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A Beautiful Mind: The Mathematician John Forbes Nash, Jr.

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John Nash in 2011.

John Nash in 2011.

Who Was John Nash?

At an early age his family noticed young John Nash had a gift for mathematics. By age 19 he had already completed a BS and an MS in mathematics. During his Ph.D. work at Princeton, Nash made significant progress in the advancement of a branch of mathematics known as “game theory.” Though the term game theory may seem trivial, it is not. The application of game theory in areas such as economics have revolutionized the fields of industrial organization, monetary policy, and international trade.

By the age of 30, Nash seemed to have it all, a beautiful wife, a Ph.D. from the prestigious Princeton University, and Fortune magazine had named him one of the rising stars in mathematics. Just a year later, his world fell apart as signs of the crippling mental illness, paranoid schizophrenia, began to appear. He spent the next three decades in and out of hospitals battling the disease, ultimately regaining his sanity and a place in the annals of mathematics.

In 1998, the story of John Nash’s brilliant rise as a star mathematician and his descent into mental illness was brought to the attention of the public in the biography A Beautiful Mind and the subsequent blockbuster film by the same title. This article is the story of a man who rose to the top ranks of his field just to be brought to poverty and isolation in his own personal hell by schizophrenia, then to recover and be awarded for his work with a Nobel Prize.

Early Life

Born in Bluefield, West Virginia, in the heart of coal country on June 13, 1928, John was the son of an electrical engineer who worked for the Appalachian Electric Power Company. His mother Margaret—everyone called her Virginia—was a former school teacher and native of Bluefield. He also had a younger sister, Martha, who was born about two and a half years after him.

Nash went to public school for most of his life, and he supplemented his formal education through books his parents and grandparents gave him. As he got older and his parents began to realize his potential, they helped him further his education by enrolling him in a few classes at their local community college when he was in the final year of high school. Nash studied advanced mathematics at college during that year.

Undergraduate School

After completing high school, he received a full scholarship to attend the Carnegie Institute of Technology in Pittsburgh, majoring in chemical engineering. He quickly switched his major to chemistry after a few months at college, before settling on mathematics after having a few important conversations with his favorite teacher, John Lighton Synge. Nash’s grasp of mathematics allowed him to graduate college with both his Bachelor and Master of Science at the age of 19.

Game Theory

A branch of mathematics that models human, animal, and computer behavior in interactions that involve conflict and cooperation between intelligent rational decision makers. Game theory has found applications in economics, political science, psychology, computer science, biology, and board games.

Graduate School at Princeton University

After his undergraduate studies were complete, Nash attended Princeton University to pursue a Ph.D. in mathematics. Even though he was such a young man at the time, all his professors and advisers could see his potential. This is evident from the letter of recommendation written for Nash by his adviser Richard Duffin, who called Nash a “mathematical genius.”

Picking a college for his graduate studies was a tough decision for Nash, because he gained acceptance to both of the top schools of Princeton and Harvard. However, the chair of the mathematics department at Princeton desperately wanted Nash to attend his school, offering him the John S. Kennedy scholarship, which convinced Nash that Princeton was more interested in having him as a student. It was at Princeton where Nash started to work on the advancement of a branch of mathematics known as “game theory.”

Even though the vast majority of Nash’s work on modern day game theory took place at Princeton, the origins of his interest in the subject go back to Carnegie, where he took an international economics course and began to learn about the various choices and consequences present in economic theories. He published a paper at Carnegie called “The Bargaining Problem,” which eventually made its way into the Econonometrica publication. He continued to develop this interest at Princeton, where he studied non-cooperative games and the theory behind those games.

When Nash earned his Ph.D. from Princeton in 1950, it was largely based on his 28-page dissertation on the topic of non-cooperative games. These games are defined as a game where players make decisions independently. In other words, players can cooperate, but the cooperation is self-enforced. His paper contained the framework and theory for what we know as the Nash Equilibrium, which is a specific solution-concept for non-cooperative games. In the Nash Equilibrium, it is stated that within a non-cooperative game involving two or more people, where the players know the equilibrium strategies of everyone else who is competing, no player can gain anything by changing their strategy. When he first came up with the theorem, he had no idea of its wide-ranging uses and how it would evolve with modern day economics.

Nash was a loner at Princeton and had few friends. He had the odd behavior of spending hours riding his bike in the commons area making figure eights while whistling Bach. A fellow graduate student, Lloyd Shapley, who would also later receive a Nobel Prize, said of Nash: “What redeemed him was a clear, logical, beautiful mind.”

John Nash at graduation from Princeton in 1950.

John Nash at graduation from Princeton in 1950.

