Mathematics legend 3: the destruction of myth
2020-12-07 14:00:18 【osc_ wxsc35it】
If we want to select three of the greatest mathematicians from the long history of mankind , So Carl · Friedrich · gaussian （Carl Friedrich Gauss） The name must be on the list . One of the founders of modern mathematics , Gauss's life has been very fruitful , There are many achievements named after him 110 individual , It's one of the mathematicians .
1777 year 4 month 30 Japan , Later called “ Prince of Mathematics ” Gauss was born into a Jewish family in Prussia . As a world-renowned genius , About his wisdom and intelligence , There's a story going on all the time . It is said that , gaussian 10 When I was young, I used a very short time to calculate the problems raised by primary school teachers ： Natural number from 1 To 100 Sum of . And his genius is reflected in the ingenious methods he uses ： Yes 50 To construct and 101 Sum the sequence of numbers of （1＋100,2＋99,3＋98……）, So it's easy to get the end result 5050.
Little Gauss is famous for his intelligence , Carl · William · The Duke of Ferdinand also learned about this gifted child , From Gauss 14 He began to support his study and life from the age of .1795 year , At the age of 18, Gauss transferred to Gottingen University for further study . When Gauss was nineteen , He used only a ruler and a compass to construct a regular 17 sided figure . Thus, it provides the first important supplement to Euclidean geometry which has been handed down for two thousand years . From now on , Gauss embarked on the road of mathematical research . His research covers almost all fields of mathematics , In number theory 、 Algebra 、 Non Euclidean geometry 、 Complex function and differential geometry have made pioneering contributions .
Gauss's academic attitude is very rigorous , Is a typical perfectionist . Despite his high academic attainments , But it rarely publishes articles . All his life , Gauss has only published it publicly 155 Papers , He just published what he thought was a very mature work . In Gauss's opinion ,“ When the building is finished, the scaffold must be removed .” This is the opposite of Euler , So people who read Gauss's article will only be surprised by his genius , But it's hard to grasp the context of his thoughts . Abel said when talking about Gauss's succinct writing style that “ He's like a fox, wiping its tracks with his tail ”.
One of Gauss's important contributions to algebra is to prove the basic theorem of algebra , His proof of existence opened up a new way of mathematical research . in fact , Before Gauss, many mathematicians thought that they had given proof of this result , But none of the proofs are rigorous . Gauss pointed out the deficiencies in the previous work , And then put forward my own opinion , He gave four different proofs in his life .
1806 year , Napoleon, the first emperor of France · The French army under the command of Bonaparte broke out a fierce battle with the fourth anti French Alliance , It's called the battle of Jena . In this fierce battle against the French , Carl · William · The Duke of Ferdinand died in battle , It's a big blow to Gauss . He was devastated , There was a deep hostility towards the French for a long time . Although Gottingen, where Gauss lives, is very close to France , But Gauss never set foot in France in his life .
Genius is always alone . In Gauss's time , His ideas are far ahead of others , It's hard to find someone who can match and talk to themselves , So Gauss became more aloof and rigid in character . Gauss's aloofness made him often ignore the creation of other young mathematicians . Abel once wrote his own article “ There is no general algebraic solution to the univariate quintic equation ” The paper was sent to Gauss , Hope to get the guidance of this master of Mathematics . Gauss, who is aloof and arrogant, threw the letter into the garbage can as waste paper . Gauss rarely even instructs students , For all that , Among his few students, there are still several outstanding mathematicians , One of them is Frederick · Bonhardt · Riemann （Friedrich Bernhard Riemann）.
1855 year , Gauss, a great mathematician, died in Gottingen . Smaller than Gauss 28 Gustav, 18 · Legena · Dirichlet （Gustav Lejeune Dirichlet） Selected as Gauss's successor to teach at the University of Gottingen .
Now we usually say that Gauss was a German mathematician . But the place name used when talking about his birthplace is “ Prussia, ”. in fact , For a long time in history , Germany is not a unified country in the present sense . Prussia as a historical place name , Generally refers to the northern part of Germany . The division of Germany has slowed economic development , The economic level also restricts the development of basic science such as mathematics .
Except for the extraordinary genius like Gauss , Many of the early German mathematicians were deeply influenced by the French mathematicians in their neighborhood . For example, Leibniz used to learn from Huygens, an academician of the French Academy of Sciences, as a diplomat in France , Even though Huygens is Dutch , But he had been in Paris for a long time in the field of astronomical observation . And Dirichlet was a student of Poisson and Fourier . Dirichlet was the founder of analytic number theory , He also created the formal definition of modern functions . His first paper was Fermat's theorem n=5 The situation of 1, It was later proved that n=14 The situation of . Fourier once claimed that ：“ Any function of a variable , Whether continuous or discontinuous , Can be expanded into a series of sine functions , And the parameter of the sine function is a multiple of the variable ”. But the result is not perfect , He obviously did not consider the condition of convergence of Fourier series . Dirichlet was the first to give , Under limited conditions , A demonstration of satisfaction with the result .
