John Nash: an eccentric genius

John Nash: an eccentric genius

Current PhD student, Josue Ortega, shares his thoughts on an inspiring Economist, Nobel Prize winner John Nash, who passed away this year.

John Nash, one of the greatest scientists of the last century, passed away on May 23rd 2015. He became famous after the movie A Beautiful Mind depicted his life. However, his scientific contributions go beyond fiction and have laid the foundation for some of the most innovative research in mathematics, economics, and computer science.

‌Nash received the Nobel Prize in Economics in 1994, and the Abel Prize in 2015, which is considered the mathematics equivalent of the Nobel prize.  This puts him on the same ground as Marie Curie - the only person who has received two Nobel awards in different areas of science.

This article provides a short biography of this remarkable man and describes in plain words his most sound contributions to three different fields: game theory, geometry, and cryptography.

The Genius

Nash published his first academic paper, with his father, when he was 17. He completed his PhD at Princeton University in the US, a place surrounded by great minds, including Albert Einstein. It is a famous anecdote that his professor’s recommendation for the doctorate was a single sentence: “this man is a genius”.

He graduated after only 3 years, at the age of 22, having already written the work in game theory that would earn him the Nobel Prize and establish him as the central figure in game theory.  His doctoral thesis was only 27 double-spaced pages.

After finishing his studies, he took positions as a professor at MIT and a researcher at the RAND Corporation. In this period, Nash wrote a letter to the National Security Agency of the US (which was kept confidential for 57 years) proposing the P=NP problem. This is the most important open question in computer science and has a reward of a million US dollars for its solver. He also gave a formal introduction to modern cryptography. It was not until 2012 that scientists realised that in his letter, Nash anticipated many discoveries in computer science by at least a decade.

In 1954 and 1956, Nash wrote his embedding theorem papers, which are considered his most prestigious work in mathematics and are the reason he received the Abel Prize.  He made all his major scientific contributions before the age of 30.

The Eccentric

From his early days at Princeton, Nash was known as an odd and arrogant person, but those were no strange features of a great mathematician.

In 1951, he had a brief romance with a nurse, from which he had his first child. However, he left the girl after knowing of her pregnancy. A couple of years later, he was accused of indecent exposure in a public bathroom, to which he argued that he was merely observing behavioural characteristics. The charges were dropped, but the scandal cost him his job at RAND.

The first signs of schizophrenia and mental disorders started in 1958. He rejected a position as a professor at the University of Chicago alleging his forthcoming announcement as Emperor of Antarctica. A year later, when giving a seminar, he could not articulate his ideas, and his disease was evident.  He had delusional ideas and heard imaginary voices. He wrote to the Pope, the UN, and the FBI to explain his concerns about aliens and communists, which he could distinguish because they were wearing red ties. He believed he was receiving messages from outer space via the New York Times, and during a trip to Europe opened many bank accounts under fake names, until he was deported back to the US. He was hospitalised and resigned his position as a professor.  He went through several treatments, including insulin coma therapy, for 11 years.

After 1970, he rejected continued medical treatment. He left all scientific efforts for more than 25 years and remained unemployed, supported only by his lifetime companion Alicia, whom he married in 1957, divorced in 1963, and remarried in 2001. They died together in a car accident on their way back home, shortly after receiving the Abel prize in Norway.

He disappeared from the scientific community, telling other scholars to use his work as if he was dead, and many thought this was the case.  Around 1980 his disease became mild, what he attributed to the natural process of aging.  He returned to Princeton University, where he walked around writing mathematical formulas on boards.

After his retirement from academia, he lived in severe austerity. Once, after being asked how his life changed after receiving the Nobel Prize, he said that after receiving the award he was able to afford a two dollar Starbucks coffee.

It is a well-believed rumour that his mental illness was a serious concern to the Nobel committee as they doubted that he would behave appropriately in front of the King. After the award, a reception was held in his honour at Princeton, where his only words were “the cookies are better than usual today”.

Nash and Game Theory

We call a game any interaction between rational agents in which what happens to each of the agents depends on what everybody else does.  Examples include people playing chess, firms determining prices, birds choosing mating partners, or computers selecting downloading channels.

Game theory is a field that tells us what agents should do in those interactions if they were rational. We are interested in equilibrium situations, i.e. those that are stable, from which no agent wants to deviate.

Before Nash, we did not know whether these equilibria existed, but he proved that any finite game has at least one equilibrium point. This point is called Nash Equilibrium, and it is taught in any economics degree. Nash’s work triggered a revolution and expansion of the field and developed many applications. For example, predicting how much buyers will bid on Google ad auctions, or the construction of stable algorithms to match students and schools in the US and Europe. Other applications include terrorism prevention, kidney transplantation and predictions of animals’ behaviour, among many others.

Geometry, Cryptography and Complexity

There is no easy way to explain Nash’s work on geometry and differential equations. 

An embedding in mathematics is a particular relationship between two objects that share some, usually not so evident, properties. What Nash did was to show that an embedding could be constructed between two mathematical objects (a Riemann manifold and a Euclidean space), in a way that distances are preserved.  He did this by brilliantly solving a system of partial differential equations. A differential equation is a relationship between a variable, say the speed of an object, and its derivative or rate of change, which in our example is the acceleration.

His hidden work in cryptography and complexity was all contained in a letter that Nash sent to the US government, proposing an encrypting device that was ultimately rejected.  He anticipated the P=NP problem, which asks whether any problem that is easy to be verified by a computer can also be easily solved. The P=NP problem remains unsolved.

Based on his conjectures about P not equal to NP, he proposed encryption schemes that are hard to crack, the fundamental basis of modern cryptography. Cryptography is the discipline that studies secure information transmission, i.e. communication that is inaccessible to third parties, and its applications include the design of credit cards or computer passwords.

Nash managed to become one of the greatest scientists in history by completing 14 influential papers before he reached the age of 30. One cannot stop wondering how much more he would have done if he was not stopped by his battle with mental health issues.

To learn more:

Erica Goode (2015), “John Nash dies at 86”, The New York Times Obituary.

John Milnor (1998), “John Nash and A Beautiful Mind”, Notices of the AMS, 25 (10), pp. 1329-1332.

Sylvia, Nasar (1998),  A Beautiful Mind, Simon and Schuster.

Noam, Nisan (2012), “John Nash’s Letter to the NSA”, Turing’s Invisible Hand.