quinta-feira, 23 de maio de 2013

Um texto de Béla Bollobás

Béla Bollobás é um matemático húngaro (sim, esse é um nome de menino, pelo menos na Hungria) da Universidade da Memphis e da Universidade de Cambridge. Nasceu em Budapeste em 3 de agosto de 1943, hoje tem 69 anos. Desde muito jovem se interessou por matemática e foi um dos primeiros medalhistas da IMO ( International Mathematical Olympiads), condecorado com duas medalhas de ouro e uma de bronze (veja). Conheceu Paul Erdös graças ao seu desempenho na IMO e aos 19 anos publicou o seu primeiro artigo junto com o Erdös.

Tem publicado artigos, livros e teses em Análise Funcional, Combinatória, Teoria dos Grafos e Percolação. É, atualmente, uma das principais referências em Percolação e em Combinatória Extremal.

Neste post irei publicar um texto de Béla Bollobás (foto abaixo) que pode ser encontrado no livro The Princeton Companion to Mathematics.

Béla Bollobás

“There is no permanent place in this world for ugly mathematics,” wrote Hardy; I believe that it is just as true that there is no place in this world for unenthusiastic, dour mathematicians. Do mathematics only if you are passionate about it, only if you would do it even if you had to find the time for it after a full day’s work in another job. Like poetry and music, mathematics is not an occupation but a vocation.

Taste is above everything. It is a miracle of our subject that there seems to be a consensus as to what constitutes good mathematics. You should work in areas that are important and unlikely to dry up for a long time, and you should work on problems that are beautiful and important: in a good area there will be plenty of these, and not just a handful of well-known problems. Indeed, aiming too high all the time may lead to long barren periods: these may be tolerated at some stage of your life, but at the beginning of your career it is best to avoid them.

Strive for a balance in your mathematical activity: research should and does come first for real mathematicians, but in addition to doing research, do plenty of reading and teach well. Have fun with mathematics at all levels, even if it has (almost) no bearing on your research. Teaching should not be a burden but a source of inspiration.

Research should never be a chore (unlike writing up): you should choose problems that you find it difficult not to think about. This is why it is good if you get yourself hooked on problems rather than working on problems as if you were doing a task imposed on you. At the very beginning of your career, when you are a research student, you should use your experienced supervisor to help you judge problems that you have found and like, rather than working on a problem that he has handed to you, which may not be to your taste. After all, your supervisor should have a fairly good idea whether a certain problem is worth your efforts or not, while he may not yet know your strength and taste. Later in your career, when you can no longer rely on your supervisor, it is frequently inspiring to talk to sympathetic colleagues.

I would recommend that at any one time you have problems of two types to work on.
  1. A “dream”: a big problem that you would love to solve, but you cannot reasonably expect to solve.
  2. Some very worthwhile problems that you feel you should have a good chance of solving, given enough time, effort, and luck.

In addition, there are two more types you should consider, although these are less important than the previous ones.
  1. From time to time, work on problems that should be below your dignity and that you can be confident of doing rather quickly, so that time spent on them will not jeopardize your success with the proper problems.
  2. On an even lower level, it is always fun to do problems that are not really research problems (although they may have been some years ago) but are beautiful enough to spend time on: doing them will give you pleasure and will sharpen your ability to be inventive.

Be patient and persistent. When thinking about a problem, perhaps the most useful device you can employ is to bear the problem in mind all the time: it worked for Newton, and it has worked for many a mortal as well. Give yourself time, especially when attacking major problems; promise yourself that you will spend a certain amount of time on a big problem without expecting much, and after that take stock and decide what to do next. Give your approach a chance to work, but do not be so wrapped up in it that you miss other ways of attacking the problem. Be mentally agile: as Paul Erd˝os put it, keep your brain open.

Do not be afraid to make mistakes. A mistake for a chess player is fatal; for a mathematician it is par for the course. What you should be terrified of is a blank sheet in front of you after having thought about a problem for a little while. If after a session your wastepaper basket is full of notes of failed attempts, you may still be doing very well. Avoid pedestrian approaches, but always be happy to put in work. In particular, doing the simplest cases of a problem is unlikely to be a waste of time and may well turn out to be very useful.

When you spend a significant amount of time on a problem, it is easy to underestimate the progress you have made, and it is equally easy to overestimate your ability to remember it all. It is best to write down even your very partial results: there is a good chance that your notes will save you a great deal of time later.

If you are lucky enough to have made a breakthrough, it is natural to feel fed up with the project and to want to rest on your laurels. Resist this temptation and see what else your breakthrough may give you.

As a young mathematician, your main advantage is that you have plenty of time for research. You may not realize it, but it is very unlikely that you will ever again have as much time as you do at the beginning of your career. Everybody feels that there is not enough time to do mathematics, but as the years pass this feeling gets more and more acute, and more and more justified.

Turning to reading, young people are at a disadvantage when it comes to the amount of mathematics they have read, so to compensate for this, read as much as you can, both in your general area and in mathematics
as a whole. In your own research area, make sure that you read many papers written by the best people. These papers are often not as carefully written as they could be, but the quality of the ideas and results should amply reward you for the effort you have to make to read them. Whatever you read, be alert: try to anticipate what the author will do and try to think up a better attack. When the author takes the route you had in mind, you will be happy, and when he chooses to go a different way, you can look forward to finding out why. Ask yourself questions about the results and proofs, even if they seem simpleminded: they will greatly help your understanding.

On the other hand, it is often useful not to read up everything about an open problem you are about to attack: once you have thought deeply about it and apparently got nowhere, you can (and should) read the failed attempts of others.

Keep your ability to be surprised, do not take phenomena for granted, appreciate the results and ideas you read. It is all too easy to think that you know what is going on: after all, you have just read the proof.  Outstanding people often spend a great deal of time digesting new ideas. It is not enough for them to know a circle of theorems and understand their proofs: they want to feel them in their blood.

As your career progresses, always keep your mind open to new ideas and new directions: the mathematical landscape changes all the time, and you will probably have to as well if you do not want to be left behind. Always sharpen your tools and learn new ones.

Above everything, enjoy mathematics and be enthusiastic about it. Enjoy your research, look forward to reading about new results, feed the love of mathematics in others, and even in your recreation have fun with mathematics by thinking about beautiful little problems you come across or hear from your colleagues.

If I wanted to sum up the advice we should all follow in order to be successful in the sciences and the arts, I could hardly do better than recall what Vitruvius wrote over two thousand years ago:
Neque enim ingenium sine disciplina aut disciplina sine ingenio perfectum artificem potest efficere.
For neither  genius without learning nor learning without genius can make a perfect artist.

[1] Timothy Gowers (Editor), June Barrow-Green (Editor), Imre Leader (Editor); The Princeton Companion to Mathematics. Princeton Reference, 2008.

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