The students of math may wonder where the word “math” comes from. Math is a Greek word, and, by origin or etymologically, it means “something that must be learnt or understood”, perhaps “acquired knowledge” or “knowledge acquirable by learning” or “general knowledge”. The word “math” is a contraction of all these phrases. The celebrated Pythagorean School in ancient Greece had both regular and incidental members. The incidental members were called “auditors”. The regular members were named “mathematicians” as a general class and not because they specialized in math. For them math was a mental discipline if science learning. What is math in the modern sense of the term, its implications and connotations? There is no neat, simple, general and unique answer to this question.
Math as a science, viewed as a whole, is a collection of branches. The largest branch is that which builds on the ordinary whole numbers, fractions, and irrational numbers, or what, collectively, is called the real number system. Arithmetic, algebra, the study of functions, the calculus, differential equations, and various other subjects, which follow the calculus in logical order, are all developments of the real number system. This part of math is termed the math of number. A second branch is geometry consisting of several geometries. Math contains many more divisions. Each branch has the same logical structure. It begins with certain concepts, such as the whole numbers or integers in the math of number, and such as point, line, and triangle in geometry. These concepts must verify explicitly stated axioms. Some of the axioms of the math of number are the associative, commutative, and distributive properties and the axioms about equalities. Some of the axioms of geometry are that two points determine a line, all right angles are equal etc. From the concepts and axioms theorems are deduced. Hence, from the standpoint of structure, the concepts, axioms and theorems are essential components of any compartment of math. We must break down math into separately taught subjects, but this compartmentalization taken as necessity, must be compensated for as much as possible. Students must see the interrelationships of the various areas and the importance of math for other domains. Knowledge is not additive but an organic whole and math is an inseparable part of that whole. The full significance of math can be seen and taught only in terms of intimate relationships to other fields of knowledge. If math is isolated from other provinces, it loses importance.
The basic concepts of the main branches of math are abstractions from experience, implied by their obvious physical counterparts. But it is noteworthy, that many more concepts are introduced which are, in essence, creations of human mind with or without any help of experience. Irrational numbers, negative numbers and so forth are not wholly abstracted from the physical practice, for the man’s mind must create the notion of entirely new types of numbers to which operations such as addition, multiplication and the like can be applied. The notion of variable that represents the quantitative values of some changing physical phenomena, such as temperature and time, it also at least one mental step beyond the mere observation of change. The concept of a function, or a relationship between variables, is almost totally a mental creation. The more we study math, the more we see that the ideas and conceptions involved become more divorced and remote from experience, and the role played by the mind of mathematician becomes larger and larger. The gradual introduction of new concepts which more and more depart from forms of experience finds its parallel in geometry and many of the specific geometric terms are mental creations.
As mathematicians nowadays working in any given branch discover new concepts, which are less and less drawn from experience, and more from human mind the development of concepts is progressive and later concepts are built on earlier notions. These facts have unpleasant consequences. Because the more advanced ideas are purely mental creations rather than abstractions from physical experience and because they are defined in terms of prior concepts it is more difficult to understand them and illustrate their meanings even for a specialist in some other province of math. Nevertheless, the current introduction of new concepts in any field enables math to grow rapidly. Indeed, the growth of modern math is, in part, due to the introduction of new concepts and new systems of axioms.
Axioms constitute the second major component of any branch of math. Up to the XIX century axioms were considered as basic self-evident truths about the concepts involved. We know now that this view ought to be given up. The objective of mathematical activity consists of the theorems deduced from a set of axioms. The amount of info that can be deduced from some sets of axioms is almost incredible. The axioms of number give rise to the results of algebra, properties of functions, the theorems of the calculus, the solutions of various types of differential equations. Mathematical theorems must be deductively established and proved. Much of the scientific knowledge is produced by deductive reasoning; new theorems are proved constantly, even in such old subjects as algebra and geometry and the current developments are as important as the older results.
