He is a former French Prime Minister and a forgotten mathematical genius

Throughout his life he switched between the different roles of a mathematician by birth and a politician by involvement, and although he achieved the highest positions in his political career, his contribution as a mathematician was even more brilliant.

Written by | Fan Ming

Since the 17th century, many first-class mathematicians have appeared in French history, including Descartes, Vieta, Pascal, Fermat, Lagrange, Laplace, d'Alembert, Legendre, Monge, Poncelet, Cauchy, Fourier, Galois, Poincare, Adama, Grothendieck and other masters, as bright as the stars in the sky, countless. Different from the stereotype of mathematicians in people's impression, many French mathematicians are enthusiastic about social and political activities, and there is a tradition of mathematicians in France. For example, in 1799, Laplace served as the Minister of the Interior for six weeks for the mathematics enthusiast Napoleon, in 1831 Galois was imprisoned twice for political reasons, and C. Villani, the 2010 Fields Medal winner, served as a member of the French National Assembly. Mathematician, politician and aviation sponsor Paul Painlevé (1863-1933, formerly translated as "Ban Lewei") is also such a strange man.

Left: Standard portrait of Painlevé, right: Painlevé’s portrait on the cover of Time magazine on November 9, 1925 | Source:
https://mathshistory.st-andrews.ac.uk/

One of the most famous mathematicians of his time

Painlevé was born in a family of artisans in Paris. His childhood was during the turbulent years in France. He was gifted in literature and science since childhood. Before graduating from high school, Painlevé had not yet decided on his life direction. He was undecided between politics and engineering, but finally chose a scientific career. In 1883, Painlevé entered the École Normale Supérieure in Paris to study mathematics. He was deeply attracted by mathematics under the influence of professors such as P. Appell, G. Darboux, C. Hermite, É. Picard, H. Poincaré and J. Tannery. At the suggestion of his doctoral thesis supervisor, Picard, one of the most outstanding mathematicians in France at the time, he went to the University of Göttingen in Germany in 1886 to study with F. Klein and HA Schwarz. The following year, he obtained a doctorate from the University of Paris with a thesis entitled "On the Singular Lines of Analytic Functions".

The standard career path for leading French scholars at the time was to obtain a first teaching position in the provinces and then try to return to Paris. After graduating with a doctorate, Painlevé was hired as a lecturer in mathematics and applied mechanics at the Université de Lille. He returned to Paris in 1892 and taught at the University of Paris, the École Polytechnique, and the Collège de France. In 1903, he became a professor of mathematics at the University of Paris and a professor of mechanics at the École Polytechnique in 1905. Painlevé's main research areas involved differential equations and analytical mechanics. His earliest interest in mathematics was rational transformations of algebraic curves and surfaces. He proposed the concept of biuniform transformations and showed great interest in nonlinear analysis theory. Painlevé had a keen mathematical intuition and he famously said, "The simplest and shortest path between two truths in the domain of real numbers is usually through the domain of complex numbers."

With the mediation of G. Mittag-Leffler, the father of modern Swedish mathematics, in 1888, Poincaré was awarded the Mathematics Prize by Oscar II, King of the United Kingdom of Sweden and Norway, for his research on the three-body problem. From September to November 1895, Painlevé, who was also interested in the three-body problem, was invited by the king to give lectures at Stockholm University. Painlevé's lecture notes, "A Course in the Analytical Theory of Differential Equations," were published two years later, including the first systematic study of the singularities of the n-body problem. For example, he proved that "the singularities of the three-body problem are all collision singularities," and proposed the famous "Painlevé conjecture": when n>3, there are non-collision singularities in the n-body problem. In layman's terms, if there are more than three planets in the system, one of them can be thrown to infinity. Chinese mathematicians Xia Zhihong (1992) and Xue Jinxin (2014), who studied in the United States, proved that the Painlevé conjecture holds when n ≥ 5 and n=4, respectively.

One of Painlevé's most important achievements was the discovery of nonlinear ordinary differential equations, which were later named after him, and new transcendental functions. It is well known that linear ordinary differential equations can be solved using elementary functions or classical special functions, while solving nonlinear differential equations is much more difficult than linear equations. The elliptic functions discovered in the 19th century expanded the family of special functions and can be used to solve a class of second-order nonlinear ordinary differential equations. Painlevé used the ideas of K. Weierstrass, L. Fuchs and SV Kovalevskaya to study a class of second-order nonlinear ordinary differential equations whose general solutions have second-order derivatives that are rational functions of themselves and first-order derivatives, which are locally analytic in the complex plane and have no movable critical singularities. This type of equation is called "Painlevé characteristic", and the definition in some articles (such as Wikipedia) that "the only movable singularities are poles" is wrong.

