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MATHEMATICS, ITS THEOREMS, AXIOMS AND PARADOXES
Л.А. Хамитова Научный руководитель - С.А. Ермолаева
Тема доклада «Математика, ее разделы и нерешенные проблемы». Статья носит обзорный характер. В статье описываются основные разделы математики, упоминается о парадоксах математики.
Introduction
The students of math may wonder where the word ''mathematics'' comes from. 'Mathematics' 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 of 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 is 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 maths 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 the essential components of any compartment of math. We must break down maths into separately taught subjects, but this compartmentalization taken as a necessity, must be compensated for as much as possible. Students must see the interrelationships of the various areas and the importance of maths 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 its intimate relationships to other fields of knowledge. If math is isolated from other provinces, it loses importance [3].
Concepts, Axioms, Theorems
The basic concepts of the main branches of maths 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 the 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 which operations such as addition, multiplication, and the like can be applied. The notion of a variable that represents the quantitative values of some changing physical phenomena, such as temperature and time, is 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 the 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 geometrical 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 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 19th 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 math activity consists of the theorems deduced from a set of axioms. The amount of information 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 solution of various types of differential equations. Math 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 they 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 numbers 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 viewpoint, is a human creation in every respect. Some mathematicians claim that pure math is the most original creation of the human mind [2].
Geometry
No doubt the Greeks deserve the highest praise in all these fields. Euclid deducted all the most important results of the Greek masters of the classical period and therefore the Elements constituted the math history of the age as well as the logical presentation of geometry. The effect of this single book on the future development of geometry was enormous and is difficult to overstate.
The creation of Euclidean geometry is more than the contribution of numerous useful theorems. It reveals the power of reason. No other human creation demonstrates how much knowledge can be derived by reasoning alone as have the hundreds of proofs in Euclid's Elements. The necessity for accurate and exact definitions, for clearly stated assumptions and for rigorous proof became evident in Euclid's Elements.
We know much of the material of Euclid's Elements through our high school studies. By studying Euclid, hundreds of generations from Greek times learned how to reason, how perfect logical reasoning must proceed, how to master the procedure, how to distinguish exact reasoning from vague pretence of proof. Even nowadays this masterpiece of Euclid serves as a logical exercise and as a model of reasoning and the art of the mind [1].
Algebra
Algebra is not only a part of math; it also plays within math the role which math itself had been playing for a long time with respect to physics. What does the algebraist have to offer to other mathematicians? Occasionally, the solution of a specific problem; but mostly a language in which to express math facts and a variety of patterns of reasoning, put in a standard form. Algebra is not an end in itself; it has to listen to outside demands issued from various parts of math. This situation is of great benefit to algebra; for, a science, or a part of science, which exists to solve its own problems only, is always in danger of falling into peaceful slumber and from there into a quiet death.
But in order to take full advantage of this state of affairs, the algebraist must have the ability to derive profit from what he perceives is going on outside his own domain. Algebra, like every other modern branch of math and science, continues to proliferate with the vitality and expansiveness of a tropical forest and every particular part of algebra has much new math knowledge that is being discovered, so that the algebraist should keep his eyes open for the small piece that may be of great value to him [2].
Cybernetics
The word "cybernetics" originated from the Greek "Kibernetike", the Latin "gubernator" and the English "governor" all meaning, in one sense or another, "control", "management" and "supervision''. More recently Norbert Wiener has used the word to name his book, which deals with the activity of a group of scientists engaged in the solution of a wartime problem and some of the math concepts involved. Nowadays the word has become associated with the solution of problems dealing with activities for computers. As such, the discipline must rely on the exact sciences as well as sciences such as biology, psychology, biochemistry and biophysics, neurophysiology and anatomy.
Before studying computer systems it is necessary to distinguish between computers and calculators. These terms have, by connotation, two distinctly different meanings. The term "calculator" will refer to a machine which can perform arithmetic operations, which is mechanical, which has a keyboard input, which has manually - operated controls. The term "computer" will refer to automatic digital computers which can solve complete problems, are generally electronic, have various rapid input - output devices, have internally - stored control programs. Speed and general usefulness make a computer equivalent to thousands of calculators and their operators. The ability of electronic computers to solve math and logical problems, thereby augmenting the efficiency and productivity of the human brain, has made the sphere of their application practically boundless.
