brief introduction

In a sense, mathematics in the modern sense (that is, pure mathematics as a deductive system) comes from Pythagoras school in ancient Greece. This school flourished around 500 BC and is an idealistic school. They attach importance to the study of the invariable factors in nature and society, and call geometry, arithmetic, astronomy and music "four arts", in which they pursue the harmony and law of the universe. They believe that "everything is important" and that mathematical knowledge is reliable and accurate and can be applied to the real world. Knowledge of mathematics is acquired through pure thinking, without observation, intuition and daily experience. Pythagoras' numbers refer to integers, and one of their great discoveries in mathematics is to prove Pythagoras' theorem. They know the general formula to satisfy the three-sided length of right-angled triangles, but they also find that the three-sided ratio of some right-angled triangles cannot be expressed by integers, that is, the hook length or the node length is not commensurable with the chord length. This negates the Pythagorean creed that all phenomena in the universe can be attributed to integers or the ratio of integers.

give rise to

The discovery of incommensurability triggered the first mathematical crisis. Some people say that this property was discovered by hippasus around 400 BC, so his companions threw him into the sea. But it is more likely that Pythagoras knew the fact that hippasus was executed for leaking information. In any case, this discovery had a great influence on the concept of mathematics in ancient Greece. This shows that some truths of geometry have nothing to do with arithmetic. Geometric quantities cannot be completely expressed by integers and their ratios, but numbers can be expressed by geometric quantities. The lofty position of integers was challenged, so geometry began to occupy a special position in Greek mathematics. At the same time, it also reflects that intuition and experience are not necessarily reliable, but reasoning proves to be reliable. From then on, the Greeks set out from the axiom of "self-evident" and established the geometric system through deductive reasoning, which was a great revolution in mathematical thought and a natural product of the first mathematical crisis. Looking back at the previous mathematics, it is nothing more than "calculation", that is, providing algorithms. Even in ancient Greece, mathematics was applied to practical problems from reality. For example, Thales predicted the solar eclipse, calculated the pyramid height by using the shadow distance, and measured the offshore distance of ships. All belong to the category of computing technology. As for Egypt, Babylon, China, Indian and other countries, mathematics has never experienced such a crisis and revolution, so it still stays in the stage of "mathematization". Greek mathematics, on the other hand, took a completely different road, forming the axiomatic system of Euclid's Elements of Geometry and Aristotle's logical system.

The product of the first crisis

Classical logic and Euclid geometry Aristotle's methodology have great influence on mathematical methods, and he pointed out the correct definition principle. Aristotle inherited the thought of his teacher Plato and distinguished definition from existence. Something defined by some attributes does not necessarily exist (such as an octahedron). In addition, the definition must be defined by what has been defined, so there must be some original definitions, such as points and lines. The method of proving existence needs to be defined and restricted. Aristotle also pointed out the necessity of axiom, because it is the starting point of deductive reasoning. He distinguishes between axiom and postulate, and thinks that axiom is the common truth of all sciences, and postulate is only the most basic principle unique to a certain discipline. He also listed the laws of logic (law of contradiction, law of excluded middle, etc. ) is an axiom. Aristotle deeply studied the process of logical reasoning, obtained syllogism and expressed it as an axiomatic system, which was the earliest axiomatic system. His research on logic not only makes logic an independent subject, but also has a good influence on the development of mathematical proof. Aristotle expounded the contradiction between discrete and continuous. Distinguish between potential infinity and real infinity. He thinks that positive integers are potentially infinite, because any integer plus 1 can always get a new number. But he believes that the so-called "infinite set" does not exist. He believes that space is potentially infinite and time is potentially infinite in extension and subdivision. Needless to say, the role of Euclid's Elements in the development of mathematics. However, it should be pointed out that Euclid's contribution is that he summed up the previous Greek mathematical knowledge for the first time in history and formed a standardized deductive system. This influence on mathematics, philosophy and natural science continued until the19th century. The mathematical principles of Newton's natural philosophy and Spinoza's ethics all adopt the style of Euclid's Elements of Geometry. Euclid's plane geometry is the first four and sixth in the Elements of Geometry. There are seven original definitions, five axioms and five postulates. He stipulated that the proof of existence depends on structure. The Elements of Geometry has become the most widely circulated book in the western world after the Bible. It has always been a standard work of geometry. However, it still has many shortcomings, which are constantly criticized by people, such as the definition of point, line and surface is not strict: "Point is an object without parts", "Line is a length without width (line refers to a curve)" and "Surface is an object with only length and width". Obviously, these definitions can't play the role of logical reasoning. The definition of straight line and plane is particularly intuitive ("a straight line is a straight line aligned with all points in it"). In addition, his axiom 5 is that "the whole is greater than the parts" and does not involve the problem of infinite quantity. In his proof, the original axiom is not enough, and new axioms must be added. In particular, whether the parallel postulate can be deduced from other axioms and postulates is an interesting issue. Nevertheless, the systematic characteristics of modern mathematics have basically taken shape.

