quickly may LOOK more impressive,
but a knowledtable problem maker who is better off reading stuff
is just as valuable.

bill gasarch

20 June, 2007 at 1:13 pm

John Armstrong

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.

Is math better done in quiet solitude or screaming across the alkali flats in a jet-powered, monkey-navigated…

You’re right as usual in your description, but surely you must recognize that some sides of these contrasts are selected for and some are selected against in the marketplace. To get a job it helps to be more of a problem solver than a theory builder, and woe betide those who hitch their wagons to the star of categories instead of analysis.

20 June, 2007 at 10:11 pm

Anonymous

“Do not disbelieve the prophecies simply because you had a hand in carrying them out.”
J.R.R. Tolkein

Isn’t what you are really saying just that genius is an emergent phenomenon?
(See: http://en.wikipedia.org/wiki/Emergence)

Then is it not the case that whether one worships it or finds it romantic
is a simple matter of theology?

26 January, 2013 at 3:04 pm

Stephen

Just… 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

Kamalayka

You have too, many, commas
Short of breath, brah?

7 March, 2013 at 2:57 pm

Anonymous

haha… 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

regina

I am teacher of math and , actually I am writing books that help me to share my teachings: math + art + Science and history

It is a way to demystify those subjects

I would like to present to you the series of books entitled, “Caius Zip – The Time Traveller,”
The main idea behind the “CAIUS ZIP – The Time Traveller” series is to show the history made by great men and how mathematics and other subjects were important in their decisions. Caius Zip is a young man that participates in these discoveries and in the great battles. In each adventure, he acquires maturity and learns that to get out of trouble he must use his most important ability that he unknowingly uses very well: the power of deduction

The first book, ” Einstein, Picasso, Agatha and Chaplin:, How to explain the theory of relativity, cubism, travelling in time and unmask a murderer ” has been published

Description

Caius Zip, the young time traveller, arrives at Paris in 1905. The turn of the 20th century is a period that sizzles with ideas and realizations and the Universe is about to be contemplated as it never was before.

In this fiction, Einstein was resting in Paris before his innovating Theory of Relativity enlightened him. At that same time, Picasso was just starting on his idea of breaking with conventional perspective.

Both characters seek the same concept: space-time relation. The encounter between art and science is finally possible by means of a limitless imagination.

There are the descriptions of interesting places of the belle époque in Paris and the memorable dialogue between Caius, Einstein, Picasso, Agatha, André Salmon, the poet and Getrude Stein, the sponsor of the novice Picasso, at the Spanish painter’s atelier on how art, literature, science, travelling in time and mystery are intertwined.

Caius penetrates the birth of the theory of relativity and cubism and also manages to solve a murder mystery with the help of his two teenage friends, Agatha Christie, with her investigative mind and Charlie Chaplin, who provides a touch of magic to this surprising work of fiction.

After all and as Einstein once said: “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed”.

See a passage from the book in: http://www.caiuszip.com/relativiting.htm

best regard
Regina

21 June, 2007 at 10:34 am

Terence Tao

Dear John,

It’s interesting that you perceive analysis and problem solving to be so fashionable these days; this seems to me to be a relatively recent phenomenon. In the height of the Bourbaki era, for instance, I would have said that the emphasis in mathematics was more in algebra and in theory-building (and for quite a long time, this focus was spectacularly productive). And, of course, much of modern mathematics also involves a lot of geometry, topology, combinatorics, physics, etc. If anything, I think the marketplace these days favours mathematics which has some interdisciplinary component – need one hitch one’s wagon to merely one star?

Dear anonymous,

I would say that the perception of genius is an emergent phenomenon, and has non-trivial impact. But the concept of genius itself appears to play a minor role at best in any accurate model of mathematical ability and achievement. (The same, incidentally, appears to also be true for prophecy and models of prediction.)

2 October, 2014 at 4:14 am

Ashwin

Dear 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 Cohen

As 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 Armstrong

I think the marketplace these days favours mathematics which has some interdisciplinary component

In truth I have only a limited data set, but I see my colleagues making calculations of known invariants getting more and better job offers than those who extend the domains of invariants or push for new ones. At the very start of one’s career, being a problem-solver means not having to convince hiring committees that the problems you work on are interesting. Being a theory-builder means you have to do exactly that.

I have not seen an analyst of my acquaintance struggle like the algebraic topologists often do. Fully half of the job offers I saw this last cycle said “analysis”, “functional analysis”, “applied analysis”, “dynamical systems”, and so on. They weren’t all exclusive, but they were strongly preferred. A scant handful claimed to want a topologist.

Interdisciplinarity is all well and good in theory, but in practice people want to put you in a nice, tidy box, and “category” is an easy box for many hiring committee members to put on the shelf and forget, even if you’re using it as a tool for studying something much more down-to-Earth than many older mathematicians think categories are used for.

