Scientist John Nash. American mathematician John Nash: biography, achievements and interesting facts
John Nash born June 13 1928 in Bluefield, Virginia, in a strict Protestant family. My father worked as an engineer at Appalachian Electric Power, and my mother worked as a school teacher for 10 years before her marriage. At school I was an average student, and I didn’t like mathematics at all - they taught it in a boring way at school. When Nash was 14, he came across Eric T. Bell's book, Great Mathematicians. “After reading this book, I was able to prove Fermat’s little theorem myself, without outside help,” Nash writes in his autobiography. This is how his mathematical genius declared himself.
Studies
This was followed by studies at the Carnegie Polytechnic Institute (now the private Carnegie Mellon University), where Nash tried to study chemistry, took a course in international economics, and then finally became convinced of his decision to study mathematics. IN 1948 year, having graduated from the institute with two diplomas - a bachelor's and a master's - he entered Princeton University. Nash's institute teacher Richard Duffin provided him with one of the most laconic letters of recommendation. It contained a single line: “This man is a genius!”
Works
At Princeton, John Nash heard about game theory, then only introduced by John von Neumann and Oscar Morgenstein. Game theory captured his imagination, so much so that at the age of 20, John Nash was able to create the foundations of a scientific method that played a huge role in the development of the world economy. IN 1949 year, a 21-year-old scientist wrote a dissertation on game theory. Forty-five years later he received the Nobel Prize in Economics for this work. Nash's contribution was described as follows: for fundamental analysis of equilibrium in the theory of non-cooperative games.
Neumann and Morgenstein were concerned with so-called zero-sum games, in which the victory of one side inevitably means the defeat of the other. IN 1950 - 1953 gg. Nash published four groundbreaking papers that provided insightful analysis of “non-zero-sum games,” a special class of games in which all participants either win or lose. An example of such a game would be negotiations on a salary increase between the trade union and the company management.
This situation can end either in a long strike in which both sides suffer, or in the achievement of a mutually beneficial agreement. Nash was able to discern a new face of competition by modeling a situation that was later called the “Nash equilibrium” or “non-cooperative equilibrium”, in which both parties use an ideal strategy, which leads to the creation of a stable equilibrium. It is beneficial for the players to maintain this balance, since any change will only worsen their situation.
IN 1951 In the same year, John Nash began working at the Massachusetts Institute of Technology (MIT) in Cambridge. His colleagues did not particularly like him, because he was very selfish, but they treated him patiently, because his mathematical abilities were brilliant. There, John began a close relationship with Eleanor Stier, who was soon expecting his child. So Nash became a father, but he refused to give his name to the child to be written on the birth certificate, and also refused to provide any financial support.
IN 1950 's Nash was famous. He collaborated with the RAND Corporation, an analytical and strategic development company that employed leading American scientists. There, again thanks to his research in game theory, Nash became one of the leading experts in the field of Cold War warfare. In addition, while working at MIT, Nash wrote a number of articles on real algebraic geometry and the theory of Riemannian manifolds, which were highly appreciated by his contemporaries.
Disease
Soon John Nash met Alicia Lard and 1957 They got married. In July 1958 Fortune magazine named Nash America's rising star in the "new mathematics." Soon Nash's wife became pregnant, but this coincided with Nash's illness - he became schizophrenic. At this time, John was 30 years old, and Alicia was only 26. At the beginning, Alicia tried to hide everything that was happening from friends and colleagues, wanting to save Nash’s career. However, after several months of insane behavior, Alicia forcibly committed her husband to a private psychiatric clinic in the suburbs of Boston, McLean Hospital, where he was diagnosed with paranoid schizophrenia.
After being discharged, he suddenly decided to go to Europe. Alicia left her mother's newborn son and followed her husband. She brought her husband back to America. Upon their return, they settled in Princeton, where Alicia found work. But Nash's illness progressed: he was constantly afraid of something, spoke about himself in the third person, wrote meaningless postcards, and called former colleagues. They listened patiently to his endless discussions about numerology and the state of political affairs in the world.
The deterioration of her husband's condition depressed Alicia more and more. IN 1959 He lost his job. In January 1961 Years ago, a completely depressed Alicia, John's mother and his sister Martha made a difficult decision: to admit John to Trenton State Hospital in New Jersey, where John underwent a course of insulin therapy - a harsh and risky treatment, 5 days a week for a month and a half. After his discharge, Nash's colleagues from Princeton decided to help him by offering him a job as a researcher, but John again went to Europe, but this time alone. He sent home only mysterious letters. IN 1962 year, after 3 years of turmoil, Alicia divorced John. With the help of her mother, she raised her son herself. Later it turned out that he also had schizophrenia.
