One: Three crises in the history of mathematics.
Pythagoras was a famous mathematician and philosopher in ancient Greece in the fifth century BC. He once founded a school of mysticism: Pythagoras School, which integrates politics, scholarship and religion. Pythagoras' famous proposition "Everything is a number" is the philosophical cornerstone of this school. "All numbers can be expressed as integers or the ratio of integers" is the mathematical belief of this school. Dramatically, however, the Pythagorean theorem established by Pythagoras has become the "grave digger" of Pythagoras' mathematical belief. After the Pythagorean theorem was put forward, hippasus, a member of his school, considered a question: What is the diagonal length of a square with a side length of 1? He found that this length can not be expressed by integer or fraction, but only by a new number. Hippasus's discovery led to the birth of the first irrational number √2 in the history of mathematics. The appearance of small √2 set off a huge storm in the mathematics field at that time. It directly shook the Pythagorean school's mathematical belief and made the Pythagorean school panic. In fact, this great discovery is not only a fatal blow to Pythagoras school. This was a great shock to the thoughts of all the ancient Greeks at that time. The paradox of this conclusion lies in its conflict with common sense: any quantity can be expressed as a rational number within any precision range. This is a widely accepted belief not only in Greece at that time, but also in today's highly developed measurement technology. However, the conclusion that is convinced by our experience and completely in line with common sense is overturned by the existence of a small √2! How contrary to common sense and ridiculous this should be! It just subverts the previous understanding. To make matters worse, people are powerless in the face of this absurdity. This directly led to the crisis of people's understanding at that time, which led to a big storm in the history of western mathematics, known as the "first mathematical crisis."
The second mathematical crisis stems from the use of calculus tools. With the improvement of people's understanding of scientific theory and practice, calculus, a sharp mathematical tool, was discovered independently by Newton and Leibniz almost simultaneously in the seventeenth century. As soon as this tool came out, it showed its extraordinary power. After using this tool, many difficult problems have become easy. But Newton and Leibniz's calculus theory is not strict. Their theories are all based on infinitesimal analysis, but their understanding and application of the basic concept of infinitesimal is confusing. Therefore, calculus has been opposed and attacked by some people since its birth. Among them, the most violent attack was British Archbishop Becquerel.
Russell paradox and the third mathematical crisis.
/kloc-In the second half of the 9th century, Cantor founded the famous set theory, which was severely criticized by many people when it was first produced. But soon this groundbreaking achievement was accepted by mathematicians and won wide and high praise. Mathematicians found that starting from natural numbers and Cantor's set theory, the whole mathematical building could be established. Therefore, set theory has become the cornerstone of modern mathematics. The discovery that "all mathematical achievements can be based on set theory" intoxicated mathematicians. 1900, at the international congress of mathematicians, poincare, a famous French mathematician, declared cheerfully: "… with the help of the concept of set theory, we can build the whole mathematical building … today, we can say that we have reached absolute strictness …"
However, the good times did not last long. 1903, a shocking news came out: set theory is flawed! This is the famous Russell paradox put forward by British mathematician Russell.
Russell built a set S: S is made up of all elements that don't belong to him. Then Russell asked: Does S belong to S? According to law of excluded middle, an element belongs to a set or not. Therefore, for a given set, it is meaningful to ask whether it belongs to itself. But this seemingly reasonable question, the answer will be in a dilemma. If s belongs to s, according to the definition of s, s does not belong to s; On the other hand, if S does not belong to S, then S also belongs to S by definition. It is contradictory in any case.
In fact, this paradox was discovered in the set theory before Russell. For example, in 1897, Burali and Folthy put forward the paradox of maximum ordinal number. 1899, Cantor himself discovered the paradox of maximum cardinality. However, because these two paradoxes involve many complicated theories in the set, they have only produced small ripples in the field of mathematics and failed to attract much attention. Russell paradox is different. Very simple and easy to understand, only involving the most basic things in set theory. So Russell's paradox caused a great shock in mathematics and logic at that time when it was put forward. For example, after receiving a letter from Russell introducing this paradox, G Frege said sadly, "The most unpleasant thing that a scientist encounters is that his foundation collapses at the end of his work. A letter from Mr. Russell put me in this position. " Dai Dejin therefore postponed the second edition of his article "What is the Nature and Function of Numbers". It can be said that this paradox is like throwing a boulder on the calm water of mathematics, which caused great repercussions and led to the third mathematical crisis.