21 June, 2007 at 12:42 pm

steven katz

I do not know whether you have already explained yourself on the subject and I am afraid my post is not at its proper place here, but I wish you could one time express your views concerning the nature of mathematics. I mean many people who could be described as great genius (es ? English is not my native language..) like Gödel seem to be on the platonic side as if their experience lead them there. Do you feel yoursel in this case, I mean do you really feel like evolving in a real mathematical universe independ of your existence, and does “mathematical intuition’ seem to you a kind of objective perception, like vision for physical objects, which is not equally distributed among people or is it just a romantic and naive point of view ?

21 June, 2007 at 3:42 pm

Anonymous

I don’t believe that the concept of genius is intended as a model to describe
how mathematical ability and achievement come about, a microscopic model, if you will. Rather, it is intended as a kind of macroscopic model for predicting what sort of mathematical achievements one can expect and from whom.

I believe it works out pretty well for this purpose as certain foundations which provide you with support might be willing to attest.

21 June, 2007 at 3:46 pm

Anonymous

N.B. Tolkein’s point about the prophecies was the same. They
were not intended to say how things would come about, only that they
would. Knowing more details about how they came about did not invalidate them.

21 June, 2007 at 8:32 pm

Sitting Duc » What it takes

[…] Terry Tao wrote in his article that mathematics is not a sport and it does not take a genius in order to do math. One needs plenty of patience and hard work. Reading his article, I feel a lot better in myself. I truly believe that if I don’t land on the moon, I would still land on another moon. Here I come, UFL!!!! […]

27 June, 2007 at 8:38 am

Terence Tao

Dear Steven,

My view is that mathematics is primarily a language for modeling the physical world, or various abstractions of the physical world(*). So in one sense it is purely formal, in much the same way that English is a formal combination of letters of the alphabet. On the other hand, as our understanding of mathematics improves, our models fit the physical world better (both in terms of predictive power, and in terms of agreement with physical intuition) and so the mathematical objects we study begin to more closely resemble physical objects, though of course they are never actually physical in nature. It is certainly helpful though, when trying to create new mathematics, to think of mathematical objects as being analogous to physical objects; for instance, a mathematical object may “obstruct” another mathematical operation from taking place, and thinking about obstructions is a very useful way to make progress in mathematics.

(*) The physical world generally refers to tangible objects, but one can also consider abstractions of these objects, abstractions of abstractions, and so forth. For instance, a children’s ball is a physical object; it might be red. The property of “redness” is then an intangible abstraction, but still physical. The phenomenon of “colour” is then an abstraction of an abstraction, but again still physical; the concept of a “sense” is a yet further abstraction; and so forth. Somewhat analogously, mathematics tends to start with “primitive” objects such as numbers or points, then moves up to sets, spaces, operations and relations, then functions, then operators, (and then functors and natural transformations, in category theory), etc. [It’s true that in set theory, all of these mathematical objects can be described as sets – much as all parts of speech in English can be described as strings of letters – but this is only one of many equally valid interpretations of these objects, and not one which corresponds perfectly to physical intuition.]

10 April, 2012 at 10:37 pm

hcai

Very 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&Omega

May 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

Chris

I 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

Shibi

Dear Terry,

Pardon me if i am wrong in asking this question(or asking it the wrong way). But do you mean to say that there is no such thing as genius? I mean i always thought that someone like Ramanujan was a genius. Also, i would like to ask you about the relative levels of understanding. I mean some people have to read several books and solve endless number of problems before they understand a topic, which might take months or years, while someone like Riemann was purported to have read and understood a great classic book in just a week’s time. Do you think that the “time” it takes to understand things(i guess this significantly varies amongst various people) affects their research. Also about understanding a concept the “correct” way. i mean a few (brilliant?) people just learn stuff in a clear crisp manner. the fundamental concepts just stick in their head . while for others they tie themselves up in hopeless logical knots and endless circles of misunderstanding and understanding things the wrong way. this obviusly affects them and their research;
Also about people who seem to have “physical insight” like Geoffrey Taylor in Fluid mechanics. He was probably not as great a Mathematician but he had the uncanny knack of separating the “essence” of a problem and reducing it to a manageable portion so as to solve it. He himself claimed that he was no good at things like Chess or Crossword puzzles but he seems to have this amazing amazing ability to solve Fluid Dynamic problems. What do you have to say about those kinds of people;

As a humdrum workaday(and defintiely not brilliant) graduate student in Aerospace Engineering this essay has given me immense solace and comfort. i really want to thank you for posting this essay.