Edit the birth of non-Euclidean geometry.

Euclid's Elements of Geometry is the product of the first mathematical crisis. Despite its shortcomings and mistakes, it has been a recognized model for more than 2,000 years. Many philosophers, in particular, put Euclidean geometry in the position of absolute geometry. In the18th century, most people thought Euclidean geometry was the correct idealization of graphic properties in material space. In particular, Kant believes that the principle of space is a priori comprehensive judgment, the material world must be Euclidean, and Euclidean geometry is unique, inevitable and perfect. Since it is perfect, everyone wants axioms and postulates to be simple and straightforward. Other axioms and postulates satisfy the above conditions, but the parallel postulate is not concise enough, like a theorem. Euclid's parallel postulate means that whenever a straight line intersects with two other straight lines, the sum of two internal angles on the same side is less than two right angles, and the other two straight lines intersect with the side with internal angles on the same side less than two right angles. In the Elements of Geometry, it is proved that the first 28 propositions do not use this postulate, which naturally causes people to consider whether this verbose postulate can be derived from other axioms and postulates, that is, the parallel postulate may be redundant. More than two thousand years later, many people tried to prove it. Some people thought it was successful at first, but after careful investigation, they found that all the proofs used some other assumptions, which could be deduced from the parallel postulate, so they just got some propositions equivalent to the parallel postulate. /kloc-in the 0 th and 8 th centuries, some people began to try to prove by reduction to absurdity, that is, to assume that the parallel postulate is not established in an attempt to draw contradictions from it. They drew some inferences, such as "there are two straight lines intersecting at infinity, and the intersection points have a common vertical line" and so on. In their view, these conclusions are unreasonable, so they cannot be true. However, the significance of these inferences is not clear, and it is hard to say that it leads to contradictions, so it cannot be said that the parallel postulate is proved. From the old Euclidean geometry concept to the establishment of new geometry concept, we need to emancipate our minds to some extent. First of all, from the failure process of proving parallel postulate in 2000, it should be seen that this proof is impossible and this impossibility can be verified; Secondly, we should choose other postulates that are contradictory to parallel postulates, and we can also establish geometries that are not logically contradictory. This is mainly Lobachevsky's pioneering work. We should realize that Euclidean geometry is not necessarily the geometry of material space, but only one of many possible geometries. Geometry should change from a space science tested by intuition and experience to a pure mathematics, that is, its existence is only determined by non-contradiction. Although people like Lambert also have these ideas, it is Hilbert who really turns geometry into such a pure mathematics. This process is long, and the most important step is that Lobachevsky and Boyer independently established non-Euclidean geometry, especially their non-contradiction is original in history. Later generations implicitly changed the non-contradiction of Roche geometry into that of Euclid geometry. This idea of proving "relative non-contradiction" with "model" runs through the future basic research of mathematics. Moreover, this reduction of non-Euclidean geometry to Euclidean geometry, which everyone has always believed, also played an important role in the process of accepting non-Euclidean geometry. It should be pointed out that the acceptance of non-Euclidean geometry in mathematics has experienced several difficult struggles. First of all, we must prove that denying the fifth postulate will not lead to contradictions. Only in this way can we say that the new geometry holds, and the fifth postulate is independent of other axiomatic postulates, which is the minimum requirement. The method of proof at that time was to prove "relative non-contradiction". Because at that time, everyone admitted that Euclidean geometry had no contradiction, and if non-Euclidean geometry could be explained by Euclidean geometry, there would be no contradiction. This requires translating points, lines, surfaces, angles and parallelism in non-Euclidean geometry into corresponding things in Euclidean geometry, and axioms and theorems can also be explained by corresponding axioms and theorems in Euclidean geometry, which is the so-called Euclidean model of non-Euclidean geometry. For Luo Barczewski geometry, the most famous Euclidean model is the negative curvature constant surface model proposed by Italian mathematician Bethlem in 1869; The projection plane model proposed by German mathematician Klein in 187 1 and the unit circle internal model explained by automorphism function proposed by Poincare in 1882. These models do confirm the relative non-contradiction of non-Euclidean geometry, and some of them can be extended to more general non-Euclidean geometry, that is, elliptic geometry founded by Riemann, and also to high-dimensional space. Therefore, from the late 1960s to the early 1980s, most mathematicians accepted non-Euclidean geometry. Although some people still insist on the uniqueness of Euclidean geometry, many people clearly point out that the era of equality between non-Euclidean geometry and Euclidean geometry has arrived. Of course, there are a few diehards, such as Frege, the founder of mathematical logic, who refuse to admit non-Euclidean geometry until their death, but this has nothing to do with the overall situation. The establishment of non-Euclidean geometry has a great shock to mathematics. Mathematicians began to care about the basic problems of geometry. Since 1980s, axiomatization of geometry has become the focus of attention, from which the new axiomatization movement of Hilbert emerged.