Despite his divorce from Alicia, his fellow mathematicians continued to help Nash - they gave him a job at the University and arranged a meeting with a psychiatrist, who was prescribed anti-psychotic medications. Nash's condition improved and he began spending time with Eleanor and his first son, John David. “It was a very encouraging time,” recalls John’s sister Martha. - It was quite a long period. But then things started to change.” John stopped taking the medication, fearing that it might affect his thinking and the symptoms of schizophrenia reappeared.
IN 1970 Mr. Alicia Nash, being sure that she had made a mistake by betraying her husband, took him in again, and now as a boarder, this possibly saved him from a state of homelessness. In subsequent years, Nash continued to go to Princeton, writing strange formulas on the boards. Princeton students nicknamed him "The Phantom."
Then in 1980 gg. Nash felt noticeably better - his symptoms subsided and he became more involved in life around him. The disease, to the surprise of the doctors, began to recede. More precisely, Nash began to learn to ignore her and took up mathematics again. “Now I think quite sensibly, like any scientist,” Nash writes in his autobiography. “I won’t say that this gives me the joy that anyone recovering from a physical illness experiences. Sound thinking limits man’s ideas about his connection with the cosmos.”
Confession
IN 1994 , at the age of 66, John Nash received the Nobel Prize for his work on game theory. However, he was deprived of the opportunity to give the traditional Nobel lecture at Stockholm University, as the organizers feared for his condition. Instead, a seminar was organized (with his participation) to discuss his contributions to game theory. After this, Nash was invited to give a lecture at the University of Uppsala, since he was not given such an opportunity in Stockholm. According to Christer Kiselman, a professor at the Institute of Mathematics at Uppsala University who invited him, the lecture was dedicated to cosmology.
IN 2001 year, 38 years after their divorce, John and Alicia remarried. Nash returned to his office in Princeton, where he continues to learn about mathematics and understand this world - the world in which he was initially so successful; a world that forced him to go through a very difficult illness; and yet this world accepted him again.
IN 2008 John Nash gave a presentation on the topic “Ideal Money and Asymptotically Ideal Money” at the international conference Game Theory and Management at the Graduate School of Management of St. Petersburg State University.
IN 2015 year, for his contributions to the theory of nonlinear differential equations, John was awarded the highest award in mathematics - the Abel Prize.
"Mind games"
IN 1998 year, American journalist (and economics professor at Columbia University Sylvia Nasar wrote a biography of Nash entitled “A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash”). The book instantly became a bestseller. .
IN 2001 year, under the direction of Ron Howard, based on the book, the film “A Beautiful Mind” was shot, and “A Beautiful Mind” was released in Russia. The film received four Oscars (for best adapted screenplay, director, supporting actress and, finally, best picture), a Golden Globe award and was awarded several Bafta prizes (British Film Award).
Death
May, 23rd 2015 86-year-old John Nash died in a car accident along with his 82-year-old wife Alicia. The driver of the taxi they were traveling in lost control and crashed into a median barrier.
His father was an electrical engineer, his mother a school teacher. At school, Nash did not show outstanding success, was withdrawn, and read a lot.
In 1945, he entered the Carnegie Institute of Technology (now Carnegie Mellon) to study chemical engineering. Then he became interested in economics and mathematics.
He received his bachelor's and master's degrees in mathematics in 1948, after which he began working at Princeton University.
In 1949 he wrote his doctoral dissertation on the mathematical principles of game theory.
In 1951 he left Princeton and took up teaching at the Massachusetts Institute of Technology. While at the university, Nash developed a method of iteration, later refined by Jürgen Moser, which is now known as the Nash-Moser theorem.
In the early 1950s, he worked as a consultant for the RAND Corporation in Santa Monica, California, funded by the US Department of Defense.
In 1956, he won one of the first Sloan Fellowships and took a year's sabbatical at the Institute for Advanced Study in Princeton. During this period he lived in New York, collaborating with the Richard Courant Institute of Applied Mathematics at New York University.
In 1959, Nash began to suffer from schizophrenia and severe paranoia, which eventually forced him to quit his job.
In 1961, at the insistence of his relatives, he was sent for treatment to Trenton State Hospital in New Jersey. After completing the course of therapy, he traveled extensively throughout Europe and was engaged in individual research.
By the 1990s, Nash's mental state had returned to normal and he received a number of awards for his professional work.
In 1994, the scientist was awarded the Nobel Prize in Economics for his analysis of equilibrium in the theory of non-cooperative games. Nash shared the prize with Hungarian economist John C. Harsanyi and German mathematician Reinhard Selten.
In 1996 he was elected to the National Academy of Sciences.