29 June, 2007 at 1:58 pm

Terence Tao

Dear Shibi,

It is true that some mathematicians can be vastly more efficient than others at learning material, but I feel this is more due to experience and an efficient means of study than to any innate genius ability, though of course innate talent is still a contributing factor. For instance, among the graduate students I have advised, the first paper they read in a subject often takes a month or so to read (and they have a question on almost every page on the paper); but after a few years, they can get the gist of a new paper in the subject within a day, skimming past all the “standard” (or at least “plausible”) portions of the argument and focusing on the key new ideas. The key, I think, is to find one or more efficient ways to internalise the subject – either by using formalism, or geometric intuition, or physical intuition, or some other analogy or heuristic. Each mathematician has their own different way of doing this. Ramanujan, for instance, apparently performed a tremendous number of numerical computations, and derived much of his intuition from the patterns he observed from those computations. The intuition wasn’t always correct (for instance, he famously gave an incorrect formula for the n^th prime), but he did discover a number of amazing results this way, some of which took a long time to prove rigorously.

31 October, 2014 at 12:57 pm

Anonymous

Wonder 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.

29 June, 2007 at 8:00 pm

Jonathan Vos Post

Of course, I mean “a person’s speed of performance of a task INCREASES as the RECIPROCAL of the log of the number of repetitions” or “a person’s length of time to complete the performance of a task declines as the log of the number of repetitions.”

*sigh* Sorry.

6 July, 2007 at 10:33 am

averageJOE

I would tend to agree with those who believe in genius. Certainly , a person of my limited talents, given a large finite amount of time could not hope to come close to understanding the works created by brilliant mathematicians ( yourself included). And then there is creativity which seems to defy all measurable tests ( IQ exams included) … perhaps this is what is meant by intuition and insight … the benefit of due dilligence and repetition with opens eyes to see paterns and to generate insight into otherwise seemingly unrelated problems… anyways… I found this discussion interesting and thought I should drop my 2 cents worth… To Terrance Tao … your son, does he possess the innane gifts that you have or is he like the rest of us .. ( or maybe somewhere in between?) … just curious … the question hints of inuendo of what percentage of genius is environmental vs. genetic.

26 January, 2013 at 2:39 pm

Stephen

Creativity 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

franck

Dear Professor Tao
Even from an outside point of view (I am not a mathematician) I agree with most of your point. The word “genius” is not precise enough, and even if there are some real genius in each generation – and in many ways you seem to be the nearest of a mathematical genius humanity has those days – the mathematical field is far too great to be explored only by genius.
But in the other hand, I don’t see why, nowaday, a medium gifted mathematical student would choose a mathematical career. In my case, for exemple, I was a schoolboy with some gifts in mathematics – nothing like competing in International Olympiads or anything, but easy with the concepts in high school and university levels. Being a frenchman, I “naturally” went to one of the best “grande école” and got an engineering degree with a lot of mathematical insights. There I met many people more gifted than I was, and undestood (maybe late !) that even if I could make a living in mathematical research and teaching, I probably would not become a well known figure in that area, and would earn much less money (with a factor of 3 or 4) than I would in industry.
So I got a master in Telecommunications and IT and now, at 40, I am deputy CIO of a medium french company. The job is usually interesting, very involving and I make good money – not much according to US Standards, but we are in France :).
But in the back in my mind there is always the idea that what I do has some futility, and that in the big picture scientific research, especially mathematics, is, probably with art, the only activity we human must me proud.

15 July, 2007 at 10:11 pm

Johm Park

Dear prof Tao,

I thank you for posting this article. But I guess I should not use this objective article as false encouragement.
Would you please define what is the ‘reasonable amount of talent’?
I was born outside of America, am a 25 year-old economics graduate enrolled, think I love math, though I think I am not much brilliant or especially knowledgeable, and am considering the options of either studing econ in America or pursuing Law career in my own country(I didnt major in math as undergraduate).
Since this ‘natural’ talent question’s the main psychological blockage to my choosing the first option, your help would make my decision a lot more objective and lead me to better chance of offering something good of my own.

17 July, 2007 at 2:20 am

kaiming

I believe it is true that one does not have to be a genius to do maths.

But if you are smart as a genius, you have a greater chance to make discoveries or solve mathematical problems as important as those by Albert Einstein or Isaac Newton. The less capable persons will only make the less important achievements.

17 July, 2007 at 8:50 am

Terence Tao

Dear Johm,

It is true that graduate economics these days does require a certain amount of mathematics (notably several-variable calculus, probability, and game theory). See this post:

http://gregmankiw.blogspot.com/2006/09/why-aspiring-economists-need-math.html

As for “reasonable intelligence”, it is hard to define precisely, but roughly speaking I refer to the ability to analyse situations (particularly hypothetical or other abstract situations) at a level deeper than a superficial level, e.g. by drawing analogies, asking questions, formulating and then testing hypotheses, reasoning logically, finding ways to double-check one’s conclusions, identifying any limitations to one’s analysis, and so forth. It’s not a skill that is restricted to scientific or mathematical situations – the same type of skills are also needed to a diagnose a mysterious computer bug, solve a crossword clue that lies outside your direct knowledge base, or dissect the complicated plot of a novel or movie. But one does need some level of higher mental faculties in order to do mathematics; it’s not a subject that can really be learned by, say, memorising flash cards.