Growth of math is possible in still another way. Mathematicians are sure now that sets of axioms, which have no bearing on the physical world, should be explored. Accordingly, mathematicians nowadays investigate algebras and geometries with no immediate applications. There is, however, some disagreement among mathematicians as to the way they answer the question: Do the concepts, axioms, and theorems exist in some objective world and are merely detected by man or are they entirely human creations? In ancient times the axioms and theorems were regarded as necessary truths about the universe already incorporated in the design of the world. Hence each new theorem was a discovery, a disclosure of what already existed. The contrary view holds that math, its concepts and theorems, are created by man. Man distinguishes objects in the physical world and invents numbers and number names to represent one aspect of experience. Axioms are man’s generalizations of certain fundamental facts and theorems may very logically follow from the axioms. Math, according to this view-point, is a human creation in every respect. Some mathematicians claim that pure math is the most original creation of human mind.
Mathematics – the language of science
What distinguishes the language of science from language as we ordinarily understand the word? How is it that scientific language is international? The super-national character of scientific concepts and scientific language is due to the fact that they are set up by the best brains of all countries and all times.
One of the foremost reasons given for the study of math is, to use a common phrase, that “math is a language of science”. This is not meant to imply that math is useful only to those who specialize in science. No, it implies that even a layman must know something about the foundations, the scope and the basic role played by math in our scientific age.
The language of math consists mostly of signs and symbols, and, in sense, is an unspoken language. There can be no more universal or more simple language, it is the same throughout the civilized world, though the people of each country translate it into their own particular spoken language. For instance, the symbol 5 means he same to a person in England, Spain, Italy or any other country; but in each country it may be called by a different spoken word. Some of the best known symbols in math are the numerals 1..0 and the signs of addition, subtraction, multiplication, division, equality and the letters of the alphabets: Greek, Latin, Gothic and Hebrew (rather rarely).
Symbolic language is one of the basic characteristics of modern math for it determines its true aspect. With the aid of symbolism mathematicians can make transitions in reasoning almost mechanically by the eye and leave their mind free to grasp the fundamental ideas of the subject matter. Just as the music uses symbolism for the representation and communication of sounds so mathematicians express quantitative relations and spatial forms symbolically. Unlike the common language, which is the product of custom, as well as social and political movements, the language if math is carefully, purposefully and often ingeniously designed. By virtue of its compactness, it permits a mathematician to work with ideas which when expressed in terms of common language are unmanageable. This compactness makes for efficiency of thought.
Mathematical language is precise and concise, so that it is often confusing to people unaccustomed to its forms. The symbolism used in mathematical language is essential to distinguish meanings often confused in common speech. Mathematical style aims at brevity and formal perfection. Let us suppose we wish to express in general terms the Pythagorean theorem, well familiar to every student through his high-school studies. We may say: “We have a right triangle. If we construct two squares each having an arm of the triangle as a side and if we construct a square having the hypotenuse of the triangle for its side, then the area of the third square is equal to the sum of the areas of the first two”. But no mathematician expresses himself that way. He prefers: “The sum of the squares on the sides of a right triangle equals the square on the hypotenuse”. In symbols this may be stated as follows: c2 = a2 + b2. This economy of words makes for conciseness of presentation, and mathematical writing is remarkable because it encompasses much in few words. In the study of math much time must be devoted 1) to the expressing of verbally stated facts in mathematical language, that is, in the signs and symbols of math; 2) to the translating of math expressions into common language. We use signs and symbols for convenience. In some cases the symbols are abbreviations of words, but often they have no such relation to the thing they stand for. We cannot say why they stand for what they do; they mean what they do by common agreement or by definition.
The student must always remember that the understanding of any subject in math presupposes clear and definite knowledge of what precedes. This is the reason why “there is no royal road” to math and why the study of math is discouraging to weak minds, those who are not able and willing to master the subject.