VI. Выберите правильный перевод подчеркнутой части предложения.

III. Заполните пропуски подходящими по смыслу словами.

1. New questions open up new... of mathematics.

a) solutions b) divisions

c) problems d) experiences

2. It shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of ….

a) hypotheses b) solutions

c) deductions d) requirements

3. The demand for logical … by means of finite number of processes is simply the requirement of rigour in reasoning.

a) reason b) deduction

c) division d) solution

4. The theory of algebraic … experienced a considerable simplification.

a) curves b) deduction

c) methods d) solution

Text IV

In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new divisions of mathematics and whilewe seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus simultaneously advance most successfully the old theories, thanks' to this ever-recurring interplay between pure thought and experience.

It remains to discuss briefly what general requirements may be proposed and laid down for the solution of a mathematical problem. I want first of all say this: that it shall be possible to establish the correctness of the solution by means of.a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This demand for logical deduction by means of a finite number of processes is simply the requirement of rigour in reasoning. Indeed, this requirement of rigour, which became proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding; and on the other hand, only by satisfying this claim do the problems attain their full effect.

Besides it is an error to believe that rigour in the proofis the enemy of simplicity. On the contrary, we find it proved by numerous examples that the rigorous method is at the same time the simpler and worthy in the long run and easier to understand. The very effort for rigour helps us come across a simpler method of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigour. Thus, the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of a more rigorous function-theoretical methods and the introduction of transcendental curves.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. What play does the outer world come into?

a) It forces upon us new questions from actual experience.

b) It opens up new divisions of mathematics.

c) It gives us answers to the old questions.

d) It does not come into play.

2. What does the outer world open up?

a) new questions from actual experience.

b) new divisions of mathematics.

c) new field of knowledge.

d) old unsolved problems.

3. What does the very effort for rigour help us to do?

a) to solve mathematical problems.

b) to come across a simpler method of proof.

c) to find new problems.

d) to make an exact formulation of the problem.

4. What is the main demand for the solution of a mathematical problem?

a) it must be exactly formulated.

b) it must be solved.

c) it must not be understandable.

d) it must be based on experience.

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. The outer world again comes into play

b. The solution of mathematical problem

c. The very effort for rigour helps us come across a simpler method of proof

d. The theory of algebraic curves

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