1859 After Dirichlet died in , Riemann took over his teaching position at the University of Gottingen . Besides the establishment of Riemannian Geometry , He was also one of the founders of the theory of functions of complex variables , He has made important contributions to mathematical analysis and differential geometry . Riemann studied mathematics under Gauss and Dirichlet . stay 1854 In the paper of , He defined the Riemann integral , The Riemannian condition for the convergence of trigonometric series is given , The direction of integral theory is pointed out , And laid the foundation of analytic number theory , The famous “ Riemann's conjecture ”. He first introduced “ Riemannian surface ” Concept , It had a great influence on modern topology . He followed Gauss , Opened up a new way of differential geometry , Founded Riemann Geometry , The curvature of Riemannian space is defined , It has been widely used in physics . He's mapping conformally 、 Elliptic function theory 、 Multiperiodic functions , And partial differential equations and other aspects have made pioneering contributions .
On the whole , Gauss has been studying at the University of Gottingen at the age of 18 , He lived here all his life 、 Work , With outstanding achievements, he broke the long-standing cold situation of German mathematics after Leibniz's death , One man created a pinnacle of German Mathematics , It also created the tradition of Gottingen's mathematics . Gauss is undoubtedly a bright star over Gottingen , Start with Gauss , Gottingen school began to come into people's sight . And Dirichlet and Riemann continued and advanced his mathematics career in Gottingen after Gauss died , It laid the foundation for the rise of Gottingen school . But it's not hard to see , Gauss himself did not like teaching , His conservative personality also kept him out of the ordinary mathematical activities . Although after Gauss , Gottingen and Dirichlet and Riemann , And Gottingen's mathematical influence continues to expand , But it's still far away from the academic centers of Europe .
The fundamental change of the situation took place in 19 In the 1970s and 1980s , At that time, both Dirichlet and Riemann were dead . There are two major incentives to change the situation , On the one hand, it comes from politics , On the other hand, it came from within German mathematicians .
As I said before , Gauss was born in Prussia , It's a part of northern Germany .1862 year , Bismarck becomes prime minister of Prussia . Under its leadership , After arduous internal affairs 、 Military and diplomatic struggles , Prussia in 1864 United with Austria to defeat Denmark , Take back the land occupied by Denmark in the North . thereafter , Prussia is again 1866 The PuO war was launched and won .1870 year , Bismarck led the Franco Prussian War again , Finally defeat the French .
You should be aware of Dodd's short stories 《 The last lesson 》 remember , This novel is from the French point of view , Describe the defeat of France after the Franco Prussian War , A historical event in which German was to be taught in areas where French was originally taught . After the Franco Prussian War , Germany took back several French ruled areas in the south , And finally 1871 Germany was unified and the German Empire was established . After the establishment of the German Empire , Bismarck in 1871 to 1890 He was appointed prime minister in 1912 , said “ Iron blood Prime Minister ”. The political unification strongly promoted the German economy to take off . In order to catch up with and surpass the old capitalist countries of Britain and France , The German government vigorously implements the policy of encouraging scientific development at home .
Another important reason for the rise of Gottingen school comes from the German mathematicians , And more precisely from two key people —— Felix · Klein （Felix Klein） David · Hilbert （David Hilbert）.
When Klein was young, he was eager to achieve something , Become the world's attention 、 Famous mathematicians in history . However , It was born out of time , At the same time, a great genius appeared in France —— Jules · Angley · Poincare （Jules Henri Poincaré）, Its brilliance completely covers the simultaneous algebras including Klein . It is universally acknowledged that , Poincare was a leading mathematician in the second quarter of the 19th century and the beginning of the 20th century , He was the last person after Gauss to have a comprehensive knowledge of mathematics and its applications . Adama commented on Poincare “ It changed the whole state of Mathematical Science , Opening up a new path in all directions .”
1886 year , Klein accepted the invitation from the University of Gottingen to come to Gottingen , Started his career as a mathematician . On individual strength , When the French couldn't compete with Poincare . It's like the era of Gauss , Others will look bleak compared to Gauss's brilliance . Klein, who realized this, imagined that if he could not compete with Poincare on his own , Can a group of people unite , Use the collective power to make Gottingen and Poincare and the French mathematical circle represented by it ？ So , The first person Klein thought of was Hilbert . Hilbert's cohesion to mathematics is unparalleled . Herman · Weil （Hermann Weyl） When I commented on him, I said ：“ Hilbert is like a piper in a motley suit , His sweet flute seduced so many mice , Then he jumped into the deep river of mathematics .”