In 1999, for the embedding theorem proven in 1956, together with Michael D. Crandall, he received the Steele Award for seminal contributions to research, awarded by the American Mathematical Society.
The scientist continued to collaborate with Princeton University.
In 2015 he was awarded the prestigious Abel Prize in Mathematics for his contributions to the study of differential equations.
John Forbes Nash Jr. and his wife died in a traffic accident in New Jersey. According to preliminary data, the deceased were not wearing seat belts.
Nash has been married to Alicia Larde since 1957. In 1962, the couple divorced due to the scientist's mental illness, but in 1970 the family was reunited. The scientist left behind a son.
Biography
John Forbes Nash Jr. was an American mathematician who worked in the fields of game theory and differential geometry. Winner of the 1994 Nobel Prize in Economics for “Equilibrium Analysis in the Theory of Non-Cooperative Games” (together with Reinhard Selten and John Harsanyi). Known to the general public mostly from Ron Howard's biographical drama A Beautiful Mind about his mathematical genius and struggle with schizophrenia.
John Nash was born on June 13, 1928 in Bluefield, West Virginia, into a strict Protestant family. His father worked as an electrical engineer for Appalachian Electric Power, and his mother worked as a school teacher for 10 years before her marriage. At school I was an average student, and I didn’t like mathematics at all - they taught it in a boring way at school. When Nash was 14 years old, he came across Eric T. Bell's book, The Creators of Mathematics. “After reading this book, I was able to prove Fermat’s little theorem myself, without outside help,” Nash writes in his autobiography.
Studies
After school, he studied at the Carnegie Polytechnic Institute (now the private Carnegie Mellon University), where Nash tried to study chemistry, took a course in international economics, and then finally decided to take up mathematics. In 1947, after graduating from college with two degrees - a bachelor's and a master's - he entered Princeton University. Nash's institute teacher Richard Duffin provided him with one of the most laconic letters of recommendation. It contained the line: “He is a mathematical genius.”
Job
At Princeton, John Nash heard about game theory, then only introduced by John von Neumann and Oskar Morgenstern. Game theory captured his imagination, so much so that at the age of 20, John Nash was able to create the foundations of a scientific method that played a huge role in the development of the world economy. In 1949, the 21-year-old scientist wrote a dissertation on game theory. Forty-five years later, he received the Nobel Prize in Economics for this work “for his fundamental analysis of equilibrium in the theory of non-cooperative games.”
Between 1950 and 1953, Nash published four groundbreaking papers in the field of non-zero-sum games. He discovered the possibility of "non-cooperative equilibrium", in which both parties use a strategy that leads to a stable equilibrium. This result was later called the “Nash equilibrium.”
In 1951 he joined the Massachusetts Institute of Technology (MIT). He wrote a number of articles on real algebraic geometry and the theory of Riemannian manifolds, which were highly appreciated by his contemporaries.
In 1954, he was arrested by Santa Monica police for lewd behavior in a men's locker room on the beach. The charge was soon dropped, but Nash was stripped of his security clearance at the RAND Corporation, where he was a part-time consultant.
Disease
Soon John Nash met a student, Colombian beauty Alicia Lard, and in 1957 they got married. In July 1958, Fortune magazine named Nash America's rising star in the "new mathematics." Soon Nash's wife became pregnant, but this coincided with Nash's illness - he began to develop symptoms of schizophrenia. At this time, John was 30 years old, and Alicia was 26. Alicia tried to hide everything that was happening from friends and colleagues, wanting to save Nash’s career. The deterioration of her husband's condition depressed Alicia more and more. In 1959 he lost his job. Some time later, Nash was involuntarily committed to a private psychiatric clinic in the Boston suburbs, McLean Hospital, where he was diagnosed with paranoid schizophrenia and subjected to psychopharmacological treatment. Nash's lawyer managed to get him released from the hospital after 50 days. After discharge, Nash decided to go to Europe. Alicia left her newborn son with her mother and followed her husband. Nash tried to obtain political refugee status in France, Switzerland and the German Democratic Republic and renounce his American citizenship. Biographer Sylvia Nasar reports that in March 1960 Nash visited Leipzig and stayed with the Thurmer family for several days while authorities decided on his status. Finally, the US authorities managed to achieve the return of Nash - he was arrested by the French police and deported to the United States. Upon their return, they settled in Princeton, where Alicia found work. But Nash's illness progressed: he was constantly afraid of something, spoke about himself in the third person, wrote meaningless postcards, and called former colleagues. They listened patiently to his endless discussions about numerology and the state of political affairs in the world.
