The word "ancient Greece" is familiar to us, but many people don't understand it.
If Euclid, the author of Elements of Geometry, can represent the whole ancient Greek people, then I can say that ancient Greece is the most brilliant branch of ancient culture-because mathematics in ancient Greece not only contains mathematics, but also contains rare logic and intriguing philosophy.
The mathematical work Elements of Geometry is connected with several obvious and well-known definitions, postulates and axioms, and develops a series of propositions: from simple to complex, they complement each other. We have to admire its rigorous logic.
Judging from the proposition I have visited so far, Euclid proved that the most common and basic problem about the "equal length" of line segments is to draw a circle: because all radii of a circle are equal. General mathematical thinking is very complicated. Just a little bit here, and then I went there again. The Elements of Geometry is easily accepted by me, probably because Euclid repeatedly used an idea to make readers accept it.
But what I want to emphasize is his philosophy.
There are several propositions in the book: for example, "the two base angles of an isosceles triangle are equal, and the two complementary angles formed by the waist and the base are equal", and for example, "if the two angles in a triangle are equal, then the two sides are equal". When I read these propositions, I have been suffering from shocks outside geometry.
We studied geometry in the seventh grade. When we thought of doing this kind of proof and needed to prove that the two angles in a triangle were equal, we always wrote: "Because it is an isosceles triangle, the two base angles are equal"-we always habitually think that the two base angles of an isosceles triangle are equal; While reading "The Elements of Geometry", he thought about "why the two base angles of an isosceles triangle are equal". Think about it, a thought is habit, and a thought is thinking about why. Isn't this enough to explain the problems of modern people?
Most modern people seem to have lost their curiosity. Curiosity here refers not only to the kind of interest in novelty, but also to ordinary things. For example, many people will ask "why do astronauts float in the air", but they may not ask "why can we stand on the ground and not float"; Many people will ask "What can you eat to lose weight", but they may not ask "Why do sheep eat grass instead of meat".
We are so used to the things around us that we are not interested in many "ordinary" things and then think about them. Why did Newton discover gravity? A large part of the reason lies in his curiosity.
If we just read the Elements of Geometry as a math book, it would be a big mistake, because ancient Greek mathematics is permeated with philosophy, and learning mathematics is learning philosophy.
The first lesson of philosophy: people should be curious, not only to explore new things, but also to explore ordinary things around them. This is my windfall from reading the Elements of Geometry!
Reflections on geometric elements II. Today, I read a book called Elements of Geometry. It is the immortal work of Euclid, an ancient Greek mathematician and philosopher, which combines the achievements and spirit of Greek mathematicians in one book.
The Elements of Geometry contains all the contents of the original volume 13, including 5 axioms, 5 postulates, 23 definitions and 467 propositions, that is, first put forward axioms, postulates and definitions, and then prove them from simple to complex, and on this basis form Euclidean geometry system. Euclid believes that mathematics is a noble world, even as a secular monarch, there is no privilege here. Compared with the decaying matter in time, the world revealed by mathematics is eternal. The Elements of Geometry is not only a mathematical work, but also full of philosophical spirit, which completes the human understanding of space for the first time. Ancient Greek mathematics was born out of philosophy. It uses all possible descriptions to analyze our universe so that it is not chaotic and inseparable. It is completely different from the secular mathematics in China and ancient Egypt. It establishes a definite system of the material world and the spiritual world, so that people as small as human beings can gain a little confidence from it.
The proposition 1 in this book puts forward how to make equilateral triangles, from which the triangle congruence theorem is produced. That is, angles, sides, angles or sides, angles, sides or sides, sides and sides are equal, and an isosceles triangle-equilateral sides are equilateral; An equilateral is an equilateral. In this way, Euclid put forward his own geometric theory from four parts: point, line, surface and angle. The former proposition paves the way for the latter proposition; The latter proposition is derived from the former, interlocking and very rigorous.
This book is so profound that I can only understand about one tenth, which is very shocking. Euclid deserves to be the father of geometry! He is the most dazzling star in the history of mathematics. I want to learn from him and walk firmly along my own goals.
Axiomatic structure is the main feature of modern mathematics. The original version is the earliest model to complete the axiomatic structure, which was produced more than 2000 years ago and is very valuable. However, by modern standards, there are also many shortcomings. First of all, an axiomatic system has some original concepts, or undefined concepts, as the basis for the definition of other concepts. Points, lines and surfaces all belong to this category. In the Elements, definitions are given one by one, and these definitions themselves are ambiguous. Secondly, the axiom system is incomplete, and there are no axioms such as movement, sequence and continuity, so many proofs have to rely on intuition. In addition, some axioms are not independent, that is, they can be deduced from other axioms. These defects were not remedied until 1899 when Hilbert's Fundamentals of Geometry was published. Nevertheless, after all, shortcomings can't cover up the remaining flaws. The original work created the correct way of axiomatization of mathematics, and its influence on the whole development of mathematics exceeded that of any book in history.
