"The poet's life in the universe must be both internal and external. If it is involved, you can write it. Outside it, we can see it. Into it, so there is life. Beyond it, it is high. "

I only know what's inside, but I can't see the forest for the trees, and I often get lost. So help you stand up and be far-sighted. If you don't stand up, you won't know where the roots of mathematics are and what the ultimate goal and direction of your research are. If you don't stand up, you can't see the close relationship and interaction between mathematics and other disciplines. If you don't stand up, you won't see the great contribution of mathematics to human civilization.

The whole history of human civilization is like the waves of the Yangtze River, rising higher and higher and rolling forward. The giants of science stand at the forefront of the times and push human civilization from one climax to another with their courage, wisdom and diligence. We believe that the whole human civilization can be divided into three distinct levels:

(1) agricultural civilization represented by hoes;

(2) industrial civilization represented by assembly line operation of large machines;

(3) Information civilization represented by computers.

Mathematics is the deep motive force of these three civilizations. Its effect is more obvious every time.

Mathematics has always been the main cultural force of human civilization. It not only has important value in scientific reasoning, but also plays a core role in scientific research and is indispensable in engineering design. Moreover, mathematics determines the content and research methods of most philosophical thoughts, destroys and constructs many religious doctrines, provides the foundation for politics and economics, shapes many schools of painting, music, architecture and literature styles, and creates logic. Mathematics provides us with the best answer to the basic relationship between man and the universe. Mathematics, as the embodiment of reason, has penetrated into the fields previously ruled by authority, habits and customs, and replaced it as the guide of its thoughts and actions.

Here, it should also be pointed out that mathematical culture includes two aspects. First, mathematics, as a subsystem of human culture, has its own law of occurrence and development and its own structure; First of all, its relationship with other cultures and the whole human civilization. Today's report hopes to balance two aspects, but focuses on the second aspect.

We must realize that the influence of mathematics on human culture has some characteristics: from small to large, from weak to strong, from less to more, from hidden to obvious, from natural science to social science.

In short, today we are going to sing a hymn of mathematics, praising the profoundness of mathematical thought, the rational spirit derived from mathematical culture and the vigorous development of human civilization under the guidance of rational spirit.

1. Mathematics in ancient Greece. The greatest contribution of the ancient Greeks was to realize the fundamental role of mathematics in human civilization. This can be summarized by Pythagoras: natural numbers are the mother of all things.

The Pythagorean school aims to explore the eternal truth of the universe by revealing the mystery of numbers. They carefully observed the world around them and found the relationship between numbers and geometric figures, the harmony between numbers and music, and they also found that numbers are closely related to the operation of celestial bodies. They divided the whole learning process into four parts: (1) the absolute theory of numbers-arithmetic; (2) Static quantity-geometry; (3) the amount of exercise-astronomy; (4) The application of numbers-music. Together, they are called the Four Arts.

They concluded that natural numbers are the mother of all things. All phenomena in the universe depend on integers to some extent. But when they used Pythagoras theorem to find that it could not be expressed as the ratio of two integers, that is, it was not a rational number, they were greatly shocked. This broke out the first mathematical crisis. The first crisis of mathematical foundation is a milestone in the history of mathematics, and its appearance and overcoming are of great significance. The first mathematical crisis shows that Greek mathematics has developed to such a stage, which proves that it has entered mathematics and mathematics has changed from empirical science to deductive science.

Mathematics in China, Egypt, Babylonia, India and other countries has never experienced such a crisis, so it still stays in the stage of experimental science, that is, arithmetic. Greece, on the other hand, took a completely different road, formed Euclid's "geometric primitive man" and Aristotle's logical system, and became the ancestor of modern science.

2. Euclid's Elements of Geometry. The appearance of Euclid's Elements of Geometry is a great milestone in the history of mathematics. People have attached great importance to it since it came out. In the western world, apart from the Bible, there is no other work whose function, research and publication can be compared with the Elements of Geometry. Since 1482 was first printed and published, there have been more than 0/000 versions of/kloc-0. In China, Matteo Ricci, an Italian missionary in the Ming Dynasty, and Xu Guangqi in China jointly translated the first six volumes, which were published in 1607. The Chinese translation is called "Geometric Element". Xu Guangqi once spoke highly of this book. He said: "There are four things you don't have to do in this book: you don't have to doubt, you don't have to guess, you don't have to try and you don't have to change. There are four things you can't get: you can't get rid of it, you can't get it if you want to refute it, you can't get it if you want to reduce it, and you can't get it before and after. There are three kinds of abilities: seemingly dull, but clear, which can be dull by other obvious things; It seems complicated, but it is simple, so it is possible to simplify other things with its simplicity; It seems difficult, but it is easy, so it is difficult to change other things with it. Easy to be born in Jane, Jane was born in Ming, and its beauty is just in Ming. " The publication of "Elements of Geometry" had a great influence on the mathematics field in China.

