However, in hyperbolic geometry, there are more parallel lines. It can be proved from "at least two different straight lines are parallel to the known straight line at a point outside the straight line" that there are infinite parallel lines at that point. 1893, Kazan University established the world's first mathematician statue. This mathematician is a great Russian scholar and an important founder of non-Euclidean geometry-Lobachevsky (никола? й Ива? нович Лобаче? вски, Nikolai Ivanovich Lobachevski, Nikolai Ivanovich Lobachevsky).
Non-Euclidean geometry is a great creative achievement in the history of human cognition. Its establishment has not only brought great progress in mathematics in the past hundred years, but also had a far-reaching impact on modern physics, astronomy and the change of human view of time and space.
However, this important mathematical discovery was not recognized and praised by the society for a long time after it was put forward by Lobachevsky. On the contrary, it was distorted, criticized and attacked, which made the new theory of non-Euclidean geometry not recognized by academic circles.
In the process of trying to solve Euclid's fifth postulate, Lobachevsky embarked on the road of discovering from failure. Euclid's fifth postulate is one of the oldest famous problems in the history of mathematics, which was first put forward by ancient Greek scholars.
In the third century BC, Euclid, the founder of the Alexandria School in Greece, collected the achievements of previous geometric studies and compiled the Geometrical Elements, a mathematical masterpiece with far-reaching influence in the history of mathematical development.
The significance of this work lies in that it is the earliest model to establish a scientific theoretical system by axiomatic method. In this book, Euclid gave five axioms (applicable to all sciences) and five postulates (only applicable to geometry) as the premise of logical deduction in order to deduce all propositions of geometry. The annotators and critics of Geometry Elements are very satisfied with these five axioms and the first four postulates, except the fifth postulate (that is, the parallel axiom).
The fifth postulate involves parallel lines. It says: If a straight line intersects with two straight lines and the sum of two internal angles on the same side is less than two right angles, then if these two straight lines are extended, they must intersect at one side of those two internal angles. Mathematicians do not doubt the truth of this proposition, but think that it is not like a postulate in the length or content of the sentence, but more like a provable theorem. It was only because Euclid didn't find its proof that he had to put it in the postulate.
In order to prove the fifth postulate and complete the work that Euclid failed to complete, mathematicians devoted endless energy from the 3rd century BC to the beginning of19th century. They tried almost all possible methods, but all failed.
Lobachevsky began to study the theory of parallel lines from 18 15. At first, he followed the thinking of his predecessors and tried to prove the fifth postulate. In the saved lecture notes of his students, there are some proofs he gave in geometry teaching in1816 ~1817 school year. However, he soon realized that his proof was wrong.
The failures of his predecessors and himself inspired him from the opposite side, which made him boldly think about the opposite formulation of the problem: there may be no proof of the fifth postulate at all. So, he turned to look for the unprovable answer of the fifth postulate. This is a brand-new way of exploration that is completely opposite to traditional thinking. It was along this road that Lobachevsky discovered a brand-new geometric world in the process of proving that the fifth postulate was unprovable.
So, how does Lobachevsky prove that the fifth postulate is unprovable? How to discover a new geometric world from it? It turned out that he creatively used a common logical method to deal with complex mathematical problems-reduction to absurdity.
The basic idea of this reduction to absurdity is that in order to prove that "the fifth postulate cannot be proved", the fifth postulate is negated first, and then a new axiomatic system is formed with this negative proposition and other axiomatic postulates, and a logical deduction is made from it.
Firstly, it is assumed that the fifth postulate is provable, that is, the fifth postulate can be derived from other axiomatic postulates. Then, there will be logical contradictions in the deduction of the new axiom system, at least the fifth postulate and its negative proposition are a pair of logical contradictions; On the other hand, if there is no contradiction in the derivation of the new axiom system of "the fifth postulate is unprovable", the hypothesis of "the fifth postulate is provable" is refuted, thus indirectly proving "the fifth postulate is unprovable".
According to this logical thinking, Lobachevsky denied the equivalent proposition of the fifth postulate-Plainfield's axiom that "if a straight line crosses a plane, only one straight line will not intersect a known straight line", and obtained the negative proposition that "if a straight line crosses a plane, at least two straight lines will not intersect a known straight line", and used this negative proposition and other axiomatic postulates to form a new axiomatic system for logical deduction.