Painlevé, B. Gambier, R. Fuchs and others found that nonlinear ordinary differential equations with Painlevé characteristics can always be transformed into one of 50 canonical forms, of which 44 equations can be solved using known functions after reduction, and only six equations require the introduction of "new" transcendental functions. These six ordinary differential equations are called "Painlevé equations", and their solutions are called "Painlevé transcendental functions", which have very different properties from classical special functions. The irreducibility of some Painlevé equations has always been a controversial topic. In the late 1980s, Japanese mathematicians K. Nishioka and H. Umemura proved that all Painlevé equations are irreducible to linear equations or solved using elliptic functions. Due to its applications in modern geometry, quantum field theory, integrable systems and statistical mechanics, Painlevé transcendental functions have regained interest in the mathematical community in recent years and have been extended to the study of high-order nonlinear ordinary differential equations and nonlinear partial differential equations.

Painlevé systematically analyzed the motion of rigid body systems, which involved dry (Coulomb) friction during sliding. He gave the general equation of motion for such systems, pointed out the paradoxical situations that may result from the use of Coulomb's law of friction, and proposed the "Painlevé paradox" in the dynamics of friction systems. Later, Painlevé also tried to create mechanical axioms, which he believed allowed the definition of absolute motion coordinate systems that only apply to linear and uniform translational motion. Similar to the Painlevé equation, the Painlevé paradox has once again returned to the public eye due to the development of nonlinear dynamics methods in recent decades. Mittag-Leffler's evaluation of Painlevé is: "He is not afraid of the most difficult problems and is a true inventor." JS Hadamard, who was a student of the same school as Painlevé, said: "Painlevé inherited the work of Poincaré and reached the limit of human power."

Painlevé's mathematical talent was soon recognized internationally, and he became one of the most famous mathematicians of that era. He won the French Academy of Sciences' Prix des Sciences Mathematicaux (1890), the Prix Bordin (1894) and the Prix Poncelet (1896), and was elected a member of the French Academy of Sciences (Académie des sciences) in 1900. At the International Congress of Mathematicians (ICM) held in Paris that same year, Painlevé served as the chairman of the analysis section. In 1904, he gave a plenary report entitled "Modern Problems of Integro-Differential Equations" at the Heidelberg ICM. One of the doctoral students supervised by Painlevé was P. Fatou, who graduated from the École Normale Supérieure in Paris in 1907 and was famous for Fatou's lemma in Lebesgue integrals and Fatou sets in complex dynamic systems.

Painlevé's books: "A Course in the Analytical Theory of Differential Equations" (left), "Aeronautics" (right) | Image source: amazon.com & omnia.ie

Aviation pioneer and multi-sector politician

If Painlevé had continued to engage in mathematical research, his future would have been limitless. However, the famous "Dreyfus Affair" at the end of the 19th century changed his life and took the first step in his political career. Dreyfus (A. Dreyfus) was a French Jewish officer who was accused of treason by the Anti-Semitic League in December 1894 and sentenced to life imprisonment. In early 1898, the famous writer E. Zola wrote a letter to support Dreyfus' innocence, which set off a more than ten-year, earth-shaking social transformation movement in France. In 1899, Painlevé testified in the new military court and continued to fight for justice for Dreyfus until he was acquitted in 1906 and officially became a national hero. Poincare and Adama, who were both teachers and friends of Painlevé, both campaigned for Dreyfus's redress.

In 1901, Painlevé married J. Petit de Villeneuve, and their son Jean was born the following year. Unfortunately, Painlevé's wife died of puerperal fever six weeks after giving birth. Jean was raised by his father's widowed sister and later became a famous documentary director and producer, directing more than 200 science and nature films. Painlevé was an idealist, humanitarian and pacifist. He stopped all teaching and research in 1910 and became a full-time politician. As a center-left Republican-Socialist, Painlevé served as a member of the French Chamber of Deputies. After the outbreak of World War I, he chaired several military committees. Painlevé joined the cabinet in 1915 and served as France's Minister of Public Education, Minister of Defense and Invention, Minister of War, Minister of Aviation, and Minister of Finance.