The second mathematical crisis

brief introduction

As early as ancient times, people were interested in the measurement of length, area and volume. Eudoxus, an ancient Greek, introduced the concept of quantity to consider things that change continuously, and treated continuous quantity strictly according to geometry. This leads to the long-term separation of number and quantity. In ancient Greek mathematics, except for integers, there was no concept of irrational numbers, or even the operation of rational numbers, but there was a proportion of numbers. They are interested in the relationship between continuity and discreteness, especially the four famous paradoxes put forward by Zhi Nuo. The first paradox is that motion does not exist, because a moving object must reach half before reaching its destination, and must reach half before reaching half ... If this goes on, it must pass through an infinite number of points, which is impossible in a limited time. The second paradox is that Ashley runs fast, but she can't catch up with the tortoise in front. Because when the tortoise is in front of him, he must first reach the starting point of the tortoise, and then use the logic of the first paradox, the tortoise is in front of him. These two paradoxes oppose the view that space and time are infinitely separable. The third and fourth paradoxes oppose the fact that space and time are composed of inseparable intervals. The third paradox is "the arrow does not move", because in a certain time interval, the arrow is always in a certain position in a certain space interval, so it is static. The fourth paradox is the military parade, which is similar in content. This shows that the Greeks have seen the contradiction between infinitesimal and tiny. Of course, they can't solve these contradictions. Although the Greeks did not have a clear concept of limit, they had strict approximation steps when dealing with the problems of area and volume. This is the so-called "exhaustive method". It relies on indirect proof to prove many important and difficult theorems.

Emerging new problems

In the 16th and 17th centuries, in addition to finding the length of the curve and the area surrounded by the curve, many new problems appeared, such as finding the speed, tangent, maximum value and minimum value. After years of hard work, 17 century finally formed calculus, which is the beginning of mathematical analysis. Newton and Leibniz are recognized as the founders of calculus. Their achievements mainly lie in: 1, unifying the solutions of various problems into one method, differential method and integral method; 2, there are clear steps to calculate the differential method; 3. Differential method and integral method are reciprocal operations. Calculus has become an important tool to solve problems because of its completeness of operation and universality of application. At the same time, the problems about the basis of calculus are becoming more and more serious. Take speed as an example. The instantaneous velocity is the value of δ s/δ t when δ t tends to zero. Δ t is zero, is it a small quantity, or what? Is this infinitesimal quantity zero? This caused great controversy and led to the second mathematical crisis. /kloc-mathematicians in the 0 th and 8 th centuries successfully solved many practical problems with calculus, so some people are not interested in discussing these basic problems. D'Alembert, for example, said that it is now "to build the house higher, not to lay a more solid foundation". Many people think that the so-called strictness is cumbersome. But because of this, the basic problems of calculus have been criticized and attacked by some people, the most famous of which is the attack on Bishop Becquerel in 1734.

Establish a strict foundation

/kloc-the mathematical thought of the 0/8th century is really not rigorous and intuitive, and pays attention to formal calculation, regardless of whether the foundation is reliable or not. In particular, there is no clear concept of infinitesimal, so the concepts of derivative, differential and integral are not clear. The concept of infinity is also unclear; Arbitrariness of summation of divergent series; Inaccurate use of symbols; Differential does not consider continuity, the existence of derivatives and integrals, and whether it can be expanded into power series. It was not until11920s that some mathematicians began to pay more attention to the strict foundation of calculus. They started with the work of Porzano, Abel, Cauchy, Dirichlet and others, and were finally completed by Wilstras, Dydykin and Cantor. After more than half a century, they basically solved the contradiction and laid a strict foundation for mathematical analysis. Porzano denied the existence of infinite decimals and infinite numbers, and gave a correct definition of continuity. Cauchy started with the definition of variables in the algebra analysis course of 182 1, and realized that functions don't have to have analytic expressions. He mastered the concept of limit, pointed out that infinitesimal and infinitesimal are not fixed quantities but variables, and defined derivatives and integrals. Abel pointed out that it is necessary to strictly limit the abuse of series expansion and summation; Dirichlet gave a modern definition of function. On the basis of these mathematical works, Wilstrass eliminated the inaccuracy, gave the limit and continuous definition of ε-δ, and strictly established the concepts of derivative and integral on the basis of limit, thus overcoming the crisis and contradiction. In the early 1970s, Willerstrass, Dai Dejin, Cantor and others independently established the real number theory, and established the basic theorem of limit theory on the basis of the real number theory, so that the mathematical analysis was finally based on the strict basis of the real number theory. At the same time, Weierstrass gives an example of a continuous function that can be differentiated everywhere. This discovery and many examples of morbid functions later fully show that intuition and geometric thinking are unreliable and must resort to strict concepts and reasoning. Results The second mathematical crisis made mathematics explore the problem of real number theory, which was the basis of mathematical analysis. This not only led to the birth of set theory, but also reduced the non-contradictory problem of mathematical analysis to the non-contradictory problem of real number theory, which was the primary problem in the mathematical foundation of the twentieth century.

