About mathematics and school mathematics

I used to be a mathematician, but when capitalism came, I had to give up regular classes in mathematics and pursue other activities. This fate befell a great many Russians who were forced to give up their profession. Friends, colleagues, and acquaintances sometimes ask me how I endured this catastrophe in my life, how I managed to give up mathematics, to which I had previously devoted myself completely? Strangely enough, it was easy. The fact seems to be that I have not given up mathematics by stopping to do mathematics regularly and to publish papers on mathematics. I think mathematically, I have not abandoned the mathematical style of expressing my thoughts orally or in writing, so I continue to consider myself a mathematician. Mathematics is not something that can be abandoned.

Mathematics is a rather mysterious phenomenon; its origin and purpose, just as the origin and purpose of humanity, are unknown. Mathematics is at the heart of technological progress, and so it is quite common to think that the main value of mathematics lies in its practical applications. This opinion, however, is inconsistent with the fact that the most striking and impressive mathematical results lie in those areas of mathematics which are as far removed as possible from any practical applications. For a science as ancient as mathematics, it is very dynamic. What are the dynamics of mathematics? Every professional mathematician strives to obtain as many new mathematical results as possible, and it might seem that the dynamics of mathematics consist mainly in the accumulation of proven results. This accumulation of results does occur, but a more important part of the dynamics of mathematics is a change in perspective, a change in the language of mathematics (toward simplification) a change in accepted notation, and finally a change in "fashion" in mathematics, that is, a change in ideas about which results are more interesting and which are less interesting. Each individual mathematician does not so much notice the progress in the accumulation of new mathematical results as he does the dynamics in the redefinition of results already known. Often the most important mathematical advances, having been discovered for the first time, are stated awkwardly, because no suitable language and suitable notation for it have yet been found. Later, either the author himself or his colleagues state the same result more adequately.

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In the first half of the twentieth century, the most important part of the dynamics of mathematics was its unification. Everyone knows that there are different branches (or areas) of mathematics: mathematical analysis, differential equations, algebra, differential geometry, partial derivative equations, theory of functions of complex variable, topology, mathematical logic, etc. After the mid-twentieth century, this division of mathematics into domains began to be conditional: in fact, mathematics is unified and indivisible in the sense that these domains are closely interrelated. Most of the most important mathematical results can be obtained (or proved) only by using the methods of several mathematical domains. The methods of each domain are used in other domains. Before the twentieth century, there was no such unity in mathematics. At that time, mathematicians did not yet know how to apply the methods of some fields to other fields. The symbol that marked this unification of mathematics was David Hilbert's famous paper delivered at the Congress of Mathematics in 1900, in which 23 Great Problems from different fields of mathematics were formulated. To be somewhat imprecise and tentative, it can be said that before Hilbert's report, mathematics consisted of different fields, which were essentially different sciences, each experiencing its own dynamics independent of the others. After Hilbert's report, all these fields merged into one science called mathematics, the dynamics of all these fields merged into one dynamics of mathematics.

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