John Forbes Nash – Death of a beautiful mind

The mathematical and economics communities mourn the death of John Nash, for both one of their respective greatest minds. A life that had begun in the midst of the great crises of the 20th century and which had born more intellectual highs and lows than hardly any other found its tragic end in an event as mundane as a car accident. The fact that Nash had been on his way home from receiving one of the world’s most prestigious mathematical awards, the Abel Prize, increases yet the tragic irony of his death.

Now, the death of a mathematician, even one as brilliant as John Nash, will normally hardly ever find a larger space in the attention of the public. That this is different in the case of Nash finds its reason, no doubt, in the many years he suffered from atragic illness, which was successfully featured (and partly disfigured) by Hollywood in the four Oscar winning movie „A Beautiful Mind“. Nash’s long-time struggle with paranoid schizophrenia which next to the genius of his mathematical creativity shaped his life was material sufficient to inspire millions of people in the movietheatres. The combination of suffering and genius can still serve this goal today, almost 15 years later, as illustrated by the recently released movies of Alan Turing’s and Stephen Hawking’s lives. Another biography that could provide stuff for a similar filmis provided by the self-reliant mathematician Kurt Gödel who among other conditions suffered from permanent poisoning fears.

However, brilliant mathematician or theoretical physicists of course do not necessarily have to suffer from mental illness in order to claim superb achievements and become famous, as shown, inter alia, by the examples of Albert Einstein and David Hilbert, one the most significant theoretical physicist, the other the greatest mathematician of the 20th century,whoboth remained healthy throughout most of their lives (however, bothhad sonsthat suffered from severe mental illness; as, by the way, did the son of John Nash). But the line between genius and madness is known to be fine.

Now, the math works of Nash may sound like madness to most people, ashe dealt with such abstract things as „real algebraic manifolds“ and „algebraic varieties”. De facto, they represent one of the most brilliant mathematical achievements of the 20th century. Among others Nash published a solution to the regularity problem of a certain class of partial differential equations which provided ananswer to one of the biggest open problems of mathematics and which Hilbert had set as the 19th problem on his famous list more than 50 years earlier.

A wider – but still quite academic – audience beyond mathematics, however, knowsJohn Nash for his work in the field of economic game theory. In his doctoral thesis from 1950 he introduced the notion of the so-called „Nash equilibrium“ showing that in situation of conflicts and competitions, in economic jargon, in „non-zero-sum games“, there exists a status in which none of the players can improve his or her situation by changing his choice. This also holds, as Nash demonstrated, for situations with more than two players. In the above film his theory, for which Nash was awarded the Nobel Prize for Economics in 1994, is illustrated as follows (not without some sexist tones): A group of testosterone driven young menfinds at the other end of a bar a group of women, one of which (a blonde) is more beautiful than the others (brunette). Of course, not all men can conquer the prettiest woman. If they now all scramble for her, most likely nobody will get her. Furthermore, the other women will also walk away from them as none of them wants to be second choice. Therefore, it is reasonable to rely on the next best choice, i.e. a brunette,as this will end up making all of them winners, so the representation of Nash’s theory goes.

Now this insight seems abundantly banal. But that is how dull a Hollywood scene todayneeds to be in order to please a wide audience. Certainly, economists did not need 174 years since Adam Smith and a brilliant mathematician to realize that it can be desirable to engage with a brunette woman. And it probably needs a little more (some may say „however not so much“) than the insight of „tit for tat“ and „your lies will soon find you out“ to receive the Nobel Prize in economics.

Mathematically, the Nash equilibrium is a little richer than what the movie example suggests, but it is still sufficiently concrete and figurative to expose students of economics to it. Unfortunately the trivial staging of Nash for it to be to the liking of Hollywood as well asalmost exclusive the emphasis on Nash’s achievements in the field of economic theory which is often more widely studied and more easily accessiblemake his great achievements in pure mathematicstoo little known. It is for these that he was honored with the mentioned Abel Prize in 2015, which after the Fields Medal (for which Nash had already been proposed in the 1950s) represents the second most important prize for mathematical excellence and is often referred to as the „Nobel Prize of Mathematics“. Were his mathematical theorems in the field of real algebraic manifolds, differential topology and partial differential equations more widely known, a more general public would obtain an idea of ​​how crazy mathematics can sound while at the same time beingpermeated by sublime brilliance. John Nash has illustrated as few other have that mathematical insights are of the finest products of the human mind. The worldhas lost a truly “beautiful mind”.