In January 1961, a completely depressed Alicia, John's mother and his sister Martha admitted John to Trenton State Hospital in New Jersey, where John underwent insulin therapy. After his discharge, Nash's colleagues from Princeton decided to help him by offering him a job as a researcher, but John again went to Europe, but this time alone. He sent home only mysterious letters. In 1962, after three years of turmoil, Alicia divorced John. With the support of her mother, she raised her son herself. He subsequently also developed schizophrenia.
Fellow mathematicians continued to help Nash - they gave him a job at the university and arranged a meeting with a psychiatrist who prescribed antipsychotic medication. Nash's condition improved and he began spending time with Alicia and his first son, John David. “It was a very encouraging time,” recalls John’s sister Martha. - It was quite a long period. But then things started to change.” John stopped taking his medication, fearing that it might impair his thinking, and the symptoms of schizophrenia reappeared.
In 1970, Alicia Nash, confident that she had made a mistake by betraying her husband, accepted him again, and this may have saved the scientist from homelessness. In subsequent years, Nash continued to go to Princeton, writing strange formulas on the boards. Princeton students nicknamed him "The Phantom." Then in the 1980s, Nash began to feel noticeably better - his symptoms subsided and he became more involved in the life around him. The disease, to the surprise of the doctors, began to recede. In fact, Nash began to learn to ignore her and returned to mathematics.
Now I think quite rationally, like any scientist,” Nash writes in his autobiography. “I won’t say that this gives me the joy that anyone recovering from a physical illness experiences. Rational thinking limits a person’s ideas about his connection with the cosmos.
Confession
On October 11, 1994, at the age of 66, John Nash received the Nobel Prize in Economics for his analysis of equilibrium in the theory of non-cooperative games.
However, he was deprived of the opportunity to give the traditional Nobel lecture at Stockholm University, as the organizers feared for his condition. Instead, a seminar was organized (with the participation of the laureate) to discuss his contributions to game theory. After this, John Nash was nevertheless invited to give a lecture at another university - Uppsala. According to Christer Kiselman, a professor at the Institute of Mathematics at Uppsala University who invited him, the lecture was dedicated to cosmology.
In 2001, 38 years after their divorce, John and Alicia remarried. Nash returned to his office in Princeton, where he continued to work on mathematics.
In 2008, John Nash gave a presentation on the topic “Ideal Money and Asymptotically Ideal Money” at the international conference Game Theory and Management at the Graduate School of Management of St. Petersburg State University.
In 2015, John Nash received the highest honor in mathematics, the Abel Prize, for his contributions to the theory of nonlinear differential equations.
A remarkable fact: having received both the Nobel and Abel Prizes, John Forbes Nash became the first person in the world to be awarded both prestigious awards.
Death
John Nash died on May 23, 2015 (age 86) along with his wife, Alicia Nash (age 83), in a car accident in New Jersey. The taxi driver in which the couple were traveling lost control while overtaking and crashed into the median barrier. Both unbelted passengers were thrown out upon impact, and arriving paramedics pronounced them dead at the scene. The taxi driver was taken to hospital with non-life threatening injuries.
Film "A Beautiful Mind"
Main article: A Beautiful Mind (2001 film)In 1998, American journalist (and Columbia University business journalism professor) Sylvia Nasar wrote a biography of Nash entitled A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash. ). The book instantly became a bestseller.
In 2001, under the direction of Ron Howard, based on the book, the film “A Beautiful Mind” was shot (in Russian box office - “A Beautiful Mind”). The film won four Oscars (for best picture, best adapted screenplay, director and supporting actress), a Golden Globe award and was awarded several BAFTA awards.
Bibliography
Books
Bargaining Problem = The Bargaining Problem. - 1950.Non-cooperative Games = Non-cooperative Games. - 1951.
Articles
Real algebraic manifolds // Ann. Math. - 1952. - Vol. 56. - P. 405-421.C1-isometric imbeddings // Ann. Math. - 1954. - Vol. 60. - P. 383-396.
Continuity of solutions of parabolic and elliptic equations // Amer. J. Math. - 1958. - Vol. 80. - P. 931-954.
Translated into Russian
J. Nash, C1-isometric embeddings // Mathematics 1957, volume 1, number 2, pp. 3-16.J. Nash, The embedding problem for Riemannian manifolds // Uspekhi Mat. Nashe, 26:4(160) (1971), 173-216.
J. Nash, Analyticity of solutions to problems on implicit functions with analytical initial data // Russian Math.
The film A Beautiful Mind, which received four Oscars, was based on the biography of John Nash. The film makes you look differently at people suffering from mysterious schizophrenia. This picture is one of the most beautiful and touching stories of madness, recovery, discovery, fame, uselessness, loneliness - everything that makes up the life of a genius. John Nash is one of the most revered and famous mathematicians in the world, working in the fields of game theory and differential geometry. In 1994 he received the Nobel Prize in Economics. Nash's dissertation, where he proved the existence of what was later called the Nash Equilibrium, was only 27 pages long. The mathematician for many years tragically struggled with his own madness, bordering on genius. Our selection includes 12 of his quotes - they will captivate you with their depth and originality.