After the crisis, mathematicians put forward their own solutions. I hope to reform Cantor's set theory and eliminate the paradox by limiting the definition of set, which requires the establishment of new principles. "These principles must be narrow enough to ensure that all contradictions are eliminated; On the other hand, it must be broad enough so that all valuable contents in Cantor's set theory can be preserved. " 1908, Tzemero put forward the first axiomatic set theory system according to his own principles, which was later improved by other mathematicians and called ZF system. This axiomatic set theory system makes up for the defects of Cantor's naive set theory to a great extent. Besides ZF system, there are many axiomatic systems in set theory, such as NBG system proposed by Neumann et al. The establishment of axiomatic set theory system successfully ruled out the paradox in set theory, thus successfully solving the third mathematical crisis. On the other hand, Russell's paradox has a far-reaching influence on mathematics. It puts the basic problems of mathematics in front of mathematicians for the first time with the most urgent needs, and guides mathematicians to study the basic problems of mathematics. The further development of this aspect has profoundly affected the whole mathematics. For example, the debate on the basis of mathematics has formed three famous schools of mathematics in the history of modern mathematics, and the work of each school has promoted the great development of mathematics.
Second, the classical mathematical problem: the seven bridges problem
One of the famous classical mathematical problems. In a park in Konigsberg, there are seven bridges connecting the Fritz fritz pregl River and two islands on its banks. Is it possible to start from any of these four places, cross each bridge only once, and then return to the starting point? Euler studied and solved this problem in 1736. He simplified the problem to the "one stroke" problem shown on the right, which proved that the above method was impossible.
Hot issues in graph theory research. 65438+ K? nigsberg, Prussia. At the beginning of the 8th century, the Fritz fritz pregl River passed through this town. Naif Island is located in the river, and there are 7 bridges on the river, connecting the whole town. Local residents are keen on a difficult problem: is there a route that can cross seven bridges without repetition? This is the problem of the seventh bridge in Konigsberg. L. Euler uses points to represent islands and land, and the connecting line between two points represents the bridge connecting them, which simplifies rivers, islands and bridges into a network and turns the problem of seven bridges into a problem of judging whether the connected networks can draw a sum. He not only solved this problem, but also gave the necessary and sufficient conditions for connected networks to be brushes, if they are connected and the odd vertices (the number of arcs passing through this point is odd) are 0 or 2.
When Euler visited Konigsberg, Prussia (now Kaliningrad, Russia) in 1736, he found that local citizens were engaged in a very interesting pastime. In konigsberg, a river called Pregel runs through it. This interesting pastime is to walk across all seven bridges on Saturday. Each bridge can only cross once, and the starting point and the ending point must be the same place.
Euler regarded every land as a point, and the bridge connecting the two lands was represented by a line.
It was later inferred that such a move was impossible. His argument is this: In addition to the starting point, every time a person enters a piece of land (or point) from one bridge, he or she also leaves the point from another bridge. So every time you pass a point, two bridges (or lines) are counted, and the line leaving from the starting point and the line finally returning to the starting point are also counted, so the number of bridges connecting each piece of land and other places must be even.
The graph formed by the seven bridges does not contain even numbers, so the above tasks cannot be completed.
Euler's consideration is very important and ingenious, which embodies the uniqueness of mathematicians in dealing with practical problems-abstracting a practical problem into a suitable "mathematical model". This research method is called "mathematical model method". You don't need to use profound theories, thinking is the key to solving problems.
Next, based on a theorem in the network, Euler quickly judged that it was impossible not to visit the seven bridges in Konigsberg at one time. In other words, for many years, the non-repetitive route that people have worked so hard to find simply does not exist. A question that stumped so many people turned out to be such an unexpected answer!
1736, Euler expounded his method of solving problems in the paper report of "Seven Bridges in Konigsberg" submitted to Petersburg Academy of Sciences. His ingenious solution laid the foundation for the establishment of a new branch of mathematics-topology.
The world of mathematics is endless, let's enjoy it!