Since childhood, Painlevé has been interested in exploring the mysteries of science and full of curiosity and passion for avant-garde technology. In 1903, he used the theory of fluid mechanics to prove the possibility of flight. In 1908, the Wright brothers, American aviation pioneers, landed in France with almost no government support to demonstrate their aircraft and negotiate patents with the French. On October 10, Painlevé boarded Wilbur Wright's plane and became the first Frenchman to fly into the sky. The plane carried 45 liters of gasoline and flew 55 kilometers at an altitude of ten meters, landing successfully after 1 hour and 9 minutes. This enthusiastic aviation scientist experienced the results of his calculations firsthand and successfully completed the feat of conquering the sky.

Painlevé was well aware of the importance of airplanes. He believed that this was a new type of transportation with broad prospects. He lobbied the French House of Representatives to establish a military department involving aviation and succeeded, laying the political foundation for the French aviation industry. In 1909, Painlevé became the first professor of aerodynamics in France. He devoted himself to the theoretical research of aviation science, served as the chairman of several air navigation committees, and took the lead in opening aerodynamics courses in universities. In 1910, Painlevé co-authored the book "Aeronautics" with his good friend, the famous French mathematician É. Borel. Borel was one of the pioneers of measure theory in the early 20th century, and the Borel set in topology was named after him. Borel was also a politician and served as Minister of the Ocean in 1925.

In 1917, as Minister of War, Painlevé said in a speech: "Science guarantees fair and reasonable laws and organization for human society. It will solve social problems by increasing industrial power and control over nature, and will constantly create new wealth, but it will not take it away from anyone. Science will ultimately soften human behavior through the training of fraternity and the development of wisdom. Its essentially collective effort has made us deeply feel in our hearts and minds the teachings of life given by a high degree of unity." From 1924 to 1925, Painlevé was elected Speaker of the House of Representatives. He also served as Prime Minister of the French Third Republic Cabinet twice. The first time was from September 12 to November 13, 1917, during the critical period of World War I, and the second time was during the financial crisis from April 17 to November 22, 1925. He resigned because his reform plan was not approved by the House of Representatives. Borel was a member of the cabinet during Painlevé's second term as Prime Minister.

On October 28, 1908, Painlevé (right) was riding in a Voisin biplane piloted by French pilot Henri Farman | Source: https://gallica.bnf.fr/

Build a bridge for cultural and scientific exchanges between China and France

Due to his scientific research, Pan Lewei was very curious about the ancient and mysterious Chinese civilization. As early as 1914, he met Cai Yuanpei, the first Minister of Education of the Republic of China, who was exiled to Paris due to the failure of the Second Revolution. During the Paris Peace Conference in 1919, Ye Gongchuo, the Minister of Transportation of the Beiyang Government, went to Europe, the United States, Japan, and Korea for inspection. With the efforts of Ye Gongchuo, Pan Lewei and Han Rujia, the Chinese Institute of the University of Paris was established on March 17, 1920. Pan Lewei was the first dean. Later, most of the Chinese students who went to the University of Paris to work and study enrolled in the institute. Pan Lewei once proposed to Ye Gongchuo that the French government was willing to use part of the returned Boxer Indemnity to print the Siku Quanshu. For this reason, he made a special trip to Shanghai in September 1919 to discuss this matter, but failed due to funding gaps and political turmoil.

From June 22 to September 11, 1920, at the invitation of the Beiyang government, Painlevé led a delegation of famous French cultural and intellectual figures to visit China. His entourage included French writer Bonnard, economics professor Martin of the University of Paris, railway engineer Nadal, and mathematician Borel. Painlevé particularly emphasized that this trip was a cultural journey, and the delegation had extensive exchanges with Chinese academic and cultural circles. On July 1, Painlevé visited Peking University and gave a speech in the Peking University Science Lecture Hall. President Cai Yuanpei delivered a welcome speech. From June 29 to July 1, the "Peking University Daily" carried out publicity for three consecutive days. On July 4, the "Shenbao" reported under the title "Peking University Welcomes Ban Lewei" and published Cai Yuanpei's welcome speech and Painlevé's speech. Painlevé said: "Three or four thousand years ago, European civilized countries had not yet been formed, but Chinese astronomy and mathematics could predict solar and lunar eclipses, which is really admirable."