In the process of deduction, he got a series of strange and very unreasonable propositions. But on closer inspection, there is no logical contradiction between the two. Therefore, Lobachevsky, a visionary, boldly asserted that this new axiomatic system "there is no contradiction in the result" can form a new geometry, and its logical integrity and rigor can be comparable to Euclid's geometry. The existence of this new geometry without contradiction is the refutation of the provability of the fifth postulate, that is, the logical proof of the unprovable of the fifth postulate. Because the prototype and analogy of the new geometry in the real world have not been found, Lobachevsky cautiously called this new geometry "imaginary geometry". 1826 On February 23rd, Lobachevsky read his first paper on non-Euclidean geometry at the academic conference of the Department of Physical Mathematics of Kazan University: the abstraction of geometric principles and the strict proof of parallelism theorem. The publication of this groundbreaking paper marks the birth of non-Euclidean geometry. However, as soon as this great achievement was made public, it met with indifference and opposition from orthodox mathematicians.
On February 23rd, all the participants in the academic forum were experts with profound mathematical attainments, including simonov, a famous mathematician and astronomer, Gupfer, who later became an academician of China Academy of Sciences, and Bolesman, who later gained great prestige in the field of mathematics. In the eyes of these people, Lobachevsky is a very talented young mathematician.
However, to their surprise, after a brief introduction, the young professor went on to say all kinds of puzzling words, such as the sum of the internal angles of a triangle is less than two right angles, which becomes infinitely small with the increase of the side length until it tends to zero; The perpendicular to one side of an acute angle cannot intersect the other side, and so on.
These propositions are not only bizarre and inconsistent with Euclidean geometry, but also deviate from people's daily experience. But the reporter pointed out seriously and confidently that they belong to a new geometry with strict logic and have the same right to exist as Euclidean geometry. These strange languages came from a sober and rigorous mathematician professor, which surprised the participants. At first, they showed a kind of doubt and shock, and later they showed various negative expressions.
After reading the paper, Lobachevsky sincerely invited the participants to discuss and propose amendments. However, no one will make any public comments, and the atmosphere of the meeting is cold. An original and important discovery has been discovered. It is a pity that the peer experts who heard the discoverer's own description of this discovery for the first time failed to understand the significance of this discovery because of their conservative thinking, but instead adopted a cold-hearted and sneering attitude.
After the meeting, the Academic Committee of the Department entrusted simonov, Gupfer and Borasman to form a three-person appraisal team to make a written appraisal of Lobachevsky's paper. There is no doubt that their attitude is negative, but they write their written opinions so slowly that they finally lose their manuscripts. Lobachevsky's groundbreaking paper failed to attract the attention of academic circles, and the paper itself seemed to sink into the sea, and I don't know where it was abandoned. But he didn't lose heart, but stubbornly continued to explore the mystery of new geometry alone. 1829, he wrote another paper entitled "Geometric Principles". This paper reproduces the basic idea of the first paper, and supplements and develops it. At this time, Lobachevsky was elected as the president of Kazan University, probably out of respect for the president. The full text of the paper was published in Journal of Kazan University.
1832, according to Lobachevsky's request, the academic committee of Kazan University submitted this paper to Petersburg Academy of Sciences for review. Academician Ostrogradski, a famous mathematician, was entrusted by the Academy of Sciences for evaluation. Ostrogradski, a newly elected academician, has made outstanding achievements in mathematical physics, mathematical analysis, mechanics and celestial mechanics, and enjoyed a high reputation in the academic circles at that time. Unfortunately, even such an outstanding mathematician failed to understand Lobachevsky's new geometric thought, even more conservative than the professors of Kazan University.
If the professors of Kazan University are "tolerant" to Lobachevsky himself, then Ostrogradski openly accuses and attacks Lobachevsky with extremely ironic language. In the same year165438+1October 7, he wrote in a sarcastic tone at the beginning of his appraisal to the academy of sciences: "It seems that the author aims to write a book that people can't understand. He has achieved his goal. " As a result, Lobachevsky's new geometric thought was distorted and belittled. Finally, I asserted rudely: "From this, I came to the conclusion that President Lobachevsky's book is full of fallacies, so it is not worthy of the attention of the Academy of Sciences."
This paper not only aroused the anger of academic authorities, but also aroused the hostile noise of reactionary forces in society. Burachek and Jerry wrote anonymously in Sons of the Motherland magazine, publicly naming Lobachevsky for personal attacks.
Lobachevsky wrote a rebuttal to this insulting anonymous article. However, Sons of the Motherland detained Lobachevsky's article on the grounds of maintaining the magazine's reputation, and it was never published. To this, Lobachevsky was furious. Lobachevsky pioneered a new field of mathematics, but his creative work was never recognized by academic circles before his death. Just two years before his death, the famous Russian mathematician Bunyakovsky criticized Lobachevsky in his book Parallel Lines. He tried to deny the truth of non-Euclidean geometry by discussing the inconsistency between non-Euclidean geometry and empirical knowledge.
The famous British mathematician Morgan's resistance to non-Euclidean geometry is even more obvious. He even said arbitrarily, "I don't think there will be another geometry that is essentially different from Euclidean geometry at any time." Morgan's words represented the general attitude of academic circles towards non-Euclidean geometry at that time.