Euclid's Elements of Geometry is called the Bible of mathematicians, which has an unparalleled lofty position in the history of mathematics and even in the history of human science. What is its main contribution to mathematics?

(1) From the basic assumption to the most complicated conclusion, the scattered mathematical theories are successfully arranged into a whole structure.

(2) Axiomatic deduction of propositions. Starting from definition, axiom and postulate, the logical system of geometry was established, which became the model of all mathematics in the future.

(3) For centuries, it has become the most powerful educational means to train logical reasoning.

(4) Deductive thinking first appeared in geometry, not algebra, which made geometry more important. This state remained until the birth of Cartesian analytic geometry.

We should also note that its influence goes far beyond mathematics, but has brought great influence to the whole human civilization. Its contribution to mankind lies not only in producing some useful and wonderful theorems, but also in nurturing a rational spirit. No other human creation can show that so much knowledge comes from only a few axioms like Euclid's hundreds of proofs. These profound deductive results made the Greeks and later civilizations understand the power of reason, thus enhancing their confidence in using this talent to succeed. Encouraged by this achievement, people apply rationality to other fields. Theologians, logicians, philosophers, politicians and all those who seek truth follow Euclid's model to establish their own theories.

In addition, the importance of Euclidean geometry lies in its aesthetic value. With the wonderful structure of geometry and the development of precise reasoning, mathematics has become an art.

3. Overview of Greek culture. The culture of ancient Greece lasted from about 600 BC to 300 BC. Ancient Greek mathematicians emphasized strict reasoning and the conclusions drawn from it. What they care about is not the practicality of these achievements, but educating people to make abstract reasoning and inspiring people to pursue ideals and beauty. Therefore, this era produced beautiful literature, extremely rational philosophy, and idealized architecture and sculpture that later generations could not surpass. The brokeback beauty, Venus de Milo (4th century BC), is the best representative of that era and a symbol of perfection and beauty. It is precisely because of the development of mathematical culture that Greek society has all the embryos of modern society.

What kind of precious heritage has Greek culture left for human civilization? It left four treasures to future generations.

First, it leaves us with a firm belief that natural numbers are the mother of all things, that is, mathematics is the core of the laws of the universe. This belief inspires people to find the ultimate cause of all phenomena in the universe and quantify it.

Secondly, it breeds a rational spirit, which has now penetrated into all fields of human knowledge.

Thirdly, it gives a template-Euclidean geometry. The brilliance of this model illuminates every corner of human culture.

Fourthly, the conic curve is studied, which lays the foundation for future astronomical research.

However, it is a pity that Roman soldiers killed Archimedes, the giant of science, with one knife. This declared the end of a moderns. Whitehead commented: "Archimedes died at the hands of Roman soldiers, which is a symbol of great changes in the world." The pragmatic Romans replaced the theoretical Greeks and led Europe ... Rome is a great nation. However, it has been criticized for emphasizing practical results without making achievements. They didn't improve the knowledge of their ancestors, and their progress was limited to the technical details of the project. They have no dreams and can't get new ideas, so they can't get new control over natural forces. "

Then there was more than a thousand years of stagnation.

4. The influence of Euclidean geometry. Euclidean geometry is a kind of reasoning mode, which is characterized by simple control of complexity, and less wins more. This book has become a model for later generations to imitate. Here are a few typical examples.

Archimedes did not experiment with heavy objects, but proved the law of lever in Euclid's way, starting from the postulate that "equal weight is in balance at equal distance from the fulcrum".

Newton called the famous three laws "axioms or laws of motion". Based on the three laws and the law of universal gravitation, his mechanical system was established. The mathematical principle of his natural philosophy has Euclidean structure.

We can find another example in 1789 Malthus' theory of population. Malthus accepted Euclid's deductive model. He took the following two assumptions as the starting point of his demography: people need food; People need to reproduce. Then he established his mathematical model from the analysis of population growth and grain supply and demand growth. This model is concise and convincing, which has a great influence on the population policies of various countries.