In view of Painlevé's enthusiasm for Sino-French cultural exchanges and his contributions in the field of mathematics, Cai Yuanpei hosted a ceremony at Peking University on August 31 to appoint Painlevé as an honorary professor of Peking University. The Peking University Academic Affairs Conference also decided to award the title of "Honorary Doctor of Science" to Painlevé, American diplomat and authority on Far Eastern affairs P.S. Reinsch, French educator and diplomat, president of the Sino-French University of Lyon P. Joubin, and famous American philosopher, educator, and psychologist J. Dewey, setting a precedent for domestic universities to award honorary doctoral titles to foreign scholars. On the day of the awarding ceremony, only Painlevé was in Beijing. Cai Yuanpei said in his speech:

"The first time Peking University awarded a degree, the recipient was Mr. Benlewei. There are two reasons why this is a special commemoration: First, the university's purpose is that anyone who studies philosophy, literature, and applied sciences must start with pure science. Those who study pure science must start with mathematics. Therefore, in the order of departments, mathematics is listed as the first department. Mr. Benlewei is a world-renowned mathematician and can represent this meaning. Second, science is for the public, and all universities naturally have common research objects. However, the location of the university must pay special attention to the society and history of the place where it is located, which is the principle of division of labor. Peking University is located in China. In addition to the common research objects of scholars around the world, it also has special responsibilities for objects unique to China. Mr. Benlewei was the most advocate of the study of Chinese knowledge, which can also represent this meaning. Therefore, I think that the first degree awarded by our university to Mr. Benlewei is not only an important commemoration for Peking University, but also a great commemoration for our country's education community."

In 1920, Painlevé (third from left), Borel (second from left) and others were at Peking University | Source:
http://mjlsh.usc.cuhk.edu.hk/

At the end of 1920, Cai Yuanpei arrived in France for an inspection and visited local celebrities. In January and February 1921, Cai Yuanpei visited his old friend Painlevé twice and asked him to recommend several French scholars to visit China. The first scientist Painlevé recommended was the world-renowned Marie Curie, and the other three were physicists JB Perrin, P. Langevin, and mathematician Adam. For this reason, Cai Yuanpei made a special trip to Marie Curie's laboratory to invite her to visit China, but unfortunately it never happened. In 1931, Langevin participated in the China Education and Science Development Investigation Team organized by the League of Nations to visit China, had extensive contacts and exchanges with Chinese physicists, and gave many academic speeches. In 1936, Adam went to Shanghai Jiaotong University and Zhejiang University to give lectures, and then was invited by Tsinghua University to Beijing to give lectures for more than three months. The four masters trained Chinese disciples such as Shi Shiyuan, Li Shuhua, Wang Dezhao, Xiong Qinglai, and Wu Xinmou, and had an important influence on the development of modern mathematics and physics in China.

The "General Relativity" episode in political career

Between 1921 and 1922, Painlevé turned his attention to general relativity. In November 1925, Einstein proposed the core of general relativity - field equations. Soon after, German physicist K. Schwarzschild proved the spherically symmetric vacuum solution called "Schwarzschild metric", whose important features are Schwarzschild radius and singularity. Painlevé and Gullstrand (A. Gullstrand) independently derived the solution of Einstein's equations without singularity at the Schwarzschild radius, which was later named Gullstrand- Painlevé coordinates. Gullstrand was a professor of ophthalmology and optics at Uppsala University in Sweden, winner of the 1910 Nobel Prize in Physiology or Medicine, and a judge of the Nobel Prize in Physics. He strongly opposed Einstein's award for relativity. In October and November 1921, Painlevé published two notes in the French Academy of Sciences, in which he considered the mathematical form of general relativity and derived the above-mentioned solution of Einstein's field equations directly from the symmetry of the problem.

At the end of 1921, Painlevé wrote to Einstein, introducing his solution and inviting Einstein to Paris for discussion. At the end of March 1922, Einstein accepted the invitation of the French Physical Society to visit Paris, becoming the first German to appear publicly in France after World War I, which caused a sensation. Einstein gave a public speech at the Collège de France and had a heated debate with Painlevé, Becquerel, Brillouin, Cartan, Adama, Langevin and others. Einstein was confused by the non-quadratic cross terms of the linear elements in Painlevé's solution, so he rejected his idea. After this debate, Painlevé published a third note, extending the geometric form he used in Newton's theory to general relativity.