In the difficult course of establishing and developing non-Euclidean geometry, Lobachevsky never met his public supporters, and even Gauss, another discoverer of non-Euclidean geometry, refused to publicly support his work.
Gauss was the first master of mathematics at that time, and he had the reputation of "the king of mathematics in Europe". As early as 1792, that is, the year when Lobachevsky was born, he had already produced the bud of non-Euclidean geometry thought, and reached a mature level in 18 17. He called this new geometry "anti-Euclidean geometry" at first, then "starry sky geometry" and finally "non-Euclidean geometry". However, Gauss was afraid that the new geometry would cause academic dissatisfaction and social opposition, thus affecting his dignity and honor. Before his death, he was afraid to make this great discovery public, but carefully wrote some of his achievements in his diary and letters with friends.
Gauss was ambivalent when he saw Lobachevsky's German non-Euclidean geometry book Geometric Research of Parallel Lines Theory. On the one hand, he privately praised Lobachevsky as "one of the most outstanding mathematicians in Russia" in front of his friends, and made up his mind to learn Russian, so as to directly read all Lobachevsky's non-Euclidean geometry works. On the other hand, his friends are not allowed to reveal his confession about non-Euclidean geometry to the outside world, and he has never publicly commented on Lobachevsky's research work about non-Euclidean geometry in any form. He actively elected Lobachevsky as a member of the School of Communication of the Royal Academy of Sciences in G? ttingen. However, in the selection meeting and notification he personally wrote to Lobachevsky, he avoided talking about Lobachevsky's most outstanding contribution to mathematics-the creation of non-Euclidean geometry.
Gauss's popularity and influence in the field of mathematics may completely relieve Lobachevsky's pressure and promote academic recognition of non-Euclidean geometry. However, in the face of stubborn conservative forces, he lost the courage to fight. Gauss's silence and weakness not only severely limited the height he could reach in the study of non-Euclidean geometry, but also objectively encouraged the conservative forces to attack Lobachevsky.
In his later years, Lobachevsky felt even heavier. He is not only suppressed academically, but also restricted in his work. According to the regulations of the Russian University Committee at that time, the term of a professor was up to 30 years. According to this regulation, in 1846, Lobachevsky submitted a petition to the Ministry of Education, demanding that he be relieved of his job in the Mathematics Teaching and Research Section, and suggested that he should make way for his student popov.
The Ministry of People's Education has long been prejudiced against Lobachevsky who disobeyed their wishes, but it can't find a suitable opportunity to dismiss him as president of Kazan University. Lobachevsky's application to resign as a professor was just used as an excuse by them, which not only removed him from the post of presiding over the teaching and research section, but also violated his own wishes and removed him from all his posts in Kazan University. Being forced to leave the university job he loved all his life made Lobachevsky suffer a serious mental blow. He expressed great indignation at this unreasonable decision of the Ministry of Education.
The misfortune of his family increased his pain. His favorite and most talented eldest son died of tuberculosis, which made him very sad. His body became weaker and weaker, his eyes gradually became blind, and finally he could see nothing.
Lobachevsky, a great scholar, finished his last journey in pain and depression. Teachers and students of Kazan University held a grand memorial service for him. At the memorial service, many of his colleagues and students highly praised his outstanding achievements in building Kazan University, improving the level of national education and cultivating mathematical talents, but no one mentioned his research work on non-Euclidean geometry, because at this time, people generally thought that non-Euclidean geometry was "nonsense".
Lobachevsky struggled for the existence and development of non-Euclidean geometry for more than 30 years, and he never wavered in his firm belief in the great future of new geometry. In order to expand the influence of non-Euclidean geometry and gain early recognition from academic circles, he published his works in French and German as well as Russian, and also carefully designed a set of astronomical observation schemes to test the geometric characteristics of large-scale space.
Not only that, he also developed the analytic and differential parts of non-Euclidean geometry, making it a complete and systematic theoretical system. Being seriously ill and bedridden, he didn't stop studying non-Euclidean geometry. His last masterpiece-about geometry-was dictated to his students when he was blind and dying.
History is the fairest, because it will eventually make a correct evaluation of all kinds of ideas, viewpoints and opinions. 1868, Italian mathematician Bertram published a famous paper "An Attempt to Explain Non-Euclidean Geometry", which proved that non-Euclidean geometry can be realized on the surface of Euclidean space. In other words, non-Euclidean geometry propositions can be transformed into corresponding Euclidean geometry propositions. If Euclidean geometry has no contradiction, non-Euclidean geometry naturally has no contradiction.
Until then, non-Euclidean geometry, which has been neglected for a long time, began to get extensive attention and in-depth research in academic circles, and Lobachevsky's original research was highly praised and unanimously praised by academic circles. At this time, Lobachevsky was known as "Copernicus in geometry".