Algebra, symbol of the West's success

PUBLISHED : Sunday, 12 February, 2012, 12:00am
UPDATED : Sunday, 12 February, 2012, 12:00am


The history and philosophy of science is an unusually dull and unpromising discipline. Yet it managed to produce at least two illuminati in the last century: Karl Popper and Thomas Kuhn.

But there is another thinker in the field whose depth and brilliance at least matched Popper and Kuhn's. Jacob Klein has been a cult hero in select academic circles but is rarely known outside of them. This is unfortunate. Though his work deals with obscure subjects in ancient Greek mathematics, astronomy and philosophy, it shines a light on how and why what people nowadays call 'The West' came to be in its best modern incarnations: its scientific outlook, its rationality and self-critiques.

These 'values', at least to me, have been the best the West has had to offer the world; everything else that Western powers have hitherto forced upon the rest of us is of questionable value.

At a time when the West seems to have lost its way and is seen by many, whether with alarm or with glee, as being in decline relative to the East, Klein should be a profound spiritual guide. Hopefully, whatever economic or military decline the West is suffering from does not mean the decline or disappearance of our respect for science, rationality and reflective self-criticism.

To study Klein, you will have to either read his seminal work, Greek Mathematical Thought and the Origin of Algebra or spend four years studying the Great Books curriculum, an undergraduate programme based on the Western canon co-founded by Klein at St John's College in Maryland. Here, let us just deal with the book.

Among Klein's followers and cognoscenti, the book, with its more-than-a-mouthful title, is simply known as 'the maths book'. Klein unfortunately did not have a genius publisher like the one who told Isaiah Berlin to forget about calling his long-winded essay 'On Tolstoy's Historical Scepticism', and retitled it The Hedgehog and the Fox; it went on to become one of the most celebrated literary works of the last century.

'The maths book' is a rich treasure trove that you can easily read and reread over a lifetime, if you are into that sort of thing. I first studied it in college, when it was taught by Eva Brann, its English translator, a former dean of St John's College and one of America's truly great living scholars.

At the core of Western rationality is science, and at the core of science is maths. And at the foundation of maths is the concept of numbers or its study, arithmetic.

Klein shows how, among ancient Greeks, a number always meant 'a concrete heap or assemblage of real things'. In other words, it always has a concrete reference. Klein's book charts the conceptual rupture from this 'concrete' Greek maths which marked the birth of modern algebra.

A political historian might argue that the Peace of Westphalia marked the dawn of the modern West, with its new concept of sovereignty. In a similar way, a science historian like Klein might argue Western modernity stemmed from the birth of modern algebra (made possible by the Arabs' preservation of the original Greek arithmetical works, such as that of Diophantus) as conceived by such geniuses as the Frenchmen Francois Vieta and Rene Descartes in the 16th and 17th centuries.

Klein bills as revolutionary the moment when Vieta conceived of symbols like a, b or x, y as being devoid of concrete references, but abstract substitutes for numbers that may refer to nothing or anything. Modern algebra established rules for the symbols' manipulation and is thus, as Klein writes, 'twice removed from the [physical] world', that is first as abstract symbols and then abstract rules for the manipulation of those symbols.

For that, Klein calls Vieta the true founder of modern mathematics. And the triumph of Western science, one might add, is precisely its ever increasing power of abstractness or remoteness, as exemplified by modern maths, from the purely physical. Modern Western science, as Klein conceived it, was rooted less in the practical than in what the ancient Greeks called theoria.

And here lies a profound historical paradox for the Chinese, who took a long time to understand. Leaders from Lin Zexu, the Qing mandarin whose anti-opium campaign triggered the first opium war, to Mao Zedong have asked why Western technology (by which they usually meant weapons) was so powerful yet their culture supposedly inferior. But scientific rationality is vital to Western culture.

Mao fancied himself an astute political realist when he said: 'Every communist must grasp the truth: political power grows out of the barrel of a gun.'

But for guns, bullets and cannons to be accurate and powerful, there first had to be a ballistics and artillery revolution in the 18th century. And it could never have taken place in China, despite gunpowder and explosives being supposedly first invented there.

This revolution was launched not by soldiers but by great mathematicians: Benjamin Robins in Britain, Leonhard Euler in Frederick the Great's Prussia and Pierre-Simon Laplace in France, applying Newtonian mechanics, differential equations and Boyle's ideal gas law. Mao would never have understood, but Klein's book might help us appreciate the true source of Western scientific genius.

Alex Lo edits the science page