(Translated from Terence Tao’s blog,
translator: liuxiaochuang)
(Original English: Does one have to be a genius to do maths?)
The answer to this question is an emphatic: NO! In order to achieve the goal of making a good and meaningful contribution to mathematics, one must work hard; learn one’s own field well, master knowledge and tools from other fields; ask many questions; communicate more with other mathematical workers; have a macro grasp of mathematics. Of course, a certain level of intelligence, requirements for patience, and mental maturity are necessary. However, mathematical workers definitely do not need some magical “genius” gene or innate insight; they do not need some supernatural ability to always have the inspiration to solve difficult problems unexpectedly.
The public has a misconception of the image of mathematicians: these people seem to be lonely and detached (even a bit crazy) geniuses. They do not pay attention to the work of other peers and do not think in conventional ways. They are always able to obtain inexplicable inspiration (or suddenly obtain it after a painful struggle), and then make breakthrough progress on a major problem when all the experts are at a loss. Such a romantic image is quite attractive, but at least in modern mathematics, such people or things basically do not exist. In mathematics, we indeed have many amazing conclusions and deep theorems, but they are all accumulated over years, decades, or even centuries, and obtained bit by bit through the efforts of many excellent or great mathematicians. Every deepening of understanding from one level to another is indeed very extraordinary, and some are even quite unexpected. But despite this, these achievements are without exception built upon the foundation of previous work and are not entirely new. (For example, Wiles’ work on solving Fermat’s Last Theorem, or Perelman’s work on solving the Poincaré Conjecture.)
Today’s mathematics is like this: some intuition, a lot of literature, plus a little bit of luck, slowly accumulating and gradually progressing through a large amount of continuous hard work. In fact, I even feel that the reality of the situation is more satisfying to me than the aforementioned romantic hypothesis, although when I was a student, I also thought that the development of mathematics mainly relied on a few geniuses and some mysterious inspirations. In fact, this “myth of genius” has its flaws because no one can generate inspiration regularly, and one cannot even guarantee the correctness of each generated inspiration (if someone claims to be able to do these, I suggest being skeptical). Believing in inspiration also creates some problems: some people will excessively devote themselves to big problems; people should have reasonable doubts about their own work and the tools they use, but the aforementioned attitude makes some people gradually lose this doubt; there are also some people who are extremely insecure in mathematics, and many, many other problems.
Of course, if we do not use the extreme word “genius,” we will find that in many cases, some mathematicians will react faster than others, be more experienced, more efficient, more careful, and even more creative. However, it is not only these so-called “best” mathematicians who should do mathematics. This is actually a very common misconception about absolute advantage and comparative advantage. The field of meaningful mathematical research is extremely vast, and it is by no means a task that can be completed by a few so-called “best” mathematicians. Moreover, sometimes the ideas and tools you possess will compensate for the mistakes of some excellent mathematicians, and these excellent mathematicians will also expose weaknesses in certain mathematical research processes. As long as you are educated, have passion, and a bit of talent, there will definitely be some aspect of mathematics waiting for you to do important, foundational work. These may not be the most glamorous parts of mathematics, but they are the healthiest parts. Often, some areas that seem dull and useless now will be more meaningful in the future than some directions that look very beautiful. And, one should first do some less glamorous work in a field until one has the opportunity and ability to solve those major difficult problems. Look at the early papers of those great mathematicians, and you will understand what I mean.
Sometimes, an abundance of inspiration and talent is actually harmful to long-term mathematical development. Imagine if early problems were solved too easily, a person might not work hard, might not ask “dumb” questions, and might not try to expand their field, which would sooner or later lead to the exhaustion of inspiration. Moreover, if a person is used to cleverness that doesn’t take much time or effort, they will not have the patience and tenacious character needed to solve truly difficult big problems. Intelligence and talent are naturally important, but how to develop and nurture them is obviously even more important.
Remember, professional mathematics is not a sports competition. The purpose of doing mathematics is not to get how many points or win how many awards. Doing mathematics is actually to understand mathematics, for oneself, for students and colleagues, and ultimately to contribute to its development and application. For this task, it really needs the joint struggle of everyone!
Original text:
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.
From: http://blog.renren.com/share/108707892/5901540435
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