Text. 5. Solvability of Mathematical Problems

Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense and for that reason do not surmount the difficulty. The problem then arises: to show the impossibility of the solution under the conditions specified. Such proofs of impossibility were effected by the ancients, for instance, when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later maths, the question of the impossibility of certain solutions plays a great part and we realize in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring the circle, the solution of equations of the fifth degree by radicals found fully satisfactory and rigorous solutions, although in a different sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares but which as yet no one supported by a proof or refuted) that every definite math problem must necessarily be settled, either in the form of a direct answer to the question posed, or by the proof of the impossibility of its solution and, hence, the necessary failure of all attempts.

Is this axiom of the solvability of every problem a peculiar characteristic of math thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable? For in other sciences there exist also old problems which were handled in a manner most satisfactory and most useful to science by the proof of their impossibility, for example, the problem of perpetual motion. The efforts to construct a perpetual motion machine were not futile as the investigations led to the discovery of the law of the conservation of energy, which, in turn, explained the impossibility of the perpetual motion in the sense originally presupposed.

This conviction of the solvability of every math problem is a powerful stimulus and impetus to the researcher. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it, for in maths there is no futile search even if the problem defies solution.

Vocabulary

1. Answer the following questions:

1) Are all mathematical problems solvable?

2) What conditions should be observed to make them solvable?

3) Is it a general law that all questions must be answerable?

4) What led the scientists to the discovery of the law of the conservation of energy?

5) What is a powerful stimulus to the researcher?

6) What mathematical problems found no solution and give the reason why?

7) What are the signs that show the impossibility of the solution of a mathematical problem?

2. Give Russian equivalents of the following word combinations:

the squaring the circle, find rigorous solution, in a different sense, the question posed, failure of attempts, there exist old problems, most useful to science, to contract a perpetual motion machine, in turn, in the sence originally presupposed, perpetual call, to seek solution, futile search, to defy solution, to give rise to the conviction, to chare the conviction, direct answer.

3. Fill in the missing word.

1) Some problems were handled in a similar ….

2) The law of the conservation of matter … the impossibility of the perpetual motion.

3) There is no futile search even if the problem … solution/

4) Every definite math problem must necessarily ….

5) The problem of perpetual motion found no ….

6) Occasionally we seek the solution in an incorrect ….