Surprisingly, Euclid's model has also been extended to political science. The American Declaration of Independence is a famous example. The declaration of independence was written to prove the complete rationality of resisting the British Empire. Jefferson, the third president of the United States (1743- 1826) was the main drafter of this declaration. He tried to convince people of the fairness and rationality of the declaration with Euclid's model. "We hold these truths to be self-evident ..." Not only all right angles are equal, but also "all men are born equal". These self-evident truths include that if any government does not abide by these preconditions, then "the people have the right to replace or abolish it." At the beginning of the main part of the declaration, the government of King George of England did not meet the above conditions. "Therefore, ... we declare that these United colonies are and should be free and independent countries according to their legal rights. By the way, Jefferson loved literature, mathematics, natural science and architectural art.

The birth of the theory of relativity is another shining example. There are only two (1) relativity principles in the axiom of relativity, and any natural law has the same form for all linear motion observation systems. (2) The principle that the speed of light is constant. For all inertial systems, light travels at a certain speed in a vacuum. Einstein established his theory of relativity on the basis of these two axioms.

Regarding the establishment of theoretical system, Einstein thought that the work of scientists can be divided into two steps. The first step is to discover axioms, and the second step is to deduce conclusions from axioms. Which step is more difficult? He believes that if the researcher has received good training in basic theory, reasoning and mathematics at school, he will succeed in the second step as long as he is "quite diligent and smart". As for the first step, that is, finding out the needed axiom, it is completely different in nature, and there is no universal method here. Einstein said: "Scientists must grasp some universal characteristics that can be expressed by precise formulas in complicated empirical facts in order to explore the universal principles of nature."

5. Distribution of votes. The distribution of votes belongs to the category of democratic politics. Whether the distribution of votes is reasonable is a hot issue that voters are most concerned about. This problem has long attracted the attention of western politicians and mathematicians and has been widely studied. So, what are the basic principles of vote distribution? The first is fairness and reasonableness. A simple way to be fair and reasonable is to distribute votes in proportion to the number of people. But there will be a problem: the proportion of people is often not an integer. What shall we do? A simple way is to round off. The result of rounding may be redundant or insufficient. Because of this shortcoming, alexander hamilton, the US Treasury Secretary under George Washington, put forward a method to solve the quota allocation in 1790, which was adopted by the US Congress in 1792.

Members of the United States Congress are allocated by state. Suppose the population of the United States is, and the population of each state is. Assume that the total number of members is. commemorate

It's called the share allocated by the I state. The specific operation of Hamilton method is as follows:

(1) Take the integer part of each state's share, and let the I-th state have a member of parliament first.

(2) Then, considering the fractional part of each part, the remaining positions are assigned to the corresponding states in the order from largest to smallest until the positions are assigned.

Hamilton's method seems reasonable, but there are still problems. Traditionally, if the population ratio of each state remains unchanged and the total number of members of parliament increases for some reason, then the number of members of parliament in each state will either remain unchanged or increase, at least it should not decrease. However, Hamilton's method can't meet this agreement. 1880, which is the case in Alabama. People call this contradiction produced by Hamilton's method Alabama Paradox. Hamilton's method violated the interests of Alabama. Later, after the census of 1890 and 1900, Maine and Colorado also strongly opposed Hamilton's method. Therefore, since 1880, the US Congress has been debating the fairness and rationality of Hamilton's method. Therefore, the Hamilton method must be improved to make it more reasonable. A new method was put forward soon, and the Alabama paradox was eliminated. But new methods lead to new problems, and new problems need to be eliminated. So newer methods, of course, more just and reasonable methods have emerged. Of course, people will ask, is there a solution once and for all?

This problem has been attracting many politicians and mathematicians to study since its birth. Especially in 1952, the mathematician Arrow proved an amazing theorem-Arrow impossibility theorem, that is, it is impossible to find a fair and reasonable election system. In other words, only more reasonable, not the most reasonable. It turns out that there is no "public" in the world. Arrow's impossibility theorem is a milestone in the application of mathematics to social science.

Arrow's impossibility theorem is not only a mathematical achievement, but also a very important economic achievement. So as a mathematician, he won the Nobel Prize in Economics of 1972. There are two main factors that attract economists to pay attention to election issues: strategy and fairness. The study of strategy led to game theory.

6. Galileo's plan. A step forward in history is often accompanied by a step back. After thousands of years of silence, Europe ushered in the great Renaissance. This is an era when giants are needed and also produced. Galileo was born in 1564. Coincidentally, Shakespeare was born in the same year. The Renaissance brought the rational spirit of Greece to people. Galileo was the first scientist to raise the banner of reason. His work has become a new starting point of modern science.