David · Hilbert （David Hilbert,1862~1943） German mathematician , yes 19 Century and 20 One of the most influential mathematicians at the beginning of the century . He put forward the theory of Hilbert space , Is one of the foundations of functional analysis . He enthusiastically supported Cantor's set theory and infinite number .1900 year , Hilbert raised a series of questions at the International Congress of mathematicians in Paris , The famous Hilbert's 23 A question , by 20 Many mathematical studies in the 20th century have pointed out the direction . With the efforts of many mathematicians , Most of the Hilbert problem is 20 In the middle of the century . It is worth mentioning that the first one in Hilbert's problem is 8 topic ： Riemann's conjecture 、 Goldbach conjecture and twin prime conjecture . Although these three conjectures are still unsolved , But at present, we have got more important breakthroughs and solved the weakening situation respectively . Among them, the best achievement about the proof of Goldbach's conjecture is by Chen Jingrun, a mathematician in China 1973 Published in “1+2”. And the latest development of twin prime conjecture , It is 2013 Zhang Yitang, a Chinese American mathematician, proved that there are infinitely many prime pairs whose differences are less than 7000 ten thousand , This is also a major breakthrough in this issue .
1895 year , It happened to be the 100th year after Gauss arrived in Gottingen , Hilbert was invited to Gottingen by Klein . Under his solicitation , Many mathematical masters gathered in Gottingen , This includes Herman · Minkowski （Hermann Minkowski）. Minzovsky was born in Russia （ Kaunas of today's Lithuania ）, His father was a successful Jewish businessman . Old Minkowski had three sons , And the three brothers are outstanding . among , Minkowski's elder brother inherited his father's business and became an excellent businessman . Second brother Oscar · Minkowski （Oscar Minkowski） He is a famous physiologist and medical scientist .1889 Oscar in 2007 · Minkowski found that there is a substance in the pancreas that controls blood sugar , So the link between diabetes and the pancreas . This study led directly to the discovery of insulin , The mission was eventually carried out by Canadian physiologist Banting （Frederick Grant Banting） With best （Charles Herbert Best） Joint completion .1921 year , Banting and best extracted insulin from animal pancreas for clinical application , Contribute to the clinical treatment of diabetes , So they both won 1923 The Nobel Prize in physiology or medicine . And then Oscar's son , Herman's nephew Rudolph · Minkowski also became a famous American astronomer .
1872 year , Old Minkowski moved his family to gunnisburg, Prussia （ Kaliningrad of today's Russia ） settle , It is only a river away from the Hilbert family, who later became a famous mathematician . Gunnisburg is also the birthplace of the famous Seven Bridge problem （ This problem was later solved by the great mathematician Euler ）. Minkowski showed great talent in mathematics as a boy , Called the prodigy . Minkowski studied at the University of Berlin , Later he went to the University of gunnisburg . At the University of gunnisburg , Minkowski and Hilbert meet again , And make a good friend . During his college years, Minkowski taught from famous mathematicians such as Kronecker and willstrass .
1885 Minkowski received his doctorate from the University of gunnisburg . After a short period of military service ,1886 He became a lecturer at the University of Bonn .1891 Minkowski was promoted to associate professor ,1894 He returned to the University of gunnisburg to teach .1895 He succeeded Hilbert as a professor at the University of gunnisburg . Then , Minkowski transferred to work at the Federal Institute of technology in Zurich, Switzerland . During this time , Einstein studied in this school in his youth , Become Minkowski's student .1902 year , Minkowski was invited by Klein , Transferred to Gottingen University as a professor of mathematics . until 1909 He died of acute appendicitis .
The main area of Minkowski's work is number theory 、 Algebra and Mathematical Physics . among , In Mathematical Physics , When Minkowski was at the University of Bonn , He has assisted the famous physicist Hertz in the study of electromagnetic waves .1907 year , Minkowski realized that the work of Lorentz and Einstein could be described in non Euclidean space , Combine time and space, which were considered independent in the past, into a four-dimensional space-time structure , Minkowski space time . Accordingly , Different descriptions of the same phenomenon can be expressed in a simple mathematical way . These works provide a framework for general relativity . Born, the Nobel Prize winner in physics, once said , He found it in minkovsky's math work “ The whole arsenal of relativity ”.
