1.High school students were more sensitive to falsehood and cheating than undergraduate students since they ere not yet used to pretending that they understood what cannot be understood at all.
2.Development of mathematics resembles a fast revolution of a wheel: sprinkles of water are flying in all directions. Fashion-it is the stream that leaves the main trajectory in the tangential direction. These streams of epigone works attract most attention, and they constitute the main mass, but they inevitably disappear after a while because they parted with the wheel. To remain on the wheel, one must apply the effort in the direction perpendicular to the main stream.
3.Poincaré said: “It is incredible how much a well-chosen word can economize thought. Often one only needs to invent a new word, and this word becomes a creator on its own right.”
4.It seems to me that modern science (i.e.m theoretical physics along with mathematics) is a new religion, a cult of truth, founded by Newton 300 years ago.
5.Fraday… “Lectures which really teach will never be popular; lectures which are popular will never teach.”
6.Mathematicians differ dramatically bu their time scale: some are very good tackling 15-minute problems, some are good with the problems that require an hour, a day, a week, the problems that take a month, a year, decades of thinking… A. N. Kolmogorov considered his “ceiling” to be two weeks of concentrated thinking.
7.An extremely important condition for serious mathematical research is good health.
8.When a problem resists a solution. I jump on mu cross country skis. Forty kilometers later a solution (or at least an idea for a solution) always comes. Under scrutiny, an error is often found. But this is a new difficulty that is overcome in the same way.
9.Some people, even though they study, but without enough zeal, and therefore live long. ——Archibishop Gennady of Novdorod in a letter to Metropolitan Simon, ca 1500.
10.All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supproted by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA.).
Cryptography has generated number theory, algebraic geometry over finite fields, algebra, combinatorics and computers.
Hydrodynamics recreated complex analysis, partial differential equations, Lie groups and algebra theory, cohomology theory and scientific computing.
Celestial mechanic is the origin of dynamical systems, linear algebra, topology, variational calculus and simplistic geometry.
The existence of mysterious reactions between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation).
11.But the pernicious character of diverging modes if thought (to which the growing specialization of mathematicians and the fragmentation of mathematics into small domains leads) becomes evident when one tries to understand the development of mathematics in the past with all its meanderings.
12.According to Sylvester, a mathematical idea should not be petrified in a formalized axiomatic setting, but should be considered instead as flowing as a river. One should be ready to change the axioms, preserving the informal idea.
13.The Russian way to formulate problems is to mention the first nontrivial case (in a way that no one would be able to simplify it). The French way is to formulate it in the most general form making impossible any further generalization.
14.Proofs are to mathematics what spelling (or even calligraphy) is to poetry.
15. Poincaré explained that only nonintersecting problems might be formulated unambiguously and solved completely.
16.In French publications there were no diagrams (indispensable to the reader) -probably to make the theory incomprehensible to the uninitiated (but more likely, due to typically careless French user-unfriendliness).
17.Vladimir Abramovich was fully aware that no matter how much time one could save using the deductive methods (“from general to speciﬁc”), the value of a lecture for a student consists of merely a number of well-explained and thoroughly understood examples. Vladimir Abramovich’s attitude towards examples was that of a respect, similar to the one held by physicists of inductive school of thought (starting with Newton), and contrary to the opinion of most of contemporary mathematicians (Sullivan once told me that he tended to avoid dealing with particular examples at all costs – they were way too complicated).
18.”It is easier to prove this statement singlehandedly than read a written proof.”