17 July, 2007 at 11:28 am

dod

No one is perfect!

19 July, 2007 at 3:56 am

Carrière-advies van Terence Tao at QED

[…] een postdoctorale carrière. Tao legt onder andere uit dat wiskunde meer is dan bewijzen, dat je geen genie moet zijn om een goede wiskundige te worden, dat je naar conferenties moet gaan, ook buiten je eigen […]

21 July, 2007 at 10:56 pm

Kristy

Several physics issues wants to ask. What is the quantum network? Relativistic how the formula is derived, which Riemannian geometry of the importance of understanding how? Earth around the sun operation, in the classical Newtonian mechanics to explain gravity, and Einstein appears, This is because the bending of space around the sun. Ordinary people do not have an advanced knowledge of mathematics can understand? Assumptions in a vacuum is a very long stick, which was one end of the rotation center, as the angular velocity unchanged, Then there will be bar a linear speed to meet or surpass the speed of light, which may be? Does this campaign need a reference system? Mathematics in the physical phenomena of the study is idealistic? Relativity applies to the macro world, and quantum mechanics is applied to the microscopic world. Physicist this thinking, as a mathematician, you will be involved in this field of physics? I used the automatic translation function of the translation, there may be statements not understand. Please forgive me! Thank you for your time

26 January, 2013 at 1:44 pm

Stephen

Just… what are you doing!?

2 May, 2015 at 11:25 pm

Kitty

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

13 October, 2007 at 8:35 am

Anonymous

Hi everyone here!!How many hours do you sleep every day??how to make yourself forget zout the sleep but concentrate to the math?

15 November, 2007 at 5:02 pm

anonymous

DEAR DR TAO,
THERE IS SO MUCH INTERESTING STUFF ON YOUR BLOG THAT I HOPE YOU WILL LEAVE IT ON THE INTERNET, ALONG WITH THE COMMENTS, PERMANENTLY. THIS IS ESPECIALLY IMPORTANT BECAUSE THERE IS SO MUCH INTERESTING STUFF TO LEARN IN MATHEMATICS AND PHYSICS AND OTHER SUCH INTERESTING THINGS THAT ITS HARD TO FIT THE TIME IN TO KEEP UP WITH YOUR WONDERFUL BLOG TOO, ESPECIALLY FOR ME BECAUSE I SPEND MOST OF MY TIME JUST STRUGGLING TO MAKE A LIVING NOW AND I BARELY HAVE THE TIME TO READ YOUR BLOG LET ALONE TO STUDY.
I HAD TO DROP OUT OF COLLEGE AND HOPE AT SOME POINT TO DO RESEARCH IN MATHEMATICS AND PHYSICS BUT I WAS AN UNDERGRADUATE IN PHYSICS WHEN BECAUSE OF DIFFICULTIES IN THINKING THAT I’M NOW SOLVING THAT I HAD TO DROP OUT. I’M 51 BUT REFUSE TO GIVE UP ALTHOUGH I DEFAULTED ON STUDENT LOANS SO THAT PAYING FOR LIVING EXPENSES AND TUITION WHILE STUDYING WOULD BE A POSSIBLY INSURMOUNTABLE PROBLEM, ESPECIALLY SINCE I GOT TERRIBLE GRADES ALTHOUGH I’M APPROACHING THE POINT THAT I COULD GET STRAIGHT A’S ALMOST EFFORTLESSLY.
THANKS FOR YOUR TIME.

18 December, 2007 at 10:14 am

Beans

“reasonable amount of talent”

That is the catch – talent!

I am being encouraged to consider doing post-graduate studies, but do not think myself capable of such a task. It is easy to look at the negatives when one evaluates such possibilities, and they can be rather numbing and painful! (I am 19 and in my second year of undergraduate studies, so still have about a year to make my mind up).

What advice would you give to someone like me, who isn’t particularly clever, but might one day want to do post-graduate studies? I mean is there anything I can do now “in preparation”? I don’t really know much maths at this moment in time, but have an idea on two people who I wouldn’t mind being my supervisors. (I have been thinking about this for way too long!) Do I look into their field and make my module choices dependent on that, or find a field and then choose a topic? [One lecturer is group theorist and the other is a number theorist – both are extremely cool! Does there exist a topic which means that both can be my supervisor?!]

I can’t seem to remove this idea that to do post-graduate studies you have to be a genius! Ah well, I best shut up now.

