“A student is not a sack to be filled but a torch to be lighted”
“Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap” .
“It seems to me that modern science (that is, theoretical physics, along with mathematics) is a new kind of religion: the cult of truth founded by I. Newton 300 years ago” .
“Jacobi noted the most fascinating property of mathematics, that in it one and the same function controls both the presentation of an integer as a sum of four squares and the real movement of a pendulum”.
“Evolution of mathematics resembles fast revolution of a wheel, so that drops
of water fly in all directions. Fashion
is this stream that leaves the main trajectory in the tangential direction.
These streams of imitation works are most noticeable, they constitute the main
part of the total volume, but they die out soon after parting with the wheel. To
keep staying on the wheel, one must apply effort in the direction perpendicular
to the main flow” .
It is worth mentioning that Arnold himself has created fashions in mathematics
more than once!
“Mistakes are an important and instructive part of mathematics, perhaps as important a part as the proofs. Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of words” .
“A remarkable property of mathematics, which one cannot help but admire, is the unreasonable effectiveness of the most abstract, and at first glance completely useless, of its branches, provided that they are beautiful”.
“According to Louis Pasteur, there exist no applied sciences – what do exist are applications of sciences”.
“A student is not a sack to be filled but a torch to be lighted”.
“I. G. Petrovskii, who was one of my teachers in Mathematics, taught me that the most important thing that a student should learn from his supervisor is that some question is still open. Further choice of the problem from the set of unsolved ones is made by the student himself. To select a problem for him is the same as to choose a bride for one’s son.”
ref: https://www.math.psu.edu/tabachni/prints/Arnoldpr.pdf
The aim of a mathematical lecture should be not the logical derivation of some incomprehensible assertions from others (equally incomprehensible): it is necessary to explain to the audience what the discussion is about and to teach them to use not only the results presented, but—and this is major—the methods and the ideas.
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Some of the mathematical events of the distant past were as lively to Arnold as if they were happening now. In particular, this concerns the second half of the seventeenth century, the time of Huygens, Hooke and Newton. Arnold read Newton’s Principia as if it were written by his contemporary and a kindred spirit. For him, the book was full of surprises and fresh ideas.
For example, Arnold connected Proposition VII, Problem II, in Principia (‘If a body revolves in the circumference of a circle; it is proposed to find the law of centripetal force directed to any given point’) with the Bohlin theorem about the duality of laws of central attraction whose strength is proportional to a power of distance to the centre.
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Arnold used to say (exaggerating, as usual) that much of what he knew in mathematics he learned from Felix Klein’s book Development of Mathematics in the 19th Century
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One of his permanent targets was the formal style of presenting and teaching mathematics that he called ‘Bourbakism’ (sometimes with the adjective ‘criminal’). The following quotation from (17) is characteristic:
Unfortunately, the simple texts of Poincaré are difficult for mathematicians raised upon set theory. Poincaré would have said ‘Pete washed his hands,’ where a contemporary mathematician would simply write instead ‘There exists a t_{1} <0 such that the image of the point t1 under the natural mapping t→Pete(t) belongs to the set of people with dirty hands, and a t_{2} ∈ (t_{1}, 0] such that Pete(t_{2}) belongs to the complement of the set mentioned above.’
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Arnold valiantly fought against the decline of the standards of mathematical
education, be this in Russia or in the West. In an unpublished draft of a
preface to (Fuchs & Tabachnikov 2007), he remarked that the book might be too
difficult for the readers, including most modern students in the USA and France,
who do not know that 1/2 + 1/3 ≠ 2/5.
His view of the future of mathematics and, in general, of our culture, was
pessimistic: he warned that the united bureaucrats of all countries may be able
to stop all kinds of creative activity.
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Another contribution of Arnold to the revival of real algebraic geometry was his elegant generalization of the famous Newtonian ‘no gravity in the cavity’ result (Proposition 70, Theorem 30, in Principia). This theorem states that a homogeneous sphere exerts zero gravitational force at every interior point.