- Good scientific ideas would not come to me if I thought like normal people.
- At times I thought differently than everyone else and did not follow the norm, but I am sure that there is a connection between creative thinking and abnormality.
- It seems to me that when people are unhappy, they become mentally ill. Nobody goes crazy when they win the lottery. This happens when you don't win it.
- Now I think quite sensibly, like any scientist. I won’t say that this gives me the joy that anyone who recovers from a physical illness experiences. Common thinking limits man's ideas about his connection with the cosmos.
- Something may be considered incredible and unrealistic, but everything is possible.
- I have never seen imaginary people, sometimes I heard them. Most people see imaginary people all their lives, having no idea about real ones.
- My main scientific achievement is that all my life I have been working on things that really interest me, and I have not spent a single day doing any nonsense.
- In mathematics, it is not so much the ability to strain the brain that is important, but the ability to relax it. I think ten out of a hundred can do this, no more. For some reason this works better in youth.
- You can't make money with math, but you can organize your brain in such a way that you can start making money. In general, those who do not know how to count money are able to earn money. Money cannot be counted rationally; its quantity almost never corresponds to your quality; this is where all the conflicts lie.
- At least three people can understand me, yes. We have a systematic language for this communication. And no one can understand another person - for example, you - precisely because you cannot formalize yourself. It is generally impossible to understand people.
- I need contact with those people who can check my results. Otherwise, I think not.
- There are no epiphanies. In my case, the task was solved the moment it was set.
In the “Main Thought” library you can read reviews of books that develop and activate creative, non-trivial thinking. For example, books
Original taken from fandorin1001 in The Mind Games of John Nash
Sometimes the line between genius and mental illness seems completely invisible. The examples of many great people confirm this sad truth. The outstanding mathematician John Nash, winner of the 1994 Nobel Prize in economics, struggled for a long time with paranoid schizophrenia...
In 2001, the film “A Beautiful Mind,” based on the book of the same name by Sylvia Nazar, was released in the United States. This film, telling the story of the tragic fate of John Nash, shocked the public and the American Academy of Motion Picture Arts, which awarded the film several Oscars. And the collections of this picture amounted to 312 million dollars.
The famous actor Russell Crowe, who played the role of the mathematician, played his character so convincingly that it seemed that all the passions and complex life conflicts of John Nash came to life on the screen. But the real story of the mathematician was even more tragic than shown in the film...
John Forbes Nash Jr. was born on June 13, 1928, in West Virginia, the son of an electrical engineer and a former schoolteacher. It is interesting that, like many future geniuses, he studied rather averagely at school, and did not like mathematics at all. In his autobiography, he said that his unusual abilities were revealed after he read Eric T. Bell’s book “Great Mathematicians” at the age of 14. And the teenager’s abilities turned out to be truly phenomenal: “After reading this book, I was able to prove Fermat’s little theorem myself, without outside help.”
After graduating from school, Nash initially intended to follow in his father's footsteps and become an electrical engineer. But instead he went to Carnegie Polytechnic Institute and majored in chemistry. However, this science did not interest the young genius at all, and he became interested in economics.
In 1948, Nash graduated from college and went to Princeton University with a short letter of recommendation from his teacher, Richard Duffin. This letter contained only one line: “This man is a genius!”...
Game time
Princeton in the late forties and early fifties was a special place. For example, Albert Einstein worked there. John von Neumann and Oskar Morgenstern, scientists who published the landmark book Game Theory and Economic Behavior in the mid-forties, also had Princeton residence permits.
Game theory has become for American science a kind of key to solving a wide variety of problems: from microeconomic issues to US foreign policy strategy.
However, having declared the enormous potential of the theoretical concept, within the framework of which almost any social phenomenon can be represented as the interaction of two players acting according to certain rules, Neumann and Morgenstern were unable to explain how it applies to everyday life.
Nash figured out how to fill this gap. His dissertation, which was only 27 pages long, focused on cooperative and non-cooperative games and the equilibrium of their strategies. He defended it at age 22 and actually received the Nobel Prize for it 45 years later.
One of Nash's main achievements is the formulation of the “Nash equilibrium”: in every game there is a certain set of strategies of its participants in which none of them can change their behavior to achieve greater success if the other participants do not change their strategies. In other words, it is not beneficial for players to give up this balance, since otherwise they will only make the situation worse.
