The name “mathematics” comes from the Greek word “mathein” – to learn, to learn. The ancient Greeks generally believed that the concepts of “mathematics” (mathematike) and “science”, “knowledge” (mathema) are synonyms.
They were characterized by such an understanding of the universalism of this branch of knowledge, which was expressed two thousand years later by Rene Descartes, who wrote: “The field of mathematics includes sciences that consider either order or measure, and it does not matter at all whether it will be numbers, figures, stars, sounds or something else …; therefore, there must be some kind of general science explaining everything related to order and measure, without entering into the study of any particular objects … ”
Another explanation of the origin of the word “mathematics” is associated with the Greek word “mathema”, which means harvest, harvest. Marking of land (geometry), determining the timing of field work (based on astronomical observations and calculations), preparing the required amount of seeds and counting the harvest required serious mathematical knowledge.
The role of mathematics in modern science is constantly growing. This is due to the fact that, firstly, without a mathematical description of a number of phenomena of reality, it is difficult to hope for a deeper understanding and development of them, and, secondly, the development of physics, linguistics, technical and some other sciences involves the widespread use of the mathematical apparatus. Moreover, without the development and use of the latter, for example, neither space exploration nor the creation of electronic computers, which have found application in various fields of human activity, would be possible.
Mathematics in the Knowledge System
During its existence, mankind has come a long way from ignorance to knowledge and from incomplete knowledge to a more complete and perfect one. Despite the fact that this path has led to the discovery of many laws of nature and to the construction of an excitingly interesting picture of the world, every day brings new discoveries, new insights into the mysteries of nature that have not been sufficiently studied and sometimes completely unknown.
But in order to advance into the area of the unknown as far as possible and put new forces of nature at the service of society, science should boldly break into those areas of knowledge that humanity was not yet seriously interested in or which, due to the complexity of the phenomena prevailing there, seemed inaccessible to our knowledge.
In the eyes of our generation, science has taken a colossal step in studying the laws of nature and in using the knowledge gained. Suffice it to say about the astounding successes in space exploration and the study of intra-atomic phenomena, as well as about the first heart operations. What was so recently still unknown, beyond the boundaries of people’s ideas and, all the more, beyond their practical activities, has now become habitual and has entered our lives. Advances in medicine have allowed many, seemingly hopelessly sick people to return to active life, for whom the joy of perceiving the beauty of the world around them has been lost.
Mathematics is becoming increasingly important in the economy, organization of production, as well as in the social sciences.
The position of mathematics in the modern world is far from what it was a hundred or even only forty years ago. Mathematics has become an everyday instrument of research in physics, astronomy, biology, engineering, production organization and many other areas of theoretical and applied activity. Many major doctors, economists, and social research experts believe that the further progress of their disciplines is closely linked to the wider and more full-fledged use of mathematical methods than they have been up to now.
Over the millennia of its existence, mathematics has come a long and difficult path, during which its character, content and presentation style have repeatedly changed.
From primary ideas about a straight line segment as the shortest distance between two points, from objective representations about integers within the first ten, mathematics came to the formation of many new concepts and powerful methods that turned it into a powerful means of studying nature and a flexible instrument of practice.
From primitive counting through pebbles, sticks and nicks on a tree trunk, mathematics has evolved into an extensive harmonious scientific discipline with its own subject of research and specific deep methods. She developed her own language, very economical and accurate, which turned out to be extremely effective not only inside mathematics, but also in many areas of its applications.
No matter how great the successes of scientific knowledge are, we notice many problems that have not yet been sufficiently studied and require additional efforts, sometimes very significant. We call the processes of thinking, the causes of the development of mental illness, the management of cognitive activity. At the same time, we are all aware of how important it is to advance our understanding of these phenomena as quickly as possible.
Indeed, if we knew the processes of thinking quite accurately, then this would facilitate and accelerate the education of children and adults, and acquire new opportunities in the treatment of mental illness. But these tasks are so complex that there are no hopes to solve them by purely experimental methods. It is necessary to attract completely different possibilities of cognition, in particular, the path of mathematical modeling of these processes and the subsequent obtaining of logical consequences that are already accessible to direct observation. This technique has proven itself in many areas of knowledge – in astronomy, physics, chemistry, etc.
We have so far talked about mathematics only as an instrument of research in other fields of knowledge and practical activity. This aspect is closely connected with the progress of mathematics itself, with the expansion of the field of its research, the development of its basic concepts and the creation of new concepts.
In the meantime, we have limited ourselves to looking at it from the standpoint of the consumer, from the standpoint of determining its value for the development of human culture and social welfare. In this regard, mathematics occupies a completely outstanding position. And although she herself does not produce material values and does not directly study the world around us, she provides invaluable assistance to humanity in this.