The Prisoner’s Dilemma

An example of the type of problems Nash’s theory explains is in a simplified example of a non-cooperative game known as “the prisoner’s dilemma.” The example is centered around the premise of two prisoners who are both offered the possibility of a lighter jail sentence if they betray the other criminal. But neither prisoner has a chance to communicate with the other before they make their decision.

Since the prosecutors in the example do not have enough evidence to convict both prisoners of the crime, they encourage each of them to betray the other. There are a few different scenarios in the game, depending on the decisions made by both prisoners. If both betray each other, each of them is going to serve five years in jail. If prisoner A betrays B, but prisoner B remains silent, A gets a one year sentence and prisoner B gets an eight-year sentence, and vice versa. If they both remain silent, they will each serve two years in jail.

In this example, the Nash equilibrium only exists at one point, when they both decide to betray each other. Why is this the Nash equilibrium, even though the scenario where both players remain silent is the optimal choice? Because it is the only point where neither player can gain anything from changing their decision, after learning about the first player’s choice.

Even though the scenario where they both stay silent is optimal in the game, if either prisoner decided to betray the other, based on learning the other prisoner was silent, they would serve one year in jail. The promise of less jail time would prompt either player to change their mind after learning of the other’s choice, which is why this scenario is not a Nash equilibrium. The example of the Nash Equilibrium as illustrated in the prisoner’s dilemma can be applied to various real-world economic problems.

Diagram of the Prisoner’s Dilemma.

Diagram of the Prisoner’s Dilemma.

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Real Algebraic Geometry

Aside from the Nash Equilibrium, Nash also made significant contributions to mathematics in the field of real algebraic geometry. He came up with the Nash embedding theorem, which highlights the fact that each abstract Riemannian manifold is isometrically realized as a submanifold of the Euclidean space. His work on the singularity theory also gained him a great deal of acclaim from fellow mathematicians.

There is an interesting story in Nash’s biography, A Beautiful Mind, which talks about his process for working on the proof of Hilbert’s nineteenth problem. When working on this theorem, which deals with elliptic partial differential equations, Nash discovered that Ennio de Giorgi, a fellow mathematician, had already published a proof of the same theorem a few months before Nash could finish his own proof. Even though Nash had used a different method for coming to the same conclusion, he was understandably disappointed.

The two men made each other’s acquaintance in New York in 1956 and the encounter was said to be amicable. Mathematicians at the time speculated that if either Giorgi or Nash had never completed the proof, the other man would have won a Fields Medal for their work. The Fields Medal is the equivalent of the Nobel Prize, except it is for mathematicians only and is awarded to those under 40 years of age and is given out every four years.

His Mental Illness Begins to Appear

After leaving Princeton, Nash took up a position on the mathematics faculty at the Massachusetts Institute of Technology (MIT). There he met his first serious girlfriend, a nurse named Eleanor Stier, when he was admitted to a local clinic as a patient. However, Nash chose not to remain with Stier when he discovered that she was pregnant, because he was angered by the pregnancy and the fact that her social status did not match his, which made her a poor match in his eyes. Nash would remain estranged from their son John David Stier until later in his life.

Nash started to develop signs of a serious mental illness in late 1958. His erratic behavior was becoming apparent to his wife, Alicia Larde, and his co-workers at MIT. Symptoms of the disease became publicly obvious when he delivered a lecture at Columbia University in 1959. Nash lectured on the Riemann hypothesis proof that he had completed, but his lecture was completely incoherent. The audience watching him speak could clearly see that something was wrong with this brilliant man. After the incident, Nash was admitted to a hospital for observation and testing. He was eventually diagnosed as a paranoid schizophrenic, which explained his intense paranoia and periodically erratic behavior.

Life as a Paranoid Schizophrenic

After his hospitalization, Nash resigned from MIT and stayed home with Alicia. The seclusion of home life helped him understand and deal with some of the psychotic delusions and episodes he was experiencing. During two particularly intense schizophrenic episodes, Nash traveled to Europe. Driven by paranoia and voices in his head, he sought to renounce his US citizenship and become a citizen of the world. With the help of government officials, John was safely returned to the United States after both episodes.

Between 1961 and 1970, John was hospitalized several times for his schizophrenia. During one stay at the Trenton State Hospital, he was subjected to insulin therapy, which was both very dangerous and provided little relief for his condition. Most of the treatments Nash received during those years did not have much of an impact on his mental state, which prompted Nash to vow in 1970 that he would never return to a hospital.

Alicia Larde Nash in 2006.

Alicia Larde Nash in 2006.