At the same time, Nash assumed that any game, in essence, can be reduced to non-cooperative - the players act on their own, without agreement. However, such a game does not assume that the opponents are initially aimed at the logic of “doom or miss.” They can pursue a dual goal - obtaining benefits for themselves and for all participants in the game. It is in the state of “Nash equilibrium” that the most successful combination of personal and collective benefits is possible.
Thanks to this point, game theory took on a new life - Morgenstern and Neumann tried to understand games that result in the absolute loss of one of the parties: ousting a competitor from the market or winning a war. Nash showed that it is wiser to seek common benefit.
In addition, the scientist developed the “bargaining theory” - a mathematical model of interaction between participants who initially have unequal knowledge and, therefore, are able to build behavioral models differently. Over time, the “bidding theory” formed the basis of modern strategies for conducting auctions and concluding transactions, where the interested party itself determines the amount of information that the “partner” in the game should know.
In the film, Nash's discovery was illustrated by an episode with five pretty girls. If all of Nash’s friends rushed to the most beautiful of them (that is, they began to play each for themselves), then, firstly, by pushing each other aside, they would not have achieved her, and secondly, by turning their backs to her friends, they would be rejected by them too, since no one wants to become a “consolation prize.” “Nash Equilibrium” offered them another option - to start courting each girl individually, as a result of which almost everyone got what they wanted.
In the scientific world, John Nash's theory is usually presented through another prominent example - the prisoner's dilemma problem, which was invented by Nash's teacher Albert W. Tucker. The problem is as follows: John and Jack are thieves who were caught by the police after committing a robbery. They are put in separate cells and asked to confess. They have two options for behavior - to confess or to deny everything. If one confesses and the other remains silent, then the first is released, and the second receives 10 years in prison. If they both confess, they will each have to serve five years. If both remain silent, then each faces 1 year in prison for illegally carrying weapons. It is important that neither of them knows which path the other has chosen.
What should they do? From the point of view of the Nash equilibrium, John and Jack must both remain silent, in which case each of them is guaranteed to receive a minimum sentence.
Such a state of balance can be found, according to game theory experts, in any area of human activity. But the gaming approach did not catch on immediately - and for several reasons.
It turned out that the “Nash equilibrium” is an excellent analytical tool for working with simple situations of interaction of two objects. However, the more complex the situation becomes, the more sets of strategies there are that satisfy the “Nash equilibrium” criterion. Which one will the players choose? Nash did not answer this.
Game theory was also not attractive because it “undermined” the foundations of classical capitalism, where the main commandment was “my interests are above all.” Concern for achieving a collective goal hinted at a planned economy, which in the fifties during the witch hunts could not be approved. It is curious that game theory would also not have harmed the Soviet economy - experts say that it could have prevented such a global, but completely unjustified project as the construction of the BAM.
Moreover, the mathematician's belief that players make decisions in isolation also turned out to be an abstraction - at least in the field of microeconomics. The seller and buyer, competitors, always have the opportunity to enter into negotiations in order to agree on a joint optimal model of behavior.
Schizophrenia
But let's return to Nash's life path. Thanks to his developments, John Nash ended up in the laboratories of the RAND Corporation, the largest US think tank during the Cold War. Americans now openly admit that game theory and the idea of equilibrium, which implies that the destruction of the enemy is not the best goal, has helped keep the “degree of war” from rising.
After RAND, Nash taught briefly at the Massachusetts Institute of Technology, climbing the academic ladder fairly quickly. There he met Alicia Larde, a promising young physicist, who eventually became his wife.
John and Alicia are newlyweds
Nash had little interest in economics and other problems of the real world, increasingly retreating into the realm of abstract mathematics. He was much more interested in Riemann spaces than in the use of the “Nash equilibrium.” He wrote several brilliant articles devoted to the most complex mathematical problems - differential equations, differential geometry, and so on. He was predicted to have a great future. In 1957, Fortune magazine named Nash an outstanding mathematician of a new generation. Nash's colleagues joked that if Nobel Prizes were awarded to mathematicians, he could become their laureate more than once.
Alicia with her son Joni
It would seem that everything was going great, Alicia was expecting a child, and Nash, at 30 years old, was supposed to become one of the youngest professors - already at Princeton. However, the mathematician reacted to this message in a completely different way than those around him expected. “I cannot take this post,” he said, “the throne of the Emperor of Antarctica awaits me.” Nash was hospitalized with a diagnosis of paranoid schizophrenia.
Hospital named after McLean - a psychiatric hospital where schizophrenic patient J. Nash was kept
Over the next 30 years he did not write a single article. Many believed that Nash was dead. Those more knowledgeable whispered that he had been lobotomized. Nash lost everything - his job, his friends, his family. In real life, Alicia could not bear this burden and divorced John in 1963
However, he had no time for this, he fled to Europe, considered himself the savior of the world, blamed communists and Jews for his troubles, was delirious, was treated and could not leave the world of illusions. Medicines didn't help.