Modern Mathematics and Style of Scientific Thinking
Consideration of the influence of mathematics on changing the very style of scientific thinking, on changing traditional methods of inference is of undoubted interest, if only because it allows you to more deeply penetrate the changes that have occurred in modern scientific thinking, to understand their causes, as well as the inevitability of this phenomenon.
Cognition of an object is not carried out suddenly, but passes through a series of successive stages. First, a person observes the phenomenon and observes some of its features. Then, in order to clarify the information obtained, it is time to conduct an experiment, i.e., to observe the phenomenon of interest to us under fairly strictly observed conditions.
At the same time, attempts are being made to explain the observed facts on the basis of existing general ideas. The foundations of the theory of this phenomenon are being created. Corollaries are derived from this theory. By the coincidence of the obtained consequences with the course of the phenomenon, they judge the compliance of the theory with the true state of affairs.
If the theory allows one to obtain information about facts that were not previously observed, and then, according to the instructions of the theory, they appear in reality, then the theory receives strong confirmation.
But the theory can be purely qualitative in nature, which does not even provide for the very possibility of producing quantitative conclusions. Until recently, medicine belonged to this type of theory. To a large extent, the economy was at this level. Pedagogy also belongs to theories of a qualitative type. This is characteristic of theories of very complex phenomena, in which it is extremely difficult to get to quantitative laws and such laws themselves are very complex.
It may happen that the familiar mathematical apparatus for studying them has not even been created. But this does not mean that attempts should not be made to use a quantitative approach to these complex phenomena, or at least to their individual, private issues. Quantitatively formulated theories provide incomparably greater opportunities for obtaining conclusions, and, moreover, conclusions that can be verified by exact methods.
In a qualitative theory, they are satisfied with conclusions of this type: “When the wires are heated, the wear of their insulation increases.” For practice of this type, the conclusion is of only limited interest, since it is important for her to know how quickly this wear increases with increasing temperature of the wires. Only knowledge of such quantitative relationships can allow us to choose the optimal mode in one sense or another.
Mankind has long noticed the effect of the lever and used it from time immemorial. However, only a quantitative theory of it made it possible to make preliminary calculations and pre-calculate the forces that must be applied in order to obtain the necessary effect. But this step in the development of our knowledge was taken at a very high stage in the progress of scientific thought.
However, the involvement of mathematical methods in science inevitably entails the necessity of attracting the very style of mathematical thinking: a clear formulation of the starting points, the completeness of the classification, the rigor of logical conclusions. These issues will now be discussed.
In mathematics, the set of initial positions in which the problem is solved is always listed. Therefore, the result obtained, generally speaking, is true only when these initial positions are satisfied. Let us take this Pythagorean theorem on the relationship between the length of the hypotenuse and the length of the legs for the sake of illustrating this statement. This theorem is true for all right triangles of the Euclidean plane. If we consider right-angled triangles on some other surface, for example, on a sphere, then the Pythagorean theorem, generally speaking, will be false.
That is why in mathematics it is necessary to enumerate all the conditions in which the result is correct, and it is not allowed to add additional assumptions that were necessary in the process of reasoning. Such a meticulous accuracy in enumerating the conditions of theorems and in the whole exposition, originating in mathematics from the time of Hellenism, was inherent only to it for a long time. In other scientific disciplines, as well as in practical activities, this honed rigor was indifferent at best.
The axiomatic method of presentation, adopted in geometry since the time of the ancient Greeks, was more widely developed in the 19th century. In the works of Italian geometers, and later in the famous work of D. Hilbert (1862-1943), “Foundations of Geometry,” the Euclidean axioms themselves were carefully studied.
Moreover, it turned out that classical axioms are far from enough for strictly logical construction of Euclidean geometry, that in the process of logical reasoning in classical geometry, in proving theorems, they resort to additional intuitive considerations that are not contained in the formulated axioms. Hilbert carefully analyzed the initial positions of the Euclidean geometry and managed to complete the process of selecting the initial positions, which began in ancient Greece.
Later, algebra, mechanics, probability theory, and a number of other areas of mathematical thought took the same path of clearly enumerating the initial principles of the theory. With this method of presentation, it is always known what is at stake, and there is no danger of introducing considerations of intuition with the right reasoning into the final result, there is no possibility of multiple judgments about the same subject.
This simple idea – to consider well-defined concepts and draw conclusions regarding them, based on certain starting points, axioms – today is widely included in the everyday life of science and practice. This approach, applied to the rules of grammar, showed that they do not have a complete definition. The situation is saved by the habit of everyday spoken language, as a result of which a certain defect in definitions does not play a serious role in the use of the native language. However, any attempt to transfer to an automaton the construction of phrases according to certain grammar rules or the translation from one language to another inevitably leads to errors, to numerous opportunities for incorrect speech turns. And there are a lot of such conversations between a person and a machine these days, and we must have confidence that the machines will correctly accept the instructions and do exactly what they are given.