the third mathematical crisis

brief introduction

After the first and second mathematical crises, people attributed the non-contradiction of the basic theory of mathematics to the non-contradiction of set theory, which has become the logical basis of the whole modern mathematics, even though the grand building of mathematics has been built. It seems that set theory is not contradictory, and the goal of strict mathematics has almost been achieved, and mathematicians are almost complacent about this achievement. The famous French mathematician Poincare (1854- 19 12) boasted at the international congress of mathematicians held in Paris in 1900: "Now it can be said that it has reached the absolute rigor." However, less than two years later, the famous British mathematical logician and philosopher Russell (1872- 1970) announced an amazing news: set theory is self-contradictory and has no absolute rigor! History is called "Russell Paradox". 19 18, Russell extended this paradox to become barber paradox. The discovery of Russell's paradox is tantamount to breaking the fog in a sunny day and waking people up from their dreams. Russell paradox and other paradoxes in set theory go deep into the theoretical basis of set theory, thus fundamentally endangering the certainty and rigor of the whole mathematical system. So it caused an uproar in the fields of mathematics and logic, and formed the third crisis in the history of mathematics. The paradox of set theory lies in the contradiction between the dialectical nature of set and the formal characteristics of mathematical method or metaphysical thinking method. For example, the cause of Russell paradox lies in the contradiction between the arbitrariness of generalization principle and the non-arbitrariness of objective rules of generating set.

The product of the third mathematical crisis

The development of mathematical logic and the emergence of a number of modern mathematics. Mathematicians have made different efforts to solve the third mathematical crisis. Due to different starting points and different ways to solve problems, different schools of mathematical philosophy were formed at the beginning of this century, namely, the logicism school headed by Russell, the intuitionism school headed by Brouwer (188 1- 1966) and the formalism school headed by Hilbert. The formation and development of these three schools have pushed the research of basic mathematics theory to a new stage. The mathematical achievements of the three schools are first manifested in the formation of mathematical logic and its modern branch proof theory. In order to eliminate the paradox of set theory, Russell put forward the first axiom system of set theory by type theory and Zemello, which was revised and supplemented by frenkel, and then the common axiom system of Zemello-frenkel set theory was obtained. Later, it was further perfected and simplified by bernays and Godel, and the axiomatic system of bernays-Godel set theory was obtained. Hilbert also established meta-mathematics. Godel's incompleteness theorem is the direct result of the study of set theory paradox. Godel, an outstanding American mathematician, put forward the incompleteness theorem in the 1930s. He pointed out that a formal system containing logic and elementary number theory is incomplete if it is coordinated, that is, no contradiction can be established in this system; If the elementary arithmetic system is harmonious, then it is impossible to prove that the arithmetic system is harmonious Godel's incompleteness theorem irrefutably reveals the limitations of the formalism system, and mathematically proves the impossibility of trying to solve the paradox problem once and for all with formalism technology. In fact, it tells people that any attempt to find an absolutely reliable foundation for mathematics and thus completely avoid paradox is futile. Godel theorem is the cornerstone of mathematical logic, artificial intelligence and set theory, and a milestone in the history of mathematics. Von Neumann, a famous American mathematician, said: "Godel's achievement in modern logic is extraordinary and immortal-it is even more immortal than a monument." It is a milestone and a monument that will always exist in a visible place and in the foreseeable future. " So far, the third mathematical crisis can not be said to have been fundamentally eliminated, because many important topics of mathematical foundation and mathematical logic have not been fundamentally solved. However, people are gradually approaching the goal of fundamental solution. It can be expected that many new and important achievements will be produced in this process.