After divorcing his wife, Nash moved into his mother's house. However, she died in 1970. Then Nash called Alicia and asked her to shelter him. To everyone's surprise, she agreed (they just recently got married again). They settled near Princeton. Nash went for walks around the university campus, entering classrooms and leaving mysterious mathematical formulas and messages to nowhere on the boards. For this, the students nicknamed him “Phantom.”
Return
However, in the early 1990s, Nash gradually began to return to the real world. His statements became logical. He began to operate with meaningful mathematical expressions. I started learning to work with a computer and became friends with some students. Doctors attributed this amazing remission to age-related changes in his body. Nash himself says that he recovered because he learned to separate illusion from the real world. This does not mean that he recovered - he learned to live with the disease. “Intellectually I abandoned it,” he wrote in his autobiography.
When the Swedish Academy of Sciences recognized his achievements in the field of game theory, Nash took this news quite calmly, however, a limited range of emotions is a characteristic feature of schizophrenics. He was more interested in the fact that he would finally be able to support his family himself. After all, besides him, Alicia also has their son, a talented young man, who also suffers from schizophrenia.
J. Nash receiving the Nobel Prize with two other laureates: John Harsanyi (far left) and Reinhard Selten (far right)
Nash received the Nobel Prize in 1994 as “a pioneer in the analysis of equilibrium in the theory of non-cooperative games.” After this, Princeton decided to give him an office and gave him the opportunity to teach students. Nash claims that, despite his age and state of health, he is ready to take on new mathematical heights.
John Nash and Paul Krugman (Nobel Prize Winner)
Nash's case lives on and...
Where are Nash's discoveries being applied today?
Having experienced a boom in the seventies and eighties, game theory has taken a strong position in some branches of social knowledge. Experiments in which Nash's team once recorded the behavior of players were regarded as a failure in the early fifties. Today they form the basis of “experimental economics.” “Nash equilibrium” is actively used in the analysis of oligopolies: the behavior of a small number of competitors in a particular market sector.
In addition, in the West, game theory is actively used when issuing licenses for broadcasting or communications: the issuing authority mathematically calculates the most optimal option for frequency distribution.
Likewise, a successful auctioneer determines what information about items can be provided to specific buyers to achieve optimal returns. Game theory is successfully used in law, social psychology, sports and politics. For the latter, a typical example of the existence of a “Nash equilibrium” is the institutionalization of the concept of “opposition.”
However, game theory has found its application not only in the social sciences. Modern evolutionary theory would be impossible without the concept of “Nash equilibrium,” which mathematically explains why wolves never eat all the hares (because otherwise they would starve within a generation) and why animals with defects contribute to the gene pool of their species (because that in this case the species may acquire new useful characteristics).
Now they don't expect any grandiose discoveries from Nash. It seems that this no longer matters, since he managed to do two of the most important things in life: he became a recognized genius in his youth and defeated an incurable disease in old age.
John Nash letter to NSA from 1955
National Security Agency US declassified the amazing letters that famous mathematician John Nash sent them in 1955
John Nash proposed a completely revolutionary idea for those times: to use the theory of computational complexity in cryptography. If you read the letter dated January 18, 1955, it is admirable how prophetic Nash's analysis of computational complexity and cryptographic strength turned out to be. It is on these principles that themodern cryptography . The first work in this area was published only in 1975.
At one time, the authorities never showed any interest in the work of the eccentric mathematics professor. Or, which is also possible, they used Nash’s ideas secretly from him.
In his letter, John Nash develops the ideatheories of communication in secret systems by Claude Shannon 1949), without mentioning it, but goes much further. He proposes that the security of cryptosystems be based on computational complexity—precisely the principle that, in 1975, formed the basis of modern cryptography two decades later. Nash goes on to clearly describe the difference between polynomial time and exponential time, which is the basis of computational complexity theory. This principle is first described in 1965 , although it is discussed in the famousGödel's letter to von Neumann from 1956 , but not in relation to cryptography.
John Nash:
“So a logical way to classify encryption processes is by the way in which the difficulty of computing the key increases as the key length increases. It's exponential at best, and probably at least a relatively small power at worst. ar 2 l ar 3, in substitution ciphers."
“My general hypothesis is as follows: for almost all fairly complex types of encryption, especially where the instructions given by different parts of the key rely on the complex interaction of the instructions with each other to determine their effect on the final result of the encryption, the average complexity of computing the key increases exponentially with key length."
