5.Read the passages and comment on the use of tense forms. Find time indicators and try to reveal the logic of the tense-forms usage

1. Let us now try, guided by the axiomatic concept, to look over the whole of the math universe. It is clear that we shall no longer recognize the traditional order of things, which just like the first nomenclatures of animal species, restricted itself to placing side by side the theories which showed greatest external similarity. In place of the sharply bounded compartments of algebra, of analysis, of the theory of numbers, and of geometry, we shall see, for example, that the theory of prime numbers is a close neighbor of the theory of algebraic curves, or, that Euclidean geometry borders on the theory of integral equations. The organizing principle will be the concept of a hierarchy of structures, going from the simple to the complex, from the general to the particular.

2. There also occurred, especially at the beginning of axiomatics, a whole crop of monster-structures, entirely without applications; their sole merit was that of showing the exact bearing of each axiom, by observing what happened if one omitted or changed it. There was, of course, a temptation to conclude that these were the only results that could be expected from the axiomatic method.

3. After the more or less evident bankruptcy of the different systems undertaken to examine the relations of maths to reality or to the great categories of thought, it looked, at the beginning of the present century, as if the attempt had just been abandoned to conceive of maths as a science characterized by a definitely specified purpose and method; instead, there was a tendency to look upon maths as "a collection of disciplines based on particular, exactly specified concepts", interrelated by "a thousand roads of communication", allowing the methods of any one of these disciplines to fertilize one or more of the others.

4. Today, we believe, however, that the internal evolution of math science has, in spite of appearance, brought about a closer unity among its different parts, so as to create something like a central nucleus that is more coherent than it has ever been. The essential aspect of this evolution has been the systematic study of the relations existing between different math theories which has led to what is generally known as the "axiomatic method".

6. Choose some complex sentences from the texts and illustrate structural analysis in terms of types of sentences, clauses, and predicate.

7. Read the text and write a summary, supplying it with a title

The Pythagorean school mathematicians were the first to find the essence of natural phenomena in number and in numerical relations. Number for them was the first principle in the explanation of nature and was the matter and form of the Universe. They also reduced the motions of the planets to number relations. They believed that bodies moving in space produce sounds and that the body which moves rapidly gives forth a higher note than the one which moves slowly. Perhaps these ideas were suggested by the sound of an object revolving on the end of a string. According to their astronomy the greater the distance of a planet from the Earth, the more rapidly it moves. Hence, the sounds produced by the planets varied with their distances from the Earth, but this "music of spheres" like all harmony was reduced by them to no more than number rela­tionships and so did the motions of the planets.

In addition, the Pythagoreans attached very interesting interpretations to the individual numbers. The number one they identified with reason, for only reason could produсe a consistent whole; two was identified with opinion; four with justice because it is the first number which is the product of equals (to the Pythagoreans one was not a number in the full sense because unity was opposed to quantity); five signified marriage because it was the union of the first odd and first even number; seven was identified with health, and eight with love and friendship.

All the even numbers were regarded as feminine, the odd numbers as masculine. From these associations it followed that the even numbers represented evil and the odd, good. The trouble with the even numbers was that they permitted bisection into more and,more even numbers as 2 into 1 and 1; 4 into 2 and 2; 8 into 4 and 4 and so on. The process of continued bisection suggested the infinite, a horrible thought to the Greeks who preferred the definite and limited. The odd numbers, on the other hand, prevented the even number from going to pieces. A number was perfect if it equalled the sum of its divisors, as 6 = 1 + 2 + 3. Two numbers were "friendly" if each was the sum of the divisors of the other. The ideal number was 10 because for one thing it was the sum of integers 1, 2,3 and 4. And because 10 was ideal, the moving bodies in the heavens must be ten in number. This ideality of 10 also required that every object in the Universe should be described in terms of 10 pairs of categories such as odd and even, bounded and unbounded, right and left, one and many, male and female, good and evil.

This philosophy is to a large extent unscientific and useless.

Nevertheless, the major thesis of the Pythagoreans, namely, that nature should be interpreted in terms of number and number relations, that number is the essence of reality, dominates modern science. This thesis was revived and perfected in the work of Copernicus, Kepler, Galileo, Newton and their successors and is represented today by the doctrine that nature must be studied quantitatively. These scientists adopted several other Pythagorean beliefs, namely, that the Universe is ordered by perfect math laws and their efforts in deducing these laws are to be credited with great success.