What is the secret of the success of modern science? Scientific activities have chosen a new goal. This goal was put forward by Galileo. Greek scientists are committed to explaining the reasons for this phenomenon. For example, Aristotle spent a lot of time explaining why objects in the air fell to the ground. Galileo was the first to realize that the mystery of cause and effect of things could not improve scientific knowledge, nor could it help people find ways to reveal and control nature. Galileo put forward a scientific plan. The plan contains three main contents:

First, find out the quantitative description of physical phenomena, that is, the mathematical formula that links them;

Second, find out the most basic physical quantity, that is, the variables in the formula;

Third, deductive science is based on this.

The core of planning is to seek mathematical formulas to describe natural phenomena. Under the guidance of this idea, Galileo found the formula of free fall and the first and second laws of mechanics. All these and other achievements are summarized in Galileo's book "Two New Scientific Methods and Mathematical Proof", which took him more than 30 years of hard work. In this book, Galileo initiated the mathematical process of physical science, established mechanical science, and designed and established modern scientific thinking mode. .

Now that the direction has been pointed out and the waterway has been opened, science will show a trend of accelerated development. However, we must have new mathematical tools to move forward.

7. Analytic geometry. The birth of analytic geometry is another great milestone in the history of mathematics. His founders were Descartes and Fermat. They all expressed their dissatisfaction with the limitations of Euclidean geometry: ancient geometry was too abstract and relied too much on graphics. They also criticized algebra, because algebra is too rigid in laws and formulas and has become an art that hinders thinking, rather than an art that is conducive to developing thinking. At the same time, they all realize that geometry provides knowledge and truth about the real world, and algebra can be used to reason abstract unknowns, which is a potential method science. So we can combine all the essence of algebra and geometry and learn from each other. Thus, a new science was born. Descartes' theory is based on two concepts: the concept of coordinates and the concept of treating any binary algebraic equation as a curve on the plane by coordinate method. Therefore, analytic geometry is a mathematical subject that studies geometric objects by coordinate method and algebraic method.

What is the significance of analytic geometry?

(1) There has been a major turning point in the research direction of mathematics: the mathematics dominated by geometry in ancient times has changed into the mathematics dominated by algebra and analysis.

(2) Mathematics dominated by constants changed into mathematics dominated by variables, which laid the foundation for the birth of calculus.

(3) The integration of algebra and geometry realizes the digitalization of geometric figures, which is the forerunner of the digital age.

(4) Algebraization of algebra and algebra of geometry make people get rid of the shackles of reality. It brings the need to understand the new space. Help people enter virtual space from real space: from three-dimensional space to higher-dimensional space.

Algebraic language plays an unexpected role in analytic geometry because it does not need to be considered geometrically. Consider this equation

As we know, it is a circle. Where does the perfect shape, symmetry and endless end of a circle exist? In the equation! For example, it has symmetry and so on. Algebra replaced geometry, thought replaced eyes! In the properties of this algebraic equation, we can find out all the properties of the circle in geometry. This fact enables mathematicians to explore deeper concepts through algebraic representation of geometric figures. That is four-dimensional geometry. Why can't we consider the following equation?

What about shape?

What about the equation? This is a great progress. Only by analogy can we move from three-dimensional space to high-dimensional space, from tangible to intangible, and from real world to virtual world. What a wonderful thing it is! This process can be accurately described by the poem of Cheng Hao, a famous philosopher in the Song Dynasty:

Tao passes through the material world and thinking enters the abnormal situation.

(5) The birthplace of algebraic geometry. You can study higher-order curves.

8. Calculus Before the birth of calculus, human beings were basically in the period of agricultural civilization. The birth of analytic geometry is a prelude to the arrival of a new era, but it is not the beginning of a new era. It summarizes the old mathematics, integrates algebra and geometry, and introduces the concept of variables. Variable is a brand-new concept, which provides a basis for the study of sports. Engels said: "The turning point in mathematics is Descartes' variable. With variables, motion enters mathematics, with variables, dialectics enters mathematics, with variables, differentiation and integration become necessary immediately. "

To derive a large number of laws of the universe, we must wait for the arrival of such an era, make ideological preparations in this respect, and produce a group of leaders like Newton, Leibniz and Laplace who can create the future, provide methods and point out the direction for scientific activities. But we must also wait for the birth of an indispensable tool-calculus, without which it is impossible to deduce the laws of the universe. In17th century, this field is the richest among all the treasures of knowledge developed by geniuses. Calculus provides a source for the establishment of many new disciplines.