The mathematician is well aware of the importance of his hypothesis for practical cryptography, because the use of new methods will put an end to the eternal “game” of cryptographers and code crackers.
“The importance of this general hypothesis, assuming its truth, is easily seen. This means that it is quite possible to create ciphers that are virtually unbreakable. As the complexity of the cipher increases, the game of cipher-breaking between skilled teams, etc., will become history."
Actually, that's exactly what happened.
It is also interesting that John Nash openly talks about using methods for which he cannot prove the theoretical basis (P = NP). Moreover, he directly says in the letter that he “does not expect its proof,” which is unusual for a mathematician.
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Interesting facts about the film
- The director's place was originally assigned to Robert Redford.
- Tom Cruise could have played John Nash.
- The bed scene between Crowe and Connelly's characters was cut from the final version of the film.
- John Nash (played by Russell Crowe in the film) was brought on set to help the actors play their roles more authentically. Russell Crowe later admitted that he was fascinated by the way John made movements with his hands, and tried to do the same during filming.
- Salma Hayek was invited to play the role of Alicia Lard.
- The Harvard scenes were actually filmed at Manhattan College.
- Two contending producers fought for the right to film the life of John Nash. Brian Grazer won this debate, while Scott Rudin lost.
- Professor Dave Bayer became the film's main consultant and even appeared in the picture. It is his hands that draw complex formulas on the windows.
- Despite the fact that the picture is a kind of biography of the life of John Nash, some details of the life of the great mathematician were deliberately omitted:
- 1) John was married several times;
- 2) in his youth, John was bisexual - he had close relationships with both women and men;
- 3) John had an illegitimate child.
- John Nash did receive the Nobel Prize, but not alone, but together with his colleagues Reinhard Selten and the Hungarian Janos Harsanyi. Moreover, the founder of Game Theory was another Hungarian, Janos Newman. Nash distinguished himself by being able to apply the principles of “game theory” to the world of business.
- Robert Redford was offered the role of director of the film, but he was not satisfied with the filming schedule.
- When Nash first sees Parker, he refers to him as "big brother" (an allusion to Orwell's 1984). Another reference to Orwell occurs later when we see the number on Nash's office door - 101.
- The manuscript that young John Nash shows to his curator, Professor Helinger, is an original copy of an article published in the journal Econometrica under the title “The Transaction Problem.”
- The screenwriter of the film, Akiva Goldsman, had considerable experience in dealing with mentally ill people: when he was a doctor, he personally developed methods for restoring the mental health of children and adults.
- The curator of the film on the mathematical part was Barnard College professor Dave Bayer - it was with his hand that Russell Crowe “displays” tricky formulas on the board. “Intricate formulas”, upon closer examination, are simply a meaningless set of Greek letters, arrows and mathematical symbols. Apparently, the professor was paid his salary in vain.
- Unlike his on-screen counterpart, who was distinguished by his rare devotion to his “other half,” the real John Nash was married several times in his life, and in his early twenties he adopted an illegitimate child.
- In the film, Jennifer Connelly plays Russell Crowe's wife. In real life, her husband is Paul Bettany, who played the role of Crowe's friend.
“I can’t say that I understand this disease,” the scientist said in one of the interviews with the film, “but I don’t think anyone understands it.”
“I didn’t hear any voices at first,” continues Nash, who is among the collegiatewas considered an eccentric mathematician. -The first deviations appeared in me in 1959, but onlyIn the summer of 1964, somewhere around that time, I started hearing voices.”
“In my madness, I thought that I had a very important role to play, and that I had been chosen to convey messages from aliens to people. In the same way, the prophet Mohammed called himself the messenger of Allah. I think this is a standard formulation,” the scientist said.
"Nobel Prizethe award opened up world recognition for me... I became an honorary member of various scientific societies and organizations ... It is clear to me thatnone of this would have happened if it weren't for her", he added self-critically.
John Nash Quotes
- But Newton was right!
- Yes, the old man had sound ideas
“If we all approach the blonde, we will block each other’s paths, and none of us will get her.” We will go to her friends, and they will turn their backs on us, because no one wants to feel second-class. What if neither of us approaches the blonde? ... We will not interfere with each other and will not offend other girls. This is the only way to victory.
- Tell me, is he real?
- Yes.
- Do you see him?
- Yes Yes.
“I’m wary of new people.”
“I don’t know what I should say to have sex with you.” But let's assume that I've already said all this and go straight to it.
I believed in numbers and terms, equations and logic, in common sense... But, having spent my life in such research, I do not know what logic is, what common sense defines... I have come a long way through physics, metaphysics, illusion... and back. And I made the most important of my discoveries - the main discovery of my life: logical foundations can only be revealed in the mysterious equations of love