John Nash, one of the greatest scientists of the last century, passed away on May 23rd of 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 Mathematics in 2015, which is considered the equivalent of the Nobel in mathematics. 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.
Nash published his first academic paper, with his father, when he was 17. He completed his Ph.D. at Princeton University in the US, an institution populated by geniuses such as Einstein. A famous anecdote of Nash’s career is 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 will gain 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 realized that in his letter Nash had 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 of why he received the Abel Prize.
He made all his major scientific contributions before becoming 30 years old.
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 woman once knowing of her pregnancy. A couple of years later, he was accused of indecent exposure in a public bathroom, to what 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 Antartica. 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 the 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 hospitalized and resigned his position as a professor. He went through several treatments, including insulin coma therapy, for 11 years.
After 1970, he rejected to continue any 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. The 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 has been dead, and many thought this was indeed true. Around 1980 his disease became mild, which 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 2 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 Norway’s 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 tell us what agents should do in those interactions if they were rational. We are interested in equilibrium situations, i.e. those which are stable and no agent wants to deviate from it.
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 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, predictions of animals’ behaviour, among many others.
Geometry, Cryptography and Complexity
There is no easy way to explain Nash work on geometry and differential equations, but we will try.
An embedding in mathematics is a particular relationship between two objects which 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. We mentioned that he anticipated the P=NP problem, which ask whether any problem that is easy to be verified by a computer can also be easily solved. The P=NP problem, as we mentioned, 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.
To summarize, Nash managed to become one of the greatest scientists of 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 would not have had to stop due to his schizophrenia problems. I conclude with a quote from John Nash itself:
“I can see there’s a connection between not following normal thinking and doing creative thinking. I would not have had good scientific ideas if I had thought more normally.”
Sources to Find Out 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.
Nasar, Sylvia (1998), “A Beautiful Mind”, Simon and Schuster.
Nisan, Noam (2012), “John Nash’s Letter to the NSA”, Turing’s Invisible Hand.
Josue is a Ph.D. researcher at the University of Glasgow and an associated member of the UK Institute of Mathematics and its Applications. He holds a M.Sc. from the Institute for Advanced Studies in Vienna.