The French Academy of Sciences was a rather conservative academic institution, and some of its most active members were hostile to general relativity until 1921, believing that it undermined Newtonian classical mechanics. After a vicious attack on general relativity by some members of the Academy, Painlevé's work was intended to "moderate" the debate and lead colleagues who were confused by Einstein's new theory to conduct a comparative study of the two theories. Given Painlevé's scientific background, it was difficult to be completely objective at the time, and he was not ready to give up the entire edifice of classical mechanics. However, his attempt was highly constructive and contributed to the illuminating debate that followed in the Academy, making Einstein's visit to Paris fruitful.

Painlevé was the first to construct a solution to Einstein's equations without a singularity at the Schwarzschild radius. Although he later expressed doubts about its validity, as a mathematician, Painlevé was convinced that the formal derivation of this controversial solution was correct. Painlevé's interest in general relativity lasted for six months before he returned to politics, but some of his advanced ideas were forgotten for decades. Although many famous physicists at the time, including Einstein, believed that the physical singularity at the Schwarzschild radius actually existed, in 1933 G. Lemaitre discovered that Painlevé's solution was actually a coordinate transformation of the Schwarzschild metric. People learned that the transformation of the coordinate system revealed that the Schwarzschild radius was just a coordinate singularity, and its more profound significance was that it represented the event horizon of a black hole. It was not until the 1960s that some more advanced mathematical tools such as differential geometry were introduced into the study of general relativity, and physicists generally recognized this.

Einstein giving a speech at the Collège de France, with Painlevé sitting in front of the left side of the blackboard | Source: astromontgeron.fr

From left to right in the front row: Langevin, Einstein, Countess Noelle, Painlevé, and Borel, second from right in the back row | Source: wellcomecollection.org/

A life full of rationality and vitality

Painlevé was born simple, energetic and full of vitality, exuding a personal charm that few people could resist even among his opponents. He switched between the different roles of a mathematician and a politician throughout his life, and wrote and published 144 academic works, textbooks and papers. His last work was "A Course in the Resistance of Inviscid Fluids" published in 1930. After resigning as French Prime Minister in 1925, Painlevé continued to serve as a senior official in the government. In 1932, he was nominated as a candidate for the French President, but withdrew before the election. Painlevé enjoyed the pleasure brought by rational thinking and scientific spirit throughout his life, becoming a model of "governing the country with mathematics". Painlevé was one of the main designers of the Maginot Line, a military fortification along the eastern border of France. He also proposed the formulation of an international convention banning the manufacture of bombers and the establishment of an international air force to maintain global peace, but it was in vain due to the fall of the French government in January 1933.

Some historians believe that although Painlevé achieved the highest positions in his political career, his contribution as a mathematician was more significant. Painlevé returned to his favorite research field in the twilight of his life, and he once said: "If I have to leave, I will try to do it gracefully!" On October 29, 1933, Painlevé died of heart failure at his home in Paris, and the prophecy came true. A state funeral was held on November 4, and Painlevé was buried in the Panthéon. France lost one of her best sons. A square in the Latin Quarter of Paris and a mathematical laboratory at the University of Lille were named after Painlevé, and the asteroid 953 in the solar system was named Painleva. A French aircraft carrier was also named "Painlevé", but it only existed on drawings. Like many of his visions, they may not be put into reality or forgotten for a long time, but Painlevé fought for them all his life and never tired of it.

Two front pages of Le Petit Journal, one of the four major French daily newspapers, in 1925

Left: Painlevé (right) and cabinet members; Right: Painlevé (standing on the plane) visiting Morocco

Image source: larousse.fr & mediastorehouse.com

References

[1] A V. Borisov & NA Kudryashov, Paul Painlevé and His Contribution to Science, Regular and Chaotic Dynamics Vol. 19, 2014.

[2] J. Fric, Painlevé in 1921, a breaking-through solution, in general relativity, totally misunderstood at that time, Paris-Diderot University 2020.

[3] Cai Yuanpei and Modern China, edited by Cai Yuanpei Research Association, Peking University Press, 2010.

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