Appendix: Eternal Master-Euler
Euler (1707- 1783) is a Swiss mathematician and natural scientist. 1707 was born in Basel, Switzerland in April, and 1783 died in Petersburg, Russia in September. Euler was born in a priest's family and was educated by his father since childhood. /kloc entered university of basel at the age of 0/3, graduated from university at the age of 0/5, and obtained a master's degree at the age of 0/6.
Euler's father wanted him to study theology, but he was most interested in mathematics. In college, he has been studying mathematics under the special guidance of John I Bernoulli until 18 years old. He gave up the idea of becoming a priest completely and specialized in mathematics. 19 (1726) started writing articles and won the prize of Paris Academy of Sciences.
1727, on the recommendation of daniel bernoulli, he went to the Academy of Sciences in Petersburg, Russia for research work. 173 1 year, he succeeded Daniel I Bernoulli as professor of physics.
During his 14 years in Russia, he devoted himself to research and made outstanding achievements in analysis, number theory and mechanics. In addition, at the request of the Russian government, Euler solved many practical problems such as drawing and shipbuilding. 1735, he became blind in his right eye because of overwork. 174 1 year, invited by frederick the great, Prussia, as the director of the Institute of Physical Mathematics, German Academy of Sciences. During his stay in Berlin, he greatly expanded his research contents, such as planetary motion, rigid body motion, thermodynamics, ballistics, demography and so on. These works and his mathematical research promote each other. At the same time, he made breakthroughs in differential equations, differential geometry of surfaces and other mathematical fields.
1766, he returned to Petersburg at the invitation of Russian Tsar Kadrin II. 177 1 year, a serious illness made his left eye completely blind. But with his amazing memory and mental arithmetic ability, he continued to engage in scientific creation. He finished a large number of scientific works through discussions with his assistants and direct dictation until the last moment of his life.
Euler is one of the most outstanding figures in the field of mathematics in the18th century. He not only made contributions in the field of mathematics, but also pushed mathematics to almost the whole field of physics. In addition, he is also the most prolific mathematician in the history of mathematics, and has written a large number of teaching materials such as mechanics, analysis, geometry and variational methods, introduction to differential analysis (1748), differential calculus principle (1755), and integral calculus principle (1768-65).
Euler's greatest achievement is to expand the field of calculus and lay the foundation for the emergence and development of some important branches of differential geometry and analysis, such as infinite series and differential equations.
Euler changed infinite series from a general operation tool into an important research topic. He calculated the value of ξ function at even points:. He proved that a2k is a rational number and can be expressed by Bernoulli number.
In addition, he also studied harmonic series and calculated the value of Euler constant γ quite accurately, which is about 0.57721566490153286060651209. ...
/kloc-In the mid-8th century, Euler and other mathematicians established differential equations in the process of solving physical problems. Among them, in terms of ordinary differential equations, he completely solved the problem of n-order linear homogeneous equations with constant coefficients, and for non-homogeneous equations, he proposed a solution to reduce the order of the equations; In terms of partial differential equations, Euler attributed the vibration problem of two-dimensional objects to the solution of one, two and three-dimensional wave equations. Euler's research on the integral method of equations is the first paper in pure mathematics to study partial differential equations.
In differential geometry (differential geometry is a branch of mathematics that studies the point-by-point variation of curves and surfaces), Euler introduced the parametric equation of spatial curves and gave the analytical expression of curvature radius of spatial curves. 1766 published the study of curves on surfaces, which is Euler's most important contribution to differential geometry and a milestone in the history of differential geometry. He expressed the surface as z=f(x, y), and introduced a series of standard symbols to express the partial derivatives of z to x, y, which are still widely used today. In addition, in this book, he also obtained the curvature formula of the section line of a surface on any section.
Euler made countless contributions to the analysis, for example, he introduced G function and B function, proved the addition theorem of elliptic integral, and first introduced double integral.
In algebra, he found that every polynomial with real coefficients must be decomposed into the product of primary or secondary factors, that is, the form of a+bi. Euler also gave three proofs of Fermat's infinitesimal theorem, and introduced the important Euler function φ(n) in number theory. His series of achievements in studying number theory laid the foundation for number theory to become an independent branch of mathematics. Euler discussed the problem of number theory by analytical method, found the functional equation satisfied by ξ function, and introduced Euler product. It also solved the problem of the famous seventh bridge in Konigsberg.
Euler studied mathematics so extensively that important constants, formulas and theorems named after him can often be seen in many branches of mathematics.