The two theoretical pillars of the original work-Proportional Theory and method of exhaustion. In order to discuss the theory of similarity, Euclid sorted out the theory of proportion and quoted eudoxus's theory of proportion. This theory is very successful. It avoids irrational numbers and establishes a correct proportional theory of commensurability and incommensurability, thus successfully establishing a similarity theory. In the history of geometric development, solving the problems such as the area surrounded by curved edges and the volume surrounded by curved surfaces has always been an important topic of concern. This is also the initial problem involved in calculus. Its solution depends on the limit theory, which has been in17th century. In ancient Greece, there was no obvious limit process in proving some important area and volume problems in the third and fourth centuries BC. Their ideas and methods to solve these problems were so advanced that they deeply influenced the development of mathematics.
The problem of turning a circle into a square was put forward by Auddock Sotheby's, an ancient Greek mathematician, and was later named after the exhaustive method. Exhaustion is based on Archimedes axiom and reduction to absurdity. In the Elements of Geometry, Euclid proved many propositions by exhaustive method, such as the ratio of the area of a circle to the square of its diameter. The volume ratio of two spheres is equal to the cubic ratio of their diameters. Archimedes was more skillful in using exhaustive method. It is used to solve some important propositions of area and volume. Of course, to prove a proposition by exhaustive method, we must first know the conclusion of the proposition, and the conclusion is often determined by speculation and judgment. Archimedes did important work here. In his article Method, he expounded the general method of finding a conclusion, which actually included the idea of integral. His contribution to mathematics established his prominent position in the history of mathematics.
Research and summary of cartographic problems. Euclid talked about the drawing methods of regular triangles, squares, regular pentagons, regular hexagons and regular pentagons in the Elements of Geometry, but did not mention other regular polygons. It can be seen that he has tried to make other regular polygons, and he has also encountered the situation that he can't do it. But at that time, it was still impossible to judge the real "impossible" or to find a drawing method for the time being.
Gauss is not satisfied with the drawing of a single regular polygon. He hopes to find a standard to judge which regular polygons can be made with rulers and compasses and which regular polygons cannot be made. In other words, he has realized that the "efficiency" of rulers and compasses is not omnipotent, and some regular polygons may not be made, which does not mean that people can't find a drawing method. In 180 1, he found a new research result, which can judge whether a regular polygon can be made. To judge whether this problem can be done, let's first turn it into an algebraic equation. Then, judge by algebraic method. The criterion of judgment is: "The necessary and sufficient condition that a geometric quantity can be made with a ruler and compasses is that the number corresponding to a known quantity can be obtained by a finite number of addition, subtraction, multiplication, division and square operations." (Pi cannot be obtained in this way, it is a transcendental number, and both E and Louisville numbers are transcendental numbers. As we know, real numbers are uncountable and can be divided into rational numbers and irrational numbers. Among them, rational numbers and some irrational numbers, such as root number 2, are algebraic numbers, and algebraic numbers are countable, so the uncountability in real numbers is due to the existence of transcendental numbers. Although there are many transcendental numbers, it is not so simple to judge whether a number is beyond. ) At this point, the "three difficult problems", that is, "turning a circle into a square, bisecting an angle, and folding a cube in half", are drawing problems that cannot be done with a ruler. Regular heptagon is ok, but its method is not easy to give. At the age of 65,438+0,796 and 65,438+0.9, Gauss gave the ruler-gauge drawing method of regular heptagon and discussed it in detail. In recognition of his discovery, after his death, a regular heptagon was carved on a monument built in his hometown of Brunswick.