18 December, 2007 at 11:49 am

Adam

Dear Beans,
Group and number theorists surely have a high school level open
source open problem (e.g., smart integration by parts). If you can solve
it, they might be impressed and even mention your work on a webpage.

22 December, 2007 at 3:29 am

goh

I am a high school student that loves Mathematics very much. I wish to major in math in university and even be a math professor. However, I do not think that I am brilliant enough in math. If I do all the things as you have mentioned in the paragraph 2, can I achieve my ambition?

Besides, can you recommend some good universities for me to major in math?

29 December, 2007 at 2:25 pm

pierre

for some definition of a genius, go here:

http://www.lygeros.org/0116-M-classification_English.htm
http://www.lygeros.org/0134-class2.html

By the way, the author of this article is a universal genuis, with an IQ of 189.

29 December, 2007 at 2:57 pm

pierre

and here for a global view:
http://www.lygeros.org/mclassi.html

After talent, patience, and works, Tao could have also mention the educational environment, which is, I think, very important. Look at for example Erdos, one of the most prominent mathematician of the all times, both of his parents where maths teachers. And I could give hundred of other examples.

I don’t want to depreciate Tao’s view since he is a great mathematician, and I am not, but I think his view is very “consensual”. It is like saying: “high intelligence is not necessary for doing research, but it helps”.

Personally, I wanted to become a mathematician, but I realized that my IQ of 130 was not high enough.

29 December, 2007 at 3:52 pm

John Armstrong

pierre, I’m not going to name any names, but I know more than a few mathematicians whose IQ (whatever that horrible linear scale actually means) is far below 130. I’ve met some who are downright morons. But they do good math despite that fact, and doing good math is all that matters.

29 December, 2007 at 10:12 pm

Andy Sanders

I don’t think there’s any precise way, beyond someone being a genius, to measure what their success will be in mathematics. Apart from a few people who are truly gifted in a unique way: it seems every other working mathematician is a combination of incredible work ethic, significant cleverness and a healthy dose of good luck (although I think this probably comes from working until the luck comes around.) I am current a graduate student in Math, and while I am certain many of the professors I walk the halls with are brilliant men, they are likely not geniuses in the sense that I think people are using that word.

I don’t know if I’ll ever do anything great, but I am going to give it a shot. From there, we’ll see if I get that dose of luck and moments of cleverness.

Nevertheless I enjoy articles like this from folks like Professor Tao. It shows everyone has to work for it, even those who seem to have more talent than just about everyone else.

30 December, 2007 at 9:59 am

Jonathan Vos Post

I strongly agree with pierre and John Armstrong on the Erdos, IQ, Tao, on genius dynamics.

I have a rather high IQ (circa 180) but am clearly not as smart as my wife, our son, or many of the undergrads I knew while at Caltech. Let alone the supergeniuses I spent extensive time with, such as Feyman, Gell-Man, Hawking, and Heman Kahn. Despite their press coverage as miraculously brainy, they were all tremendously hard workers.

More important, my ostensible intelligence and rapid grasp of simple Math (i.e. taught myself calculus at age 12, tested out of some required courses at Caltech by exams I took age 16) worked very much against me getting my B.S. and beyond. The quick and effortless learning and occasional insight led to me to seriously neglect acquiring good work habits and probem solving discipline.

Again and again, wherver I go, I see people of limited intelligence who achieve astonishing results, while people of enrmous giftedness go nowhere.

What matters is not what cards you are dealt by genetic lottery, and parental support, and early mentoring, but how well one learns to play the hand that one is dealt.

Wherever you go, there is someone smarter, prettier, richer, and/or better connected. Get used to it, and move beyond it.

“Chance favors the prepared mind.”

16 September, 2009 at 2:41 am

Anonymous

If 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

Blackacre

Sorry, 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 Trimble

As an amusing side note on IQ scores: in the book No Ordinary Genius, it’s reported that Feynman had an IQ of something like 127 (from when he took an IQ test in school). His sister, with tongue firmly in cheek, would gloat that she was smarter: hers was 128!

Please let’s not obsess about IQ scores. It’s just ridiculous. Let’s not worry about ‘genius’ either — an awful lot of romanticism tied up with that word, most of it completely wrong-headed IMO.

I’d say that the sine qua non for success in mathematics is sheer fascination with the subject and the drive for greater understanding. From that a great deal follows, including the willingness to do the necessary hard work.

30 December, 2007 at 6:42 pm

Pacha Nambi

Too much importance is given to IQ. I think it is about time the idea of IQ is discarded the way the idea of ether was thrown out.

One should simply enjoy doing what he/she finds fascinating in mathematics, physics, chemistry, biology etc. The reward and recognition will come to those who do very good work in due course of time.