Calculus is the great crystallization of human wisdom. It gives a complete set of scientific methods and opens a new era of science, thus strengthening and deepening the role of mathematics. Engels said: "Among all the theoretical achievements, nothing can be regarded as the highest victory of human spirit like the discovery of calculus in the second half of17th century. If we see the pure and unique achievements of the human spirit somewhere, it is here. " With calculus, human beings have the ability to grasp movement and process. With calculus, there will be an industrial revolution, with large-scale industrial production, there will be a modern society. Space shuttle, spacecraft and other modern means of transportation are the direct result of calculus. Mathematics suddenly came to the front desk. The role of mathematics in the second wave of human society is much more obvious than that in the first wave.

1642 65438+1On October 8th, Galileo died silently under the persecution of religion. On February 25th of the same year, 12, a weak premature baby without a father was born. He is Newton. Newton took over Galileo's career and moved on. When Galileo described nature in mathematical language, he always confined his activities to the surface of the earth or near it. His contemporaries Kepler got three mathematical laws about the motion of celestial bodies. However, these two branches of science seem to be independent. Finding the connection between them is a challenge for the greatest scientists at that time. With the help of calculus, the law of universal gravitation was discovered. Newton used the same formula to describe the action of the sun on the planet and the action of the earth on its nearby objects. In other words, these laws established by Galileo and Newton describe the motion behavior from the smallest dust to the farthest celestial body. There is not a corner of the universe that is not included in these laws. This is an unprecedented leap in the history of human cognition, which not only has great scientific significance, but also has far-reaching social influence. It strongly proves the mathematical design of the universe and destroys the mysticism, superstition and theology shrouded in celestial bodies.

Under the guidance of Galileo's planning, the success of seeking natural laws with the help of calculus goes far beyond the field of astronomy. People study sound as the movement of air molecules and draw famous mathematical laws. Hooke studied the vibration of objects. Boyle, Edm Edme Mariotte, Galileo Torricelli and Pascal measured the pressures and densities of liquids and gases. Van Helmont took an important step in modern chemistry by measuring matter with a balance. Hales began to study physiology with quantitative methods. Harvey quantitatively proved that blood flowing from the heart will flow around the whole body before returning to the heart. Quantitative research also extends to botany. All these are just the beginning of an unprecedented scientific movement sweeping the modern world.

By the middle of18th century, the infinite superiority of Galileo and Newton's quantitative methods in studying nature had been completely established. Kant, a famous philosopher, said that the development of natural science depends on the combination of its methods, contents and mathematics, and mathematics has become the golden key to open the door of knowledge and the queen of science.

The amazing achievements of the alliance between mathematics and natural science make people realize that:

(1) Rational spirit is the highest source of truth;

(2) Mathematical reasoning is the purest, deepest and most effective means of all thinking;

(3) Every field should explore the corresponding laws of nature and mathematics. Especially the concepts and conclusions in philosophy, religion, political economy, ethics and aesthetics, should be redefined, otherwise it will not conform to the laws of that field.

9. Math and painting. Throughout the history of painting, the system of painting can be roughly divided into two categories: conceptual system and optical system. Conceptual system is drawn according to certain concepts or principles. For example, most paintings and relief works in Egypt follow the conceptual system. The size of characters is not based on the principle of realism, but on their political status or religious status. Pharaoh is often the most important person, he is the biggest, his wife is younger than him, and his servants are pitiful. Optical perspective system tries to represent the image of the person himself in the eyes. It developed from western painting art, and the optical system had been developed as early as Greece and Rome. But in the Middle Ages, the influence of Christian mysticism brought artists back to the conceptual system. The background and themes painted by painters tend to show religious themes, aiming at guiding religious feelings, rather than showing real people and things in the real world. From the end of the Middle Ages to the Renaissance, the painting art has undergone a qualitative change. Its typical feature is that artists move towards realism. /kloc-at the end of 0/3, mathematics also entered the field of fine arts.

By the13rd century, Aristotle's works were widely known by translating Arabic and Greek works. Western painters began to realize that medieval painting was divorced from reality and life, and this tendency should be corrected. In fact, from the Middle Ages to the Renaissance, the first is the awakening of human nature. In the Middle Ages, art was only for "disciplining people" to become good believers. During the Renaissance, art was more about "enriching people" and "pleasing people".