Informatics
We may ask a question what "information" is. In the discussions of computers, the word "information" has a rather special definition. Information is a set of marks that have meaning. In a large automatic electronic computer, information may be recorded and manipulated as sequence of minute electrical pulses which are about a millionth of a second apart; and the presence or absence of a position where either may occur is the basic code which represents
information. "Informatics" is a collection of computer theories and novel information technologies.
In terms of computer development informatics is concerned with the design and construction of electrical or electronic analogs capable of performing processes carried out within a living entity, including the selection and evaluation, as well as the storage of information. In terms of understanding the operation of the human nervous system, informatics contributes new insight into a wide range of processes such as learning, regulation of and the emotional behavior of individual human beings as well as societies. Specifically, the problems of decision - making, thinking and synthesis, imagination and creative endeavor of people, come under the scrutiny of informatics [5].
Mathematics - the language of science
One of the foremost reasons given for the study of maths is to use a common phrase, 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 maths in our scientific age.
The language of maths consists mostly of signs and symbols, and, in a 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 the 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 of maths are the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, 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 maths for it determines its true aspect. With the aid of symbolism mathematicians can make translation in reasoning almost mechanically by the eye and leave their mind free to grasp the fundamental ideas of the subject matter. Just as music uses symbolism for the representation and communication of sounds, so maths expresses quantitatively relations and spatial forms symbolically. Unlike the common language, which is the product of custom, as well as social and political movements, the language of maths 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.
Math language is precise and concise, so that it is often confusing to people unaccustomed to its forms. The symbolism used in math language is essential to distinguish meanings often confused in common speech. Math 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 math writing is remarkable because it encompasses much in few words. In the study of maths much time must be devoted 1) to the expressing of verbally stated facts in math 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 relations 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 students 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 maths and why the study of maths is discouraging to weak minds, those who are not able to master the subject.
Myth in mathematics
There are many myths about math, e.g., that ''mathematics is the queen of the sciences''; that the Internet is the cyberspace world- a new universe- and that informatics will reign and dominate throughout the 21st century. Some people believe that only gifted, talented people can learn maths that it is only for math-minded boys, that only scientists can understand math language, that learning maths is a waste of time and efforts, etc.
Another myth in math is that "women cannot be genuine mathematicians". Female applicants must satisfy the same requirements at the entrance competitive examinations as boys should, there are no special tracks for girls. Most female applications assert to have chosen to study maths because they like it rather than as a career planning. The change of high - school maths into university maths is for many of them a real shock, especially in the amount of information covered and the skills that are being developed. Despite this shock the study of higher maths should be available to a large set of students, both male and female, and not to the selected few.
There is no reason that women cannot be outstanding mathematicians and the Russian women mathematicians have proved it. There should be affirmative action to bring women teachers onto math faculties at colleges and universities. One cannot expect the ratio to be 50/50, but the tendency should continue until male mathematicians no longer consider the presence of female mathematicians to be unusual at math department faculty or at the conferences and congresses.
Some ambitions experts claim that they think of mathematicians as forming a world nation of their own without distinctions of geographical origins, race, creed, sex, age or even time because the mathematicians of the past and "would - be" are all dedicated to the most beautiful of the arts and sciences. As far as math language is concerned, it is in fact too abstract and incomprehensible for average citizens. It is symbolic, too concise and often confusing to non - specialists. The myth that there is a great deal of confusion about math symbolism, that mathematicians try by means of their peculiar language to conceal the subject matter of maths from people at large is unreasonable and meaningless. The maths language is not only the foremost means of scientist's intercourse, finance, trade and business accounts, it is designed and devised to become universal for all the sciences and engineering, e.g., multilingual computer processing and translation [4].
Paradoxes inn mathematics
In 1926 F.P. Ramsey proposed a distinction of the paradoxes known at that time into two types: logical or mathematical paradoxes, and semantical paradoxes. Ramsey argued that paradoxes of the latter sort by virtue of making reference to language (meaning, truth, definability) cannot be stated within math, in which there is no reference to such matters, and thus there is no need to consider them at all in attempting to devise ways of avoiding paradoxes within maths. This reasoning is not as conclusive as Ramsey apparently thought, but his classification is helpful and has been widely used.