31 December, 2007 at 3:33 am

pierre

As an amusing side note on IQ scores: in the book No Ordinary Genius, it’s reported that Feynman had an IQ of something like 127
>>>No, as Jonathan means it in a post I read on internet, its an urban legend, like the one that Einstein failed a maths exam.

I am not a mathematician, but I know what a genuis is. I definitly think that you cannot expect to go far without an high IQ.

For those who speak french, go here, for an view on maths and intelligence.
http://www.lygeros.org/0218-Mathematiques_Cognitives_et_Intelligence_Extreme.htm

>>Now, does someone have an idea of Erdos’s IQ? Iam looking for it.
Thx.

31 December, 2007 at 8:08 am

Todd Trimble

Pierre: Feynman was of course a notorious embellisher of legends about himself, and this may be one of them for all I know. But, the story is told in James Gleick’s otherwise well-documented biography on Feynman (on p. 30 according to Wikipedia); I don’t have the book to hand, but it’s apparently it’s part of the Feynman lore as told by himself and his sister. (And, there may well be documentation available to prove the story correct.)

Whatever the actual truth, the moral of that story is that Feynman himself didn’t take IQ scores seriously (far from it!). If a super-genius like Feynman doesn’t take it seriously, why should you?

There are more kinds of intelligence in heaven and on earth than are dreamt of in IQ testing. And that’s the last I’ll say on this topic here.

31 December, 2007 at 9:06 am

pierre

Whatever the actual truth, the moral of that story is that Feynman himself didn’t take IQ scores seriously (far from it!). If a super-genius like Feynman doesn’t take it seriously, why should you?

Ok. Lets put aside the story about Feyman . The theory of IQ is a statistic theory, with its “outliers” like every statistic theory. Feyman was one of them

I agree that intelligence is a complex structure that cannot be characterized by a number.
But on a large scale, it is consistent with reality.

31 December, 2007 at 10:27 am

John Armstrong

But on a large scale, it is consistent with reality.

Cite? And what objective quantity is it that IQ measures? And why is it that (what with constant rescaling) everyone from the late 19th century would score as mentally retarded on modern IQ tests?

31 December, 2007 at 10:41 am

Terence Tao

It is strange that IQ has such a hold over the popular imagination, because as far as I can tell it plays no role in academia whatsoever. In professional mathematics, at least, we don’t brag about our IQs, put them in our cv’s, or try to find out other mathematician’s IQ when trying to evaluate them; it has about as much relevance in our profession as the Meyers-Briggs Type Indicator.

More generally, the skills and traits that are popularly associated with “intelligence” or “genius” become largely decoupled, after a certain point, to those that are needed to do good mathematics. For instance, a very creative person may have a hundred innovative ways to attack a mathematical problem, but what one really needs is the rigorous thinking, comparison with existing literature, intuition and experience, and knowledge of heuristics in order to winnow these hundred ways down to the two that actually have a non-zero chance of working. Indeed, being overly creative at the expense of true mathematical skill may in fact impede one’s progress on a mathematical research problem, due to all the time wasted on the ninety-eight hopeless avenues.

Similarly, a very intelligent person may be very comfortable with abstract concepts and abstruse reasoning, and a certain amount of this can indeed be an asset when learning some of the more theory-intensive portions of mathematics, but at some point one has to be able to digest this theory and connect it with more mundane, “common sense” concepts (e.g. geometry, motion, symmetry, information, etc.); there is a risk of an excessively intelligent student getting overly enchanted with the formalism and esotericism of a subject, and neglecting to keep his or her knowledge grounded in reality (and to communicate it effectively with others).

In a third direction, a very quick thinker may be able to pick up new ideas rapidly, to find snappy rejoinders to any question, and to complete tests and examinations in a remarkably short amount of time, but these attributes may in fact lead to excessive frustration when such a student encounters a genuine research problem for the first time – one that requires months of patient and systematic effort, starting with existing literature and model problems, identifying and then investigating promising avenues of attack, and so forth. In athletics, the best sprinters can often be lousy marathon runners, and the same is largely true in mathematics.

To summarise: as I said in the main article, a reasonable amount of intelligence is certainly a necessary (though not sufficient) condition to be a reasonable mathematician. But an exceptional amount of intelligence has almost no bearing on whether one is an exceptional mathematician.

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 mathematics

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

31 December, 2007 at 11:29 am

pierre

And what objective quantity is it that IQ measures?
>>Intelligence quotient measures intelligence in the following fields : >>mathematics, sciences, philosophy and literature.

And why is it that (what with constant rescaling) everyone from the late 19th century would score as mentally retarded on modern IQ tests?
>>I doubt it, altought there is that kind of “Flynn effect”.

Go here:
http://www.iqcomparisonsite.com/Cox300.aspx

31 December, 2007 at 12:48 pm

John Armstrong

Pierre: what is this “intelligence” in these fields that an IQ test purports to measure? You’ve just pushed around words like peas on your plate, but you haven’t eaten anything.

