Think of the heading for each paragraph of the text

MATHEMATICS − THE LANGUAGE OF SCIENCE

1. Human 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 a 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.