4. Do you agree or disagree with the quotations? Discuss them in pairs.

1. "There are things which seem incredible to most men who have not studied mathematics." Aristotle

2. "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Albert Einstein

3. "Mathematics is the queen of sciences and arithmetic is the queen of mathematics." Carl Friedrich Gauss

4. "Film is one of the three universal languages, the other two: mathematics and music." Frank Capra

5. "The essence of mathematics lies in its freedom." Georg Cantor

6. "To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts." Ivars Peterson

7. "For the things of this world cannot be made known without a knowledge of mathematics." Roger Bacon

8. "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."Bertrand Russell

READING

5. You need to

a) underline the stressed sound in each word as in the example and pronounce them. Practise reading.

Abbreviation, concise, purposefully, throughout, frequent, minus, precise, algebra, geometry, pervade, philosophy, magnitude, vowel, successful, successively, ambiguity, susceptible, technique.

b) pay attention to the shift of the stress in the following numerals. Practise reading.

thirteen − thirty	seventeen − seventy
fo(u)rteen − fo(u)rty	eighteen − eighty
fifteen − fifty	nineteen − ninety
sixteen − sixty

6. Tell what the following abbreviations or shortenings mean. If you don't know, see Appendix 4.

1. vs. – as against –проти; 2) cf – compare – порівняй;

3. e.g. – for example –наприклад; 4) v.v. - on the contrary – навпаки;

5) etc. – etcetera, and so on –тощо; 6) B.C. – Before Christ – до нашої ери;

7) i. e. -in other word(s), that is – тобто; 8) A.D. – нашої ери;

9) fig. – figure – малюнок, схема цифра; 10) exc. – except – (o)крім.

7. Read the text "Mathematics − the language of science" and answer the questions.

1. Is the language of mathematics universal? Why (not)?

2. What is algebra?

3. What three stages has algebra passed?

4. How is the language of mathematics designed?

5. What signs and symbols are there in mathematics?

6. What sciences does mathematics embrace?

7. What is verbal algebra?

8. Give examples of abbreviated algebra.

8. Think of the heading for each paragraph of the text. Mathematics − the language of science

1. H uman language is capable of precise statements because it is a system of symbols. But common language is a product of social development, customs and traditions. Even by the most careful choice of words the meaning concealed in them may influence our reasoning. Algebra, the language of mathematics, consists mostly of signs and symbols and is carefully and purposefully designed. It is precise, concise and universal, i.e. one and the same throughout the civilized world, though the people in each country translate it into their own spoken language.

2. Algebra in the broad sense of the term, deals with operations upon symbolic forms. In this capacity it not only permeates all of mathematics, but pervades practically all sciences including formal logic, philosophy, and even linguistics, poetry and music. In our scientific age there is a general belief that all science, as it grows to perfection, becomes mathematical in its ideas.

3. It is generally true that algebra in its development has passed successively through three stages: verbal, abbreviated and symbolic. Verbal algebra is characterized by the complete absence of any symbols, except, of course, that the words themselves are used in their symbolic sense. To this day verbal algebra is used in such a statement as “the sum is independent of the order of the terms”, which in symbols is designated by a+ b=b + a.

4. Abbreviated algebra of which the Egyptian is a typical example, is a further development of verbal one. Certain words of frequent use are gradually abbreviated. The history of the symbols “+” and “−“ may illustrate the point. In medieval Europe the latter was denoted by the full word ''minus'', then by the first letter "m" duly superscribed. Eventually the letter itself was dropped leaving the superscript only. The sign “plus" has passed through a similar metamorphosis. The abbreviation has become a symbol.

5. The turning point in the history of algebra was an essay written late in the sixteenth century by a Frenchman; it was Viete who denoted the unknown magnitudes by vowels. The given magnitudes were designated by consonants.

6. Within half a century of Viete's death there appeared Descartes's Geometry. In it, the first letters of the alphabet were used for the given quantities, the last − for those unknown. The Cartesian notation not only displaced the Vietan one, but has survived to this day.

7. It is symbols that permit of concise, clear representation of ideas which are sometimes quite complex. Consider, for example, how much is involved in the calculus symbol “Dy”. Once we have grasped the meaning and use of a symbol there is no need to think through the origin and development of the idea symbolized, each time it is used. It is due to a powerful technique based upon the use of symbols that mathematics is so effective in problems which are insoluble by other methods.

8. It is convenient because the literal notation is free from ail ambiguities of words. The letter is susceptible of operations and this enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent forms. It is this power of transformation that lifts algebra above the level of convenient shorthand.

9. It is symbolic language that is one of the basic characteristics of modern mathematics. And modern mathematics supplies a language for the treatment of the qualitative problems of physical and social sciences.

Look through the list of words and phrases and check if you know their Ukrainian equivalents. Use the Mini-Dictionary (UNIT 7) if necessary.

precise and concise statements	shorthand
to conceal the meaning	essay
spoken language	vowels and consonants
the sum is independent of the order of the terms	to deal with
in this capacity	literal notation
to permeate	calculus symbol
to pervade all sciences	to grasp
abbreviated	to be due to
verbal	insoluble by other methods
to denote	ambiguity
duly	susceptible
Dy (derivative of y)	to superscribe