Text. 3. The sources of Mathematical Problems

It often happens that the same special problem finds application in the most diverse and unrelated branches of maths. So for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And F. Klein convincingly pictured, in his work on the icosahedron, the significance which is attached to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in the theory of linear differential equations.

After referring to the general importance of problems in maths, let us return to the question from what sources this science derives its problems. Surely, the first and oldest problems in every field of maths spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with natural numbers were discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first unsolved problems of antiquity, such as the duplication of the cube, the squaring of the circle. Also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential, to say nothing of the abundance of problems properly belonging to mechanics, astronomy and physics.

But, in the further development of the special domain of maths, the human mind, encouraged by the success of its solutions becomes convinced of its independence. It evolves from itself alone, often without appreciable influence from outside by means of logical combination, generalization, specialization, by separating and collecting ideas in elegant ways, by new and fruitful problems, and the mind appears then as the real questioner and the source of the new problems. Thus arose the problem of prime numbers and the other unsolved problems of number theory, Galois’ theory of equations, the theory of algebraic invariants, the theory of Abelian and automorphic functions; indeed, almost all the nicer problems of modem arithmetic and function theory arose in this way.

Vocabulary

1. Find answers in the text to the following questions

1) From what sources does science derive its problems?

2) What did F.Klein convincingly picture in his work on the icosahedron?

3) What does the world of external phenomena suggest?

4) How does a child learn about the world?

5) What were the first unsolved problems of antiquity?

2. Give Russian equivalents of the following word combinations:

curved lines and surfaces, liner equations, solution of equations, in this fashion, to attach significance to the problem, surely, to become convinced, separate and collect ideas, squaring of the circle, such as, to say nothing of, the abundance of problems, prime numbers.

3. Translate into English.

1) Множество проблем относящихся к механике было решено таким же образом.

2) То же самое верно (относится) для первых нерешенных проблем античности.

3) Он убедительно показал значение, которое придается этой проблеме.

4) Давайте вернемся к вопросу о том, откуда наука черпает свои проблемы.

5) Проблемы возникают из человеческого жизненного опыта без взаимного влияния извне.