Recovery From Mental Illness

Then, in the early 1980s, John’s mental illness began to recede. The change was slow, but over time his lucid thoughts returned. Nash would later write of his condition, “Thus further time passed. Then gradually I began to intellectually reject some of the delusionally influenced lines of thinking which had been characteristic of my orientation. This began, most recognizably, with the rejection of politically-oriented thinking as essentially a hopeless waste of intellectual effort.”

Nash’s later writings confirmed that he would only sporadically take his prescribed medications. He believed that the benefits of psychotropic drugs were vastly overrated, and he felt the side effects of these drugs were too much for him to handle.

While Nash was slowly recovering from his mental illness, his work on non-cooperative games was beginning to be used in economics, mathematics, biology, and political science. Outside the small circle of the Princeton mathematics department, most thought the important contributor to game theory was dead. Little did he realize that researchers around the world were starting to put to work the theory he developed in graduate school.

Awards and Recognition

John Nash won many awards and accolades over the years for his work in mathematics. He received the John von Neumann Theory Prize in 1978 for discovering non-cooperative equilibria, which we call Nash equilibria. The big prize came in 1994 when John F. Nash, Jr., received the SverigesRiksbank Prize in Economic Sciences in Memory of Alfred Nobel, alongside German professor Reinhard Selten and John C. Harsanyi from the University of California, Berkeley. This prestigious award was presented to the three men for their research on game theory. Although certain groups questioned the Nobel Committee for granting the award to a person who suffered from severe mental illness, Selten and Harsanyi defended Nash. The three winners split a prize worth $930,000.00. The prize citation recognized the “pioneering analysis” of all three Nobel Laureates in the field of game theory. Assar Lindbeck, the former chair of the committee for the Nobel Prize in Economic Sciences, remarked about Nash, “We lifted him into daylight...We resurrected him in a way.”

Advertisement for the movie A Beautiful Mind.

Advertisement for the movie A Beautiful Mind.

A Beautiful Mind

The biography of Nash’s life, A Beautiful Mind written by Sylvia Nasar, brought his amazing story to the public’s attention and was the basis of a 2001 blockbuster movie by the same name. The movie starred Russell Crowe as John Nash and Jennifer Connelly as Alicia. The film was directed by Ron Howard and won four Oscars, including the Academy Award for Best Picture.

A Beautiful Mind (2001) Official Trailer - Russell Crowe Movie HD

Death

John Nash and his wife Alicia were both killed in a car accident on May 23, 2015, on their return trip from Norway. The Abel Prize recognized Nash and his collaborator Louis Nirenberg for their work in geometry and partial differential equations. The couple had traveled to Norway so John could receive the Abel Prize from the King of Norway. They had just landed at the Newark airport and were traveling home when their taxi driver struck a guardrail, with the accident resulting in both Nash and his wife being ejected from the car. Princeton University President Christopher Eisgruber called John and Alicia Nash special members of the community, lamenting, “We are stunned and saddened by news of the untimely passing of John Nash and his wife and great champion, Alicia.”

References

  • John Nash, Jr. – The Life and Legacy of One of America’s Most Influential Mathematicians. Charles Rivers Editors, 2015.
  • Nasar, Sylvia. A Beautiful Mind. Simon & Schuster, 2011.
  • John Nash (Author), Harold William Kuhn (Editor), Sylvia Nasar. The Essential John Nash. Princeton University Press, 2001.
  • Venes, Donald (Editor) Taber’s Cyclopedic Medical Dictionary, Edition 20. Philadelphia: F.A Davis Company, 2001.
  • West, Doug. The Mathematician John Forbes Nash Jr. – A Short Biography. C&D Publications, 2016.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

Comments

Mary Norton from Ontario, Canada on October 23, 2021:

I have not watched that movie but will do now that you have given me this interesting background. What a life.

Doug West (author) from Missouri on October 22, 2021:

Fran, Heidi:

Thanks for the comments. John Nash suffered greatly but in the end he was redeemed. It was a good movie too!

Doug West (author) from Missouri on October 22, 2021:

Bhattuc: Thanks for the comment.

Umesh Chandra Bhatt from Kharghar, Navi Mumbai, India on October 21, 2021:

Very interesting and elaborate article. Nice work.

Heidi Thorne from Chicago Area on October 21, 2021:

I remember reading the story about the passing of Nash and his wife. What a sad ending! But his life was so one of opposites, a brilliant mind devastated by mental illness. Though his story is relatively recent history, I wonder whether medical advances we now have would have helped curb the illnesses that plagued his life and career. We'll never know. But thank you for sharing the highlights and challenges of this mathematical genius!

fran rooks from Toledo, Ohio on October 21, 2021:

Doug, what an interesting article yet sad too. Nash had a rare mind and then through his mental illness rebounded. So sad he and his wife died tragically. Thank you for your article.

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