With the joint efforts of Hilbert and Klein , Over thirty years at the beginning of last century , Gottingen became a veritable International Mathematical center , A large number of young scholars flocked to Gottingen , Not only from Germany 、 The European , And from Asia , Especially in America . According to statistics ,1862~1934 American mathematician with a foreign degree in 114 people , among 34 People get their doctorates in Gottingen . this 34 Among them was Sanders, the later president of the American Mathematical Association · McLean （Saunders Mac Lane）, His doctoral advisor in Gottingen was Herman · Well and Paul · Bernese （Paul Bernays）. McLean has a great admiration for Gottingen in his heyday , He once said without hesitation that ：“ It's a real smart Activity Center . I was a graduate student at the University of Chicago , It's a hundred times better than Chicago , I went to Yale University , It's ten thousand times better than Yale ！”
Alfred, Jewish Hungarian mathematician · Hal （Alfréd Haar） I studied at the University of Gottingen . Hal's first college life began in Budapest , But in 1904 He turned to Gottingen University, the mathematical center of the world at that time .1909 year , Hal got his doctorate , His mentor was Hilbert . Now we know the Hal measure , The mathematical concepts of Hal wavelet and Hal transform are named after him .1912 year , Kruger University （ Now called the University of Franz Joseph ） Invite Hal and Rhys to coach . Before the end of the first World War , The fall of Austro Hungarian Empire , The kingdom of Hungary, a partner of the Austrian Empire, declared its independence . Because the Austro Hungarian Empire contains several different races , So Hungary needs to be redefined 、 The borders of Austria and other newly independent states . And then there was the famous 《 Treaty of Trianon 》. According to this treaty , Part of Transylvania was ceded from Hungary to Romania . So Hal's University had to move to nearby segod . There? , Hal and Reese set up a math center together , And founded the first internationally recognized Hungarian mathematics academic journal .
Klein was early in 1914 He began to plan for the establishment of a special mathematics research institute , The plan ended up in 1929 Years to come , Klein was dead at the time . The newly established Gottingen Institute of mathematics has rapidly become the holy land of mathematicians from all over the world . Gottingen is creating myths every day .
However , Success is nothing but failure ！ Gottingen, who rose after the political unification of Germany, fell down again because of political influence . I can't help but sigh ,“ Seeing him rise high , See him feast guests , I saw his building collapse ”. Accurately speaking , Gottingen, a popular mathematical myth , It was destroyed in the catastrophe of fascism .
1933 Hitler came to power in , There was a frenzy of racism and anti Semitism , To throw the German scientific community into chaos , Gottingen was hit hard . In Gottingen's long history of development , This holy land of mathematics has always included different nationalities 、 Mathematicians of different nationalities , Many of them are Jews . You will never forget , Gauss, the founder of Gottingen's mathematics, was Jewish . in addition , Like Hal we mentioned earlier 、 Minkowski 、 Even Minkowski's teacher Leopold · Kronecker （Leopold Kronecker） They're all Jews .
Under the apartheid policy of the fascist government , A large number of Jews fled Germany . For the University of Gottingen , This is the holy land of research that has been based on intellectual achievement for decades , The impact of the relevant laws is devastating .
Richard · Brown （Richard Courant） Left Gottingen for America . The Institute of mathematics he founded at New York University became one of the top three mathematical institutes in the United States . Herman · Well himself is not Jewish , But his wife is Jewish . Like Einstein , Well went to Princeton in the United States . Paul · Bernese was deprived of the right to teach , But he also kept his position as Hilbert's most loyal assistant , Until he left for Zurich .
At a party , Hilbert found himself sitting next to the Minister of Education . The minister asked Hilbert ：“ No Jewish influence , How's Gottingen's math now ？” He replied. ：“ Gottingen's Mathematics ？ No longer exists .”
McLean later recalled ：“ When Dorothy and I were 1933 When I left in August, 2000 , The old Gottingen , The unique model of the great mathematics department has been stored in my heart . The loss made me very sad , But I'm not just sad about science . I didn't foresee the Holocaust , But I did realize the power of national machine propaganda , I was really scared about the prospect of a world war , Even though I know I can't stop this . Now in retrospect , The whole process of development is a demonstration , Any supremacy of the masses , An indisputable demonstration of the damage to academic and mathematical life by subordination to political pressure and guiding political principles .”
The decline of Gottingen's Mathematics , It is a typical tragedy of the retrogression of science culture caused by political persecution in the history of modern science . As Gottingen's influence waned , It is difficult for German mathematics to reproduce the brilliance of yesterday . in fact , To this day , German mathematics has not returned to pre war levels . Maybe , Only by reincarnation of a genius like Gauss or a mathematician with extraordinary cohesion like Hilbert can the glory of German mathematics be revived .
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