In connection with the first steps of mankind in the conquest of outer space, the problem of humanity’s communication with other civilizations, which may have to be encountered during space flights, becomes urgent. In this case, the task of communication will inevitably arise. It is clear that French, English or Russian is not enough for this. So far, science fiction writers are primarily concerned with problems of this kind.
They offer a solution that may not be realized in reality: representatives of other civilizations are at such a high level of intellectual development that they already have perfect automatic translators that automatically adjust to the language of the astronaut who has arrived at them and conduct a conversation with him in his native language. However, scientists are also thinking about this problem. They come from a different assumption. If we have to meet with representatives of extraterrestrial civilizations, then they will possess elements of formal logic and possess the basics of geometric representations. Since the laws of the world are the same, the laws of logic and the primary geometric concepts of earthlings and representatives of extraterrestrial civilization will be the same.
However, the need for a mathematical approach to the rigor and accuracy of definitions and logical reasoning is needed not only for similar, so far distant prospects, but also for cases, regardless of whether they relate to linguistics, jurisprudence, engineering, or economics. For a number of years I was quite closely associated with doctors, doing joint research on objectifying the diagnosis of heart disease. I was struck by the presence of an almost mathematical style of thinking in the main team of doctors – employees of the Institute of Heart Diseases. The analysis of the condition of each patient was carried out with amazing logical scrupulousness, which until recently has been characteristic only of mathematical research.
The second side of the mathematization of thinking is the desire that is now being observed – to derive logical consequences from strictly formulated initial positions and then to directly observe these consequences. Moreover, those theoretical constructions that make it possible to attract a diverse apparatus of deductive mathematics to obtain logical conclusions are of particular value. At the same time, it is possible to take advantage of the huge volume of conclusions already obtained by mathematics. This has been used in mathematics for a long time.
Almost two centuries ago, mathematical physics arose, which, on the basis of the basic principles derived from observation and experience, receives extensive consequences mathematically. This is how geometric and wave optics developed, so the development of acoustics and electrodynamics proceeded. To an even greater extent, this path has proved itself in modern physics dealing with atomic and subatomic phenomena. Mathematical theory led to the conclusion that previously unobserved elements of matter should exist.
These conclusions were compared with the results of observations, and these comparisons led to interesting and important consequences: the calculation of the particle mass and charge; its relationships with previously observed particles, etc. Sometimes years passed before it was possible to confirm the conclusions of mathematical theory experimentally. Modern physics is full of such mathematical precalculations of real phenomena, about which nothing was known and which were later discovered through complex experiments, specially thought out on the basis of mathematical theory.
It is easy to give numerous examples of how the mathematical style of thinking has benefited in other areas of knowledge – biology, economics, and the organization of production. Recall, for example, that electrical engineering and radio engineering are stated as mathematical disciplines and use a diverse and very complex mathematical apparatus. This fully justifies itself, because it allows you to make timely calculations, predict the course of processes, and get the ability to control processes.
We talked about the fact that the quality of any theory of real phenomena is verified by practice and by staging appropriately organized experiments. However, mathematics intervened in the organization of the experiment itself: how should observations be organized in order to extract the maximum information with the same number of tests? This problem is important because huge material resources and human efforts are now spent on tests in industry, on experiments in scientific laboratories and design bureaus.
The foundations of the mathematical theory of the experiment have already been created, which can significantly reduce the number of necessary observations, their cost and duration to obtain reasonable conclusions. Sometimes this gain is very large – dozens of times (at the cost I spent the effort). The main idea, which is used in this case, is to take into account the result of previous tests and to carry out each subsequent test so that it clarifies the information already received.
The advent of computers has changed the attitude of people towards the possibilities of mathematics in solving vital problems. It turned out that not only the production of cumbersome computational work, but also the implementation of logical conclusions can be transferred to machines. But in order to make this possible, it is necessary to first draw up a logical-mathematical model of a phenomenon or process, to identify the relationships and the available quantitative relationships.
In other words, it is necessary to subject the process to preliminary mathematical and logical analysis. A new, very powerful research method has opened before humanity, which has found almost immediately the broadest application in the most diverse fields of knowledge, both in science and in direct practice. As a result, many people who were previously skeptical of the possibilities of mathematics became adherents of its use and with enthusiasm began to apply the mathematical style of thinking, mathematical methods to problems of interest to them.
The presence of mathematical machines also allows for fantastically short time to carry out grandiose calculations, which until recently were inaccessible to the previous means of computer technology. The difficulties of computing have shifted to the questions of creating the appropriate programming languages, to compiling computational programs, to creating techniques for automatically selecting the right program by the machine itself, developing the theory of errors in mass computing, etc. Mathematics and mathematicians have been freed from the need to perform numerous elementary purely technical operations.
