He [Nash] proved that for a certain very broad class of games of any number of players, at least one equilibrium exists – so long as one allows mixed strategies. ibid.
‘The discussion of manifolds was everywhere,’ said Joseph Koln in 1995, gesturing to the air around him. ‘The precise question that Ambrose asked Nash in the common room one day was the following: It is possible to embed any Riemannian manifold in a Euclidean space?’ ibid.
Mikhail Gromov, the geometer whose book Partial Differential Relations builds on Nash’s work ... ‘Many of us have the power to develop existing ideas. We follow paths prepared by others. But most of us could never produce anything comparable to what Nash produced. It’s like lightning striking. Psychologically the barrier he broke is absolutely fantastic. He has completely changed the perspective on partial differential equations. There has been some tendency in recent decades to move from harmony to chaos. Nash says chaos is just around the corner’. ibid.
What an irony that mathematicians, who live so much more in their minds than most of humanity, should feel so much more trapped by their bodies. An ambitious young mathematician watches the calendar with a sense of trepidation and foreboding, equal to or greater than that of any model, actor, or athlete. The Mathematician’s Apology by G H Hardy sets the standard for all laments of lost youth. ibid.
These mathematicians are very exclusive. They occupy a very high terrain, from which they look down on everyone else. That makes their relationships with women quite problematic. Zipporah Levinson, 1995
All mathematicians live in two different worlds. They live in a crystalline world of perfect platonic forms. An ice palace. But they also live in the common world where things are transient, ambiguous, subject to vicissitudes. Mathematicians go backward and forward from one world to another. They are adults in the crystalline world, infants in the real one. S Cappell, Courant Institute of Mathematics, 1998
The natural question is – the question Fermat raised – is supposing you changed from squares, supposing you replace the 2 by 3, by 4, by 5, by 6, by any whole number, and Fermat said simply that you’ll never find any solutions however far you look. Professor Andrew Wiles
I think I’m about to prove Fermat’s last theorem. Andrew Wiles
8It was so indescribably beautiful. It was so simple and so elegant. Andrew Wiles
I used to love mathematics for its own sake, and I still do, because it allows for no hypocrisy and no vagueness, my two bêtes noire. Stendhal aka Henri Beyle
Mathematics was the discipline that endowed the Greeks with ultimate truth. Stephen Hawking’s Universe: Seeing is Believing
Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation. In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means. Maurice Allais, La formation scientifique, Une communication du Prix Nobel d’économie
It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances. Hannah Arendt, The Life of the Mind, 1971
The study of mathematics is the indispensable basis for all intellectual and spiritual progress. F M Cornford, 1874-1943, summarizing central tenet of Pythagoreanism
Mathematics is not something that you find lying around in your backyard. It’s produced by the human mind. Yet if we ask where mathematics works best, it is in areas like particle physics and astrophysics, areas of fundamental science that are very, very far removed from everyday affairs. Paul Davies, cited ‘How Unique You Are! Is There a Creator Who Cares About You?’ published Jehovah’s Witnesses
Mathematics is the bold luxury of pure reason, one of the few that remain today. Robert Musil, Mathematical Man
His greatest contribution to science was his discovery of a set of universal laws to describe the cosmos. Great Scientists: Sir Isaac Newton, Channel 5 2005
He continued his work on optics and even devised the Theorem of Calculus. ibid.
Principia Mathematica: The mathematical principles of science. ibid.
One of the world’s greatest unsung heroes – ‘Boole is the father of information technology’ … and he would do this against a backdrop of Ireland’s darkest days. The Genius of George Boole, RTE 2015
Nearly two hundred years ago George Boole made a revolutionary discovery … Boole argued that almost every value or question could be reduced to either true or false. A simplification of our world as a basic statement. ibid.
He dedicates every breathing moment to mathematics and in particular to a branch of maths known as calculus. ibid.
And then Boole develops a new branch of mathematics – Invariant Theory. ibid.
To prove the existence of God … He seeks to prove this by applying to the Bible the process of logical analysis. ibid.
Boole publishes his masterpiece – The Laws of Thought. ibid.
The invention of deliberately oversimplified theories is one of the major techniques of science, particularly of the ‘exact’ sciences, which make extensive use of mathematical analysis. If a biophysicist can carefully employ simplified models of the cell and the cosmologist simplified models of the universe then we can reasonably expect that simplified games may prove to be useful models for more complicated conflicts. John Williams, The Complete Strategyst
There are two kinds of mathematical contributions: work that’s important to the history of mathematics and work that’s simply a triumph of the human spirit. Paul J Cohen, 1996
Mathematics is a young man’s game. Norbert Wiener
A military think-tank called the Rand Corporation: and the strategists at Rand used Game Theory to create mathematical models that predicted how the Soviets would behave. Adam Curtis, The Trap I: Fuck You, Buddy, BBC 2007
Underlying game theory was a dark vision of human beings who were driven only by self-interest. ibid.
Nash … was notorious at Rand for inventing a series of cruel games. ibid.
In the early ’60s Laing set up a psychiatric practice in Harley Street in London. He offered radical new treatments for schizophrenia and quickly became a media celebrity. But his research into the causes of schizophrenia convinced him that a much wider range of human problems were caused by the pressure-cooker of family life. Laing decided to investigate how power and control were exercised within the world of normal families. And to do this he would use the techniques of game theory. ibid.
Beneath the surface of the world are the rules of science. But beneath them there is a far deeper set of rules, a matrix of pure mathematics which explains the nature of the rules of science and how it is we can understand them in the first place. Dangerous Knowledge, BBC 2007
Georg Cantor: how big is infinity? ibid.
A whole hierarchy of different infinities. ibid.
After his breakdown everything about Cantor is transformed. ibid.
Like Cantor, Boltzmann’s ideas were out of step with his times. Cantor had undermined the ideal of a timeless and perfect logic in maths; Boltzmann’s formula and his destiny was to undermine the ideal of a timeless order in physics. Together their ideas were part of a general undermining of certainty in the wider world outside maths and physics. ibid.
Boltzmann in essence captured mortality in an equation. ibid.
Kurt Godel and the work that he did here brought that dream of finding the perfect system of reasoning and logic crashing down. Dangerous Knowledge II
Godel’s Incompleteness Theorem: ‘The obvious logic of Godel’s work is that Logic is a failure.’ ibid.
In 1934 he had his first breakdown. ibid.
He [Turing] is also the man who made Godel’s already devastating incompletely theorem even worse. ibid.