Vladimir Arnold

"At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste . . . Bizarre questions like Fermat's problem or problems on sums of prime numbers were elevated to supposedly central problems of mathematics."

"In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun)."

"It is almost impossible for me to read contemporary mathematicians who, instead of saying “Petya washed his hands,” write simply: “There is a t_1<0 such that the image of t_1 under the natural mapping t_1 \mapsto {\rm Petya}(t_1) belongs to the set of dirty hands, and a t_2, t_1, such that the image of t_2 under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.”"

"A person, who had not mastered the art of the proofs in high school, is as a rule unable to distinguish correct reasoning from that which is misleading. Such people can be easily manipulated by the irresponsible politicians."

"In the last 30 years, the prestige of mathematics has declined in all countries. I think that mathematicians are partially to be blamed as well—foremost, Hilbert and Bourbaki—the ones who proclaimed that the goal of their science was investigation of all corollaries of arbitrary systems of axioms."

"All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA.)."

""In almost all textbooks, even the best, this principle is presented so that it is impossible to understand." (K. Jacobi, Lectures on Dynamics, 1842-1843). I have not chosen to break with tradition."

"When you are collecting mushrooms, you only see the mushroom itself. But if you are a mycologist, you know that the real mushroom is in the earth. There’s an enormous thing down there, and you just see the fruit, the body that you eat. In mathematics, the upper part of the mushroom corresponds to theorems that you see. But you don’t see the things which are below, namely problems, conjectures, mistakes, ideas, and so on. You might have several apparently unrelated mushrooms and are unable to see what their connection is unless you know what is behind."

"Such axioms, together with other unmotivated definitions, serve mathematicians mainly by making it difficult for the uninitiated to master their subject, thereby elevating its authority."

"Mathematics is the part of physics where experiments are cheap."

"The axiomization and algebraization of mathematics, after more than 50 years, has led to the illegibility so such a large number of mathematical texts that the threat of complete loss of contact with physics and the natural sciences has been realized."

"Let me just say that Arnold was a geometer in the widest possible sense of the word, and that he was very fast to make connections between different fields."