Introduction to the axiom of continuity in geometry. The existence of "intersection point" in drawing problems cannot be deduced from Euclid's postulate and axiom. Because there is no concept of continuity (axiom). This requires adding a new axiom-continuity axiom to Euclid's axiom system. Although Fermat and Descartes discovered analytic geometry before19th century, algebra has made great progress, calculus has entered the university classroom, and topology and projective geometry have appeared. However, the theoretical basis of mathematicians' logarithmic system is still vague and has not been paid attention to. Intuitively, it is recognized that real numbers and points on straight lines are continuous and correspond to each other one by one. It was not until the end of 19 that this important issue was satisfactorily solved. Scholars engaged in this work include Cantor, Dai Dejin, Peano, Hilbert and others. At that time, Cantor hoped to establish the theory of real numbers with basic sequence, and Dai Dejin also studied the concept of irrational numbers deeply. One of his papers was published in 1872. Before 1858, when he was teaching calculus to students, he knew that the real number system had no guarantee of logical basis. Therefore, when he wants to prove that "monotonically increasing bounded variable sequence tends to a limit", he has to turn to geometric intuition. In fact, "all points on a straight line are continuum" has no logical basis. It is not clear whether there is a one-to-one correspondence between all real numbers and all points on a straight line. For example, the mathematician Polka regards the existence of at least one number between two numbers as the continuity of numbers. Actually, this is a misunderstanding. Because any two rational numbers can definitely find a rational number. However, rational numbers are not all numbers. After Dydykin's division, people realized that Polkanu's viewpoint was only the density of numbers, not the continuity. The mathematical crisis caused by irrational numbers continued until19th century. Until 1872, German mathematician Dai Dejin defined irrational numbers by dividing rational numbers, and established the theory of real numbers on a strict scientific basis, which ended the era when irrational numbers were considered "unreasonable" and the first great crisis in the history of mathematics that lasted for more than 2,000 years.
The elements also studied many other problems, such as finding the greatest common factor of two numbers (which can be extended to any finite number) and the infinite number of prime numbers in number theory.
In advanced mathematics, there is the concept of orthogonality, and the earliest concept origin should be Pythagorean theorem, which we call Pythagorean theorem, except that the hook of 3 strands, 4 chords and 5 is a special case, and Pythagorean theorem holds for any right triangle. And from Pythagorean theorem, the root number 2 of irrational number is found. Deduction is involved in mathematical methods, and reduction to absurdity (that is, reduction to absurdity) is used to prove propositions. Perhaps inspired by Diophantine's dividing a square number into integer solutions of two squares, the famous Fermat's Last Theorem was put forward by French mathematicians more than 350 years ago, which attracted mathematicians of all ages to make great efforts to prove it and effectively promoted the application of number theory to the whole process of mathematics. 1994, the British mathematician Andrew Hueros solved this epic problem.
Over the years, thousands of Qian Qian people (Newton, Archimedes, etc. They are all famous) They received training in logic by studying Euclidean geometry, thus stepping into the hall of science.
Reflections on geometric elements iv. The Elements of Geometry is regarded as mathematics. Bible, the first systematic mathematical work, Newton and Einstein wrote the mathematical principles and relativity of natural philosophy in this form. Spinoza wrote the philosophical work Ethics, which can be used as an interface between philosophy, social science and psychology, and is highly speculative.
Geometry originally has a total of 13 volumes, so it is enough to study the first six volumes, because the latter are all applied to specific fields, irrational numbers, solid geometry and so on. I think the essence of geometry is reasonable assumption and abstraction of points, lines and surfaces, so that the latter theorem can be established. The fifth postulate was later overturned, based on points, lines and surfaces, with Euclid tools as tools. Mainly the simplest geometric shape, starting with how to draw it, is also well-founded, and then the properties of various shapes and the theorem of the relationship between various shapes are deduced step by step.
In geometry, Apollonius's theory of conic secant and Newton's mathematical principle of natural philosophy are both systematic mathematical works, both of which are proved by Euclid tools. Later, the appearance of calculus tools, I think, is the process of solving pi and the idea of infinite approximation, which makes calculus tools come into being. Modern mathematics seems to have a luxurious lineup, but there are no new tools. It is only the application of calculus tools of various shapes, and mathematics mainly makes a fuss about space. There seems to be a lot of work that can be done in mathematics now, but it also benefits from the development of physics. On the one hand, mathematics has developed to the general aspect and has been forgotten. It's nothing to think about mathematical thinking, but it's a lot of mental work, especially for those who just do pure mathematical research and don't think. They are very tired and can't do meaningful work.
After reading the history of mathematics in the twentieth century, I found people's works in it. I don't want to see any of them. It's so empty.
After reading Elements of Geometry, algebra developed rapidly in Europe after the Renaissance due to the influence of Arabia. On the other hand, after17th century, the development of mathematical analysis is very remarkable. Therefore, geometry has also got rid of the state of isolation from algebra. As he said in his famous work Geometry, numbers are closely related to figures, coordinates are set in space, and figures are expressed by the relationship between numbers; Conversely, a graph can be expressed as the relationship between numbers. In this way, according to the coordinates, the figure becomes a problem of the relationship between numbers. This method is called analytic geometry. Engels spoke highly of Descartes' work in Dialectics of Nature. He pointed out: "The turning point of mathematics is Descartes' variable. With variables, movement enters mathematics. With variables, dialectics enters mathematics. With variables, differentiation and integration become necessary. "
In fact, Descartes' thought provided a strong foundation for the development of mathematical analysis in17th century. 18th century, due to the pioneering work of L. Euler and others, analytic geometry developed rapidly, and even the conic curve theory discussed by Apollonius and others in the Greek era (about 262 BC ~ about 190 BC) was once again regarded as conic curve theory and was algebraically sorted out. In addition, the mathematical analysis developed in the18th century is applied to geometry in turn. At the end of this century, gaspard monge initiated the application of mathematical analysis in geometry and became the pioneer of differential geometry. As mentioned above, many geometric problems can be discussed by analytic geometry. But it can't be said that this is the most applicable to all problems. Contrary to analytic geometry method, there are synthetic geometry or pure geometry method, which is a method of directly examining graphics without coordinates, such as mathematician Euclid geometry. Projective geometry is the product of this way of thinking.