As for the “Flynn effect”, that table is all but meaningless. Besides the fact that there’s no mention of how these numbers were derived or how they were “corrected”, a correction should act to increase the score to compensate for the fact that the scaling has gotten harder.

31 December, 2007 at 12:56 pm

David

I was admitted to several top-10 (US) math PhD programs and ended up going to a top-5 school, but once I realized I was not head and shoulders above my fellow students, I couldn’t bring myself to continue. A major factor was the fear that I would spend a year working on a problem, and then some brilliant guy who urgently needed the result would turn his attention to it and solve it in a week. Then I’d be left with nothing to show for my year of hard work. Not to mention the humiliation that an entire year of my work was worth less than a week’s worth of work for a better mathematician. I don’t think I could bear to explain that to anyone.

Maybe I had read too many biographies of famous mathematicians — like John Nash, who (in cahoots with another bright guy) used to get a kick out of solving other grad students’ thesis problems in a weekend. Anyway, I always had the impression that the vast bulk of professional mathematicians were toilers whose work went unread and was ultimately superseded by work done by the top guys. I tried to tell myself it still wouldn’t be such a bad life to be a mediocre mathematician, but unfortunately, it takes more motivation than that to get through a PhD program.

31 December, 2007 at 1:05 pm

pierre

I am not a specialist of intelligence, I am only interested in it. I dont have the time nor the space to explain everything, but here is a definition of intelligence: the creativeness,dicovery or inventiveness capacity in the fields.

Besides the fact that there’s no mention of how these numbers were derived or how they were “corrected
>>see: historiometry in wikipedia

31 December, 2007 at 2:12 pm

High IQ and Mathematics « Vishal Lama’s blog

[…] 31, 2007 in Uncategorized Tags: IQ, mathematics, Terry Tao Here is an interesting article (by Terry Tao) and the accompanying discussion on the relation between […]

31 December, 2007 at 2:12 pm

Richard

I completely agree with Terence in his last comment here, and attempted to post something similar late last night until fatigue took over. I might add that this view generalizes to other fields as well. My medical specialist with an MD PhD and many papers under his belt put it this way in regard to the medical field: “the whiz kids don’t necessarily make the best researchers.”

31 December, 2007 at 3:12 pm

John Armstrong

pierre: I fail to see the creativity, discovery, or inventiveness in a standardized test such as the WISC or the WAIS. Maybe the stumbling block is lacking your 130 IQ (no, I don’t know what mine is, though it’s in my parents’ records somewhere).

Now, as to your reference to Wikipedia, that doesn’t answer my question at all. I know how IQ scores can be generated, but I don’t know how those numbers on that page were generated. Maybe people with 130 IQs can think perfectly without citing their sources, but I’m just not that smart.

31 December, 2007 at 7:34 pm

Richard

Here are a few random thoughts.

People have different approaches to solving problems. Some will launch directly into a frontal attack, while others methodically circle and envelope the problem before going in for the kill. The later type of person is probably not ideally suited for excelling at IQ tests, or any kind of exam for that matter, yet their working method works for many kinds of mathematical problems. Remember the story of the tortoise and the hare?

I’m old enough that I remember when TV was all black and white, so my memory of that grade school IQ test is a little vague. However, I believe that it was just a long series of disconnected questions and problems. Mathematics is highly conceptual in nature, and involves understanding deep connections between these complex concepts. Moreover, a good researcher must have a good instinct for what is truly interesting and what really matters. I just don’t see how an IQ test can measure any of ability of this sort.

What is intelligence anyway? While hiking with my dog last summer in the woods in our favorite conservation area, we ran into a familiar volunteer there. Usually we discuss the complexity of nature, but that day she told me of a relative of hers who is borderline autistic, but who can multiply two long integers in his head with perfect accuracy. He just rolls his eyes up, and he soon has an answer. She said that he has some system working with “twelves”. When asked if he works in base 12 arithmetic, she said “something like that.” * Anyway, although I was one of those grade school kids who went through a period where I was factoring numbers almost compulsively in my head, and even earlier, thinking almost instantly 7 x 9 = 7^{2} + 7 x 2 = 49 + 14 = 63 rather than simply retrieving the result from a memorized multiplication table, that sort of feat is just way beyond and incomprehensible to me. Is this person intelligent? They have an extremely specialized skill to be sure, but I doubt that they will ever grasp the notions of measure or abstract topological spaces and algebras. I wonder if in fact the ability to grasp and manipulate highly abstract concepts is yet another highly specialized skill decoupled from almost everything else.

* I’d be interested to hear any comments from number theorists about the “twelves” thing. I couldn’t get much more detail from this person.

