Nothing is more fecund, all the mathematicians know it, than those obscure analogies, the blurred reflections from one theory to another... nothing gives more pleasure to the researcher. One day the illusion drifts away, the premonition changes to a certitude: the twin theories reveal their common source before disappearing; as the Gita [Bhagavad Gita] teaches it, knowledge and indifference are reached at the same time. The metaphysics has become mathematics, ready to form the subject matter of a treatise, the cold of which cannot move us anymore. (A. Weil)
The combination of these four things: beauty, exactness, simplicity and crazy ideas is just the heart of mathematics, the heart of classical music. (I. Gel'fand)
I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail. (Abraham Maslow)
Gedanken ohne Inhalt sind leer, Anschauungen ohne Begriffe sind blind. (Immanuel Kant)
Everett exhibited a trait of mind whose effects are, so to speak, non-additive: persistence in thinking. Thinking continuously or almost continuously for an hour, is – at least for me and I think for many mathematicians – more effective than doing it in two half-hour periods. It is like climbing a slippery slope. If one stops, one tends to slide back. Both Everett and Erdös have this characteristic of long-distance stamina. (S. Ulam)
I have certainly known people who are far brighter mathematicians than I am, but if they have thought about a problem for two days and can't solve it, they get bored with it and want to move on. But that is not a recipe for good research; you have to just keep going on and on. (R. Taylor)
The truth is that, to begin with, there are definite concrete problems, with all their undivided complexity, and these must be conquered by individuals relying on brute force. Only then come the axiomatizers and conclude that instead of straining to break in the door and bloodying one's hands, one should have first constructed a magic key of such and such shape and then the door would have opened quitely, as if by itself. (H. Weyl)
Science is wonderful as long as one does not have to do it for a living. (A. Einstein)
There are also many situations in which it makes tactical sense to defer, delay, delegate, or procrastinate on any given task, and go work on something else instead in the meantime; not everything is equally important, and also a given task may in fact become much easier (and be completed in a much better way) if one waits for one’s own skills to get stronger, or for other events to happen that reduce the importance or need for the task in the first place. My current papers on wave maps, for instance, have been delayed for years, much to my own personal frustration, but in retrospect I can see that it was actually a good idea to let those papers sit for a while, as the project as I had originally conceived it was a technical nightmare, and it really was necessary to wait for the technology and understanding in the field to improve before being able to tackle it in a relatively civilised manner. (T. Tao)
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. (T. Tao)
...he [Gel'fand] said to me that in any good mathematical theory there should be at least one "transcendental" element and this transcendental element should account for many of the subtleties of the theory. (B. Kostant)
He [Weil] used to say that a good mathematician must have two good ideas. "It is possible for someone to have a really good idea, but it may be just a fluke. Once the person has a second good idea, then there is a good chance for him to develop into a better mathematician". (G. Shimura)
A French gentleman’s ideal is to have three concurrent loves: the first one, whom he cares about at present; the second, a potential one, whom he has his eye on with the hope that she will eventually be his principal love; the third, the past one, with whom he hasn’t completely cut off his relations. Then he [Weil] observed: "It’s a good idea for a mathematician to have three mathematical loves in the same sense." (G. Shimura)
Long computations are, for me, a way to get into a special state of mind, into a particular mood, in which a mental picture can slowly emerge. As a preparation I go for a long walk with a particular problem in mind, and start computing in my head, before doing it in a notebook. I avoid meeting other people during such walks, especially mathematicians. I live in a remote place where I can go for a walk within a radius of 10 km around my house without meeting anyone. (A. Connes)
"So we [people in the former Moscow school] did have some contacts, and some information did reach us. Letters, just as now they take two weeks to be delivered, took the same two weeks back then... when they actually were delivered, of course. So what did bring about this unprecedented take-off of our school?
Were there any reasons particular to us? I think that isolation played a significant role. In those days we suffered this impossibility or rarity of contacts. Nowadays my students, very young, have already spent several times more time abroad than I did in my whole life. In my time, even if they did let you out, until the last moment you never knew if you would really go or not. Nevertheless, now, after many years, looking back I think that the mathematical isolation was not just the obvious evil but also, to some extent, a benefit". (A.N. Parshin, Mathematics in Moscow: We had a great epoque once)
Most mathematicians of this day, confronted with an argument requiring combinatorial thinking, react with one of two stock phrases: (a) “This is a purely combinatorial argument,” (b) “This is a difficult combinatorial argument.” Hypnotic repetition of either of these slogans is likely to have the same balming effect on the speaker: freed from all scruples, he will pass the buck and unload the work onto someone else’s shoulders.
While the end result of this oft-repeated vignette is an overwhelming variety of problems for specialists in the art, the impression grows that among mathematicians, especially “pure” mathematicians, total ignorance of combinatorial theory is as proudly flaunted as -- in bygone days -- an aristocrat’s ignorance of his country’s vernacular.
It is tempting to react to this rejection, which in the past has succeeded in finessing combinatorialists into the mathematical proletariat, by a ringing ça ira. This might well take the form of a concerted attack on one of the currently fashionable branches of mathematics. The barrage of definitions and the superstructure of grammatical gamesmanship removed, little more than a few puny combinatorial facts would be left, which would then be dealt an embarrassingly easy coup de grace by the application of standard combinatorial techniques. Fortunately, this course will not be followed, for a sound reason: combinatorialists have better fish to fry. (G.-C. Rota)
Beginning with the outside influences, it can be said that the recent development of combinatorics is somewhat of a cinderella story. It used to be looked down on by “mainstream” mathematicians as being somehow less respectable than other areas, in spite of many services rendered to both pure and applied mathematics. Then along came the prince of computer science with its many mathematical problems and needs — and it was combinatorics that best fitted the glass slipper held out.
The developments within mathematics that have contributed to the current strong standing of combinatorics are more difficult to pinpoint. One is that, after an era where the fashion in mathematics was to seek generality and abstraction, there is now much appreciation of and emphasis on the concrete and “hard” problems. Another is that it has been gradually more and more realized that combinatorics has all sorts of deep connections with the mainstream areas of mathematics, such as (to name the most important ones) algebra, geometry, probability and topology. (A. Björner and R. Stanley)
Chủ nghĩa thị trường tự do trong khoa học: người có tiền tài trợ khổng lồ và cả năm không bén mảng đến thư viện, người đi hội nghị quốc tế như đi chợ và không bao giờ đứng lớp, là nhà khoa học thành đạt.
Updated on 19-04-2024