But at the same time, a more complex and interesting set of works fell on specialists: compiling models, developing methods for communicating between a person and a machine, studying the possibility of automatically collecting experimental data and processing them, etc. The problems of mathematical research have been enriched significantly. So a change in the style of scientific thinking in the direction of its mathematization forced mathematics to progress, significantly expand the arsenal of its tools and methods of studying the various phenomena of the world around us.
Since mathematics is by its nature universal and abstract knowledge, it can and should be used in principle in all branches of science. Mathematics can be attributed to the general sciences. In fact, it is considered a universal and abstract science, since the mathematical apparatus can in principle be used and is practically used in all fields of knowledge without exception.
The task of mathematics is to describe a process with the help of some mathematical apparatus, that is, in a formal logical way. Speaking about the subject and functions of mathematics, it is obvious that in modern science the integrating role of mathematics is becoming more and more tangible, since it is a universal scientific discipline. The functions of mathematics are equally humanitarian functions, since they are aimed at improving the material and spiritual spheres of human existence.
In the study of mathematics, the development of the student’s intellect is carried out, enriching it with methods of selecting and analyzing information. Teaching any branch of mathematics has a beneficial effect on the mental development of students, as it instills in them the skills of clear logical thinking, operating with clearly defined concepts.
Mathematics contains the features of volitional activity, speculative reasoning and the desire for aesthetic perfection. Its main and mutually opposing elements are logic and intuition, analysis and construction, generality and concreteness.
The study of mathematics also contributes to the formation of the civic qualities of the individual through the upbringing of a property that we call intellectual honesty, which has a beneficial effect on the mental, moral, and aesthetic development of students.
At the same time, the strong-willed qualities of a person are brought up, without which it is impossible to master scientific theory, the skills of independent research work are formed, and finally, intellectual honesty is cultivated, which does not allow one to operate on dubious facts that have not been proved with all the necessary rigor. Moreover, this applies not only to solving mathematical problems, but also to other areas of human activity, including the analysis of the phenomena of social and political life. Mathematical education from the learning process external to the student is transformed into the cognitive process itself. Only the combined actions of these polar principles and the struggle for their synthesis provide the vitality, usefulness and high value of mathematical science.
Given the internal logical unity of mathematics, the organic interconnection of its parts, the most important requirement for the organization of its teaching should be consistency and continuity in learning, vision at all its stages of the main goal. This goal is the accumulation of special knowledge, mastering the methods of setting and solving mathematical problems and on their basis the development of students’ intellect, the formation of a culture of thinking in them, the development of strong-willed personality qualities, the ability to overcome difficulties, aesthetic development based on the ability to appreciate the beauty of scientific constructions and joy from gaining new knowledge.
Thus, mathematics, by its specific means, contributes to the solution of a whole range of humanitarian problems and is of great importance in the life of society.
There is no doubt that mathematics and the mathematical style of thinking are now making a triumphal march both in science and in its applications. Pupils, students should to some extent feel this and relate to mathematics with great interest, enthusiasm and understanding of the need for mathematical knowledge, both for their future activities and for the life of human society.
How to Write an Essay on Mathematics
Mathematics is not only formulas, equations and examples. An essay can also be written on this subject. Sample topics:
- “The mathematics around us.”
- “The beauty of mathematics.”
- “Why do we need math?”
- “Mathematics in our life.”
In such works, one needs to speculate on what mathematics is taught for; how useful this science is in everyday life (measurements, monetary calculations, etc.).
As noted above, an essay does not have a clear structure, but its outline usually includes:
Choose a topic. For instance, pick a algebra you have studied, if you find it interesting. When you create your own model, describe exactly what it is. This part should set the emotional mood, bring the reader to the issue under consideration, interest him, prompting him to read the text to the end.
Support your thoughts with a quote about the benefits of mathematics. For example, Aristotle claimed:
Mathematics reveals order, symmetry and certainty, and these are the most important types of beauty.
The main part.
At this stage, the author puts forward theses, finds them justifications, thereby proving his own point of view.
After choosing a subject, you should at least know in principle with which mathematical model you will work. The next step is to decide what questions you’ll consider.
Select some questions you want to answer when writing your essay. Divide them into appropriate parts. Briefly describe each of them.
- Make calculations
Solve the problems. This is the key part! Make fun!
Summing up what has been said in the main part, the author makes a general conclusion. If the purpose of the introduction is to interest the reader, then the conclusion should give integrity to the overall picture, not leave doubts about the legitimacy and viability of the ideas and evidence expressed by the author.
But it should also include enough words for people to understand this; Theorems and proofs may be appropriate, but certainly not sufficient.