1 January, 2008 at 2:10 am

pierre

1 January, 2008 at 2:36 am

John Armstrong

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

1 January, 2008 at 7:10 am

pierre

And how do these tests reduce “creativity” to the same linear scale as the WISC or the WAIS?
>>I dont know; I am not the author of those tests.

1 January, 2008 at 9:59 am

John Armstrong

But you’re the one who is advocating these tests as examples of how IQ measures “creativity”. And now you’re saying you have absolutely no idea how they even claim to work.

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

1 January, 2008 at 12:45 pm

Jonathan Vos Post

“Intelligence testing began in earnest in France, when in 1904 psychologist Alfred Binet was commissioned by the French government to find a method to differentiate between children who were intellectually normal and those who were inferior. The purpose was to put the latter into special schools. There they would receive more individual attention and the disruption they caused in the education of intellectually normal children could be avoided…. Binet himself cautioned against misuse of the scale or misunderstanding of its implications. According to Binet, the scale was designed with a single purpose in mind; it was to serve as a guide for identifying students who could benefit from extra help in school. His assumption was that a lower IQ indicated the need for more teaching, not an inability to learn. It was not intended to be used as ‘a general device for ranking all pupils according to mental worth.’ Binet also noted that ‘the scale, properly speaking, does not permit the measure of intelligence, because intellectual qualities are not superposable, and therefore cannot be measured as linear surfaces are measured’…. American educators and psychologists who championed and utilized the scale and its revisions failed to heed Binet’s caveats concerning its limitations. Soon intelligence testing assumed an importance and respectability out of proportion to its actual value…. According to Harvard professor Steven Jay Gould in his acclaimed book The Mismeasure of Man, these tests were also influential in legitimizing forced sterilization of allegedly ‘defective’ individuals in some states.
By the 1920s mass use of the Stanford-Binet Scale and other tests had created a multimillion-dollar testing industry. By 1974, according to the Mental Measurements Yearbook, 2,467 tests measuring some form of intellectual ability were in print, 76 of which were identified as strict intelligence tests. In one year in the 1980s, teachers gave over 500 million standardized tests to children and adults across the United States. In 1989 the American Academy for the Advancement of Science listed the IQ test among the twenty most significant scientific discoveries of the century along with nuclear fission, DNA, the transistor and flight. Patricia Broadfoot’s dictum that ‘assessment, far more than religion, has become the opiate of the people,” has come of age.’”
http://iq-test.learninginfo.org/iq01.htm

“An intelligence quotient or IQ is a score derived from one of several different standardized tests attempting to measure 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.

1 January, 2008 at 2:42 pm

t8m8r

I 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 Tao

Dear 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

David

Terry,

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

pierre

What 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

yasiru89

A 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 jian

Dear 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 Jian

Dear 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.

17 January, 2008 at 7:59 am

Steve

Sounds 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

T. 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

Anonymous

I 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 Post

So “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

Anonymous

Quoting 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 Knutson

I think the marketplace these days favours mathematics which has some interdisciplinary component

Bert 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 real danger 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 Atur

How 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

Roberta

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

29 January, 2008 at 6:55 pm

Anonymous

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?

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

Croc

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

30 January, 2008 at 10:10 am

Anonymous

Imagine 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

t8m8r

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

31 January, 2008 at 8:49 pm

jisung

This comments make me hopeful.
Thanks tao.
I’ll have to try.

1 February, 2008 at 9:14 am

Jonathan Vos Post

Math 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 Viswanathan

Thank you for this fantastic post.

3 February, 2008 at 7:14 am

Pacha Nambi

I 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 Riepe

Here an article on aging and the brain:
http://tinyurl.com/yuf2ah

3 February, 2008 at 9:04 am

Thomas Riepe

26 February, 2008 at 1:43 am

Thomas Riepe

an 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 Freeze

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.

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 peuple of 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 bourgeois academics pleading for their jobs.

16 May, 2008 at 10:23 pm

Terence Tao

Dear 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

Anonymous

John 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

Dmouth

I’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.

30 April, 2014 at 3:22 am

Anonymous

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

9 November, 2014 at 8:42 pm

Ngoc Nguyen

I 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.)

17 May, 2008 at 3:10 am

John R Ramsden

There 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

Anonymous

Oh 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 Freeze

Terry: 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 gibbered his 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 street into 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

Nirman

Dear 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 Jian

Just 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

Dmouth

Haha, 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 Freeze

The petit bourgeois style 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 even more intangible 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 bourgeoisie of 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 Freeze

Terry 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 masas which 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 attitude about 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 satisfechos that Ortega y Gasset despised more than any other class… These are the “self-satisfied little gentlemen” whose narrow technical education supposedly endows their ex nihilo discussion 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 in La rebelión de las masas.

28 December, 2010 at 2:36 am

Jacob 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

I 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?