The most famous of the logical, or math paradoxes is Russell's paradox. This paradox proceeds as follows. First we define a class K, say, as the class of all those classes that are not elements of themselves. The class of dogs, for example, is not itself a dog and this is not an
element of itself. By the definition of K, then, the class of dogs is an element of K. And so are most, if not all, of the classes that first come to mind. Now we ask whether K itself is an element of K. We see immediately that K is an element of K if, and only if, it is not an element of K. It follows by the propositional logic that the class K both is and is not an element of itself. But this is a contradiction, or a paradox. As a well-known example of the semantical paradoxes, we have the paradox of the liar. This paradox, in one form or another, goes back to ancient times. In one of its forms it proceeds as follows. Consider a man who says, "I am lying" and then says nothing further. If this man is telling the truth, then (as he says) he is lying; if, however, he is lying, then he is telling the truth in saying so. It follows by the propositional calculus of logic that he is both lying and telling the truth, which is a contradiction. This paradox is a semantical paradox because it makes reference to certain uttered words, which express a lie. True, this paradox does indeed at first seem frivolous and unworthy of serious consideration. Yet, it is as genuine a paradox as any other, and must be taken seriously if any paradoxes are taken seriously at all. These paradoxes cannot appear within maths, in particular within set theory [2].
Summary
There are two ways in which maths has become so effective in our age. The first is through its relationships with science, the second is through its connection with human reasoning. Math method is reasoning of the highest level known to man, and every field of investigation - be it law, politics, psychology, medicine or anthropology - has felt its influence and had modelled itself on maths to some extent ever since its creation. In order to gain a more comprehensive view of the relation of maths to the sciences, let us analyze the various ways in which maths has been serving scientific investigations.
Maths has been supplying a language for the treatment of the quantitative problems of the physical and social sciences. Much of this language has taken the form of math symbols. Symbols also permit concise, clear representation of ideas which are sometimes very complex. Scientists have learned to use math symbols whenever possible.
Maths has been supplying science with numerous methods and conclusions. Among the important conclusions are its formulas, which scientists have accepted and used in solving problems. The use of such formulas is so common that the contribution of maths in this direction has not been fully appreciated.
Maths has been enabling the sciences to make predictions. This is perhaps the most valuable contribution of maths to the sciences. The ability to make predictions by math means was exemplified in the most remarkable way in 1846 by the two astronomers Leverrier and Adams. As a result of calculations, they predicted, working independently, that there must exist another planet beyond those known at the time. A search for it in the sky at the predicted place and time revealed the planet Neptune. Prediction has played a part in every math solution of a quantitative problem arising in the physical and social sciences.
Maths has been furnishing science with ideas to describe phenomena. Among such ideas may be mentioned the idea for functional relation; the graphical representation of functional relations by means of coordinate geometry; the notion of a limit; the notion of infinite classes which helps us to understand motion. Of special importance are the statistical methods and theories which have led to the idea of a statistical law. The description is not complete without mentioning the fact that for many physical phenomena no exact concepts exist other than math ones.
Maths has been of use to science in preparing men's mind for new ways of thinking. The concepts of importance in science had been coming to men with great difficulty. The concepts of gravity, of energy and of limitless space took years to develop and men of genius were required to express them precisely. Great as is the genius of Einstein, it is almost certain that
he was able to achieve some of his results only because the maths of preceding decades had suggested new ways of thinking about space and time.
To summarize: Maths has been supplying a language, methods and conclusions for science; enabling scientists to predict results; furnishing science with ideas to describe phenomena and preparing the minds of scientists for new ways of thinking.
It would be quite wrong to think that maths had been giving so much to the sciences and receiving nothing in return. Physical objects and observed facts had often served as a source of the elements and postulates of maths. Actually, the fundamental concepts of many branches of maths are the ones that had been suggested by physical experiences. Scientific theories have frequently suggested directions for pursuing math investigations, thus furnishing a starting point for math discoveries. For example, Copernican astronomy had suggested many new problems involving the effects of gravitational attraction between heavenly bodies in motion. These problems had stimulated the further activities of many scientists in the field of differential equations [4].
Literature
1. Aleksandrov P.S. Geometry.M.: Astrel, 1968, 490 c.
2. Egorov I.P. The history of mathematics. M.: AST, 1970, 488 c.
3. Ribnikov K.A. Introduction to math.: Astrel, 1979, 236 c.
4. Freiman L.S. Mathematics. M.: Astrel, 1968, 520 c.
5. Whitehead A.J. Science of modern world. New York, 1967, 112 c.