As early as the Renaissance, plastic arts prevailed and developed in Italy, accompanied by the study of so-called perspective. At that time, many people, including Leonardo da Vinci, studied this perspective as practical geometry. Since17th century, G. Dezag and B. Pascal have extended and developed this perspective method, thus laying the foundation for projective geometry. Two theorems named after them became the basis of projective geometry. One is Dezag's theorem: if the connecting lines of the corresponding vertices of two triangles on a plane intersect at a point, the intersection points of their corresponding edges are on a straight line; or vice versa, Dallas to the auditorium The second is Pascal's theorem: if the vertices of a hexagon are on the same quadratic curve, then the intersections of its three pairs of opposite sides are on the same straight line; or vice versa, Dallas to the auditorium 18th century later, J.-V. Pansley, Z.N.M Gianno and J. Steiner completed this geometry.
Reflections on geometric elements 6. The oldest branch of mathematics. It is said that it originated from the survey method of land reclamation after the Nile River flooded in ancient Egypt, and its mbth geo consists of geo (land) and metry (measurement). Thales once used the equivalence of two triangles to do indirect measurement; Pythagoras school is famous for Pythagoras theorem. Pythagoras measurement existed in ancient China. The first chapter of the Classic of Parallel Calculations of Zhou Dynasty written by Han people records the questions and answers of Duke Zhou, Ji Dan and Shang Gao when the Western Zhou Dynasty was founded (about 65438 BC+0000 BC), discusses the method of measuring moments, obtains the famous Pythagorean law, and lists the examples of "Gou San, Gu Si and Xian Wu". Geometry produced in Egypt spread to Greece, and then gradually developed into pure mathematics. The philosopher Plato (429 ~ 348 BC) made a profound discussion on geometry, and established the concepts of definition, postulate, axiom and theorem in geometry today, as well as the concepts of analysis and synthesis in philosophy and mathematics. In addition, Menek Muse (about 340 BC) already had the concept of conic curve.
Greek culture reached its peak in the era of Plato School, and then gradually declined, while Alexandria School in Egypt gradually prospered and became the center of culture for a long time. Mathematician Euclid compiled the mathematical knowledge obtained until the Greek era into a thirteen-volume "Elements of Geometry", which is the mathematician Euclid Geometry (referred to as Euclid Geometry) which is still widely used as a geometry textbook today. Xu Guangqi translated the first six volumes of Elements of Geometry in 1606, and it was not until 1847 that Li finished translating the remaining seven volumes. "Geometry" is not so much a transliteration of geo as a more appropriate explanation of "size". It is true that modern geometry is a branch of mathematics about graphics, but in Greek times, it represented the whole mathematics. Mathematician Euclid first described some definitions in Elements of Geometry, and then put forward five postulates and five axioms. Among them, the fifth postulate is particularly famous: if two straight lines intersect with the third straight line, and the sum of two internal angles on the same side is less than two right angles, then the two straight lines will intersect when they extend to this side properly. Although the axiomatic system in the Elements of Geometry is not so complete, it has just become the pioneer of the basic theory of modern geometry. It was not until the end of 19 that D. Hilbert established a strict system of Euclidean geometry axioms.
Compared with other postulates, the content of the fifth postulate is more complicated, which later attracted people's attention, but the attempt to deduce it with other postulates failed. This postulate is equivalent to the following postulate: on the plane, points other than a straight line can lead to one and only one straight line does not intersect this straight line. η и Lobachevsky and J. Bolyai independently created a new geometry, in which the fifth postulate was abandoned and replaced by another postulate: on the plane, a point other than a straight line can lead to infinite straight lines that do not intersect with this straight line. The non-contradictory geometry created in this way is called hyperbolic non-mathematician Euclidean geometry. (G.F.) B. Riemann changed the fifth postulate to "On a plane, any straight line drawn by a point outside a straight line must intersect this straight line", and the non-contradictory geometry created in this way is called non-mathematician Euclidean geometry of an ellipse.