Maths can be fascinating
The Universe in Zero Words: The Story of Mathematics as told through Equations, Dana Mackenzie, Princeton University Press, Pp 224, £ 19.95
In this book, the author tries to lift the veil of mystery surrounding mathematics where ‘equations, ‘formula’ and ‘identity’, ‘axiom’, ‘theorem’, ‘hypothesis’ and ‘conjecture’ are frequently used. In mathematics, a fact can never be proven by empirical evidence, plausibility or a statistical test. This is a rule that distinguishes mathematics from the empirical sciences including physics, biology and chemistry.
iSince the arrival of Arabic numerals in Europe through to the present day, the author reveals mathematics’ hidden ‘masterpieces’ and bridges the cultural gap between those who speak the language of equations and those who, up till now, haven’t.
The book tells the fascinating and remarkably human histories of the 24 most surprising, concise, consequential and universal equations ranging from ‘1+1=2’ to the Black-Scholes equation for financial derivatives. Along the way, Mackenzie shares the stories behind what each equation means, who discovered it and how it has affected our lives. He recounts fun narratives such as the mathematical ‘duel’ between Nobel-Prize-winning physicist, Richard Feynman and an abacus salesman, explains the complicated ideas like infinity comes in different sizes and shares his thoughts on how new equations could change the future.
The book says that zero seemed to have a divine origin in mathematics. In ancient Mesopotamia, it was a gift from Nisaba, the patron goddess of scribes. “Nisaba, the woman radiant with joy, the true woman, the scribe, the lady who knows everything, guides your fingers on the clay,” wrote a scribe in the 20th century BC. “Nisaba generously bestowed upon you the reassuring rod, the surveyor’s gleaming line, the yardstick, and the tablets which confer wisdom.” On Babylonia’s mathematical tablets, the solution to a problem was never complete until the solver wrote, “Praise Nisaba!” at the end.
According to the Chinese, the originator of mathematics was Fu Xi, the legendary first emperor of China. He has been often depicted as holding a carpenter’s square. “Fu Xi created the eight trigrams in remote antiquity to communicate the virtues of the gods,” wrote the 3rd-century mathematician, Liu Hui. In addition he said, Fu Xi “invented the nine-nines algorithm to coordinate the variations in the haeograms.” The ‘trigrams’ and ‘hexagrams’ are the basic units of Chinese calligraphy; thus, in a sense, Fu Xi is being credited with the invention of writing, while the ‘nine-nines algorithm’ means the multiplication table. Thus mathematics was not only divinely inspired but was invented at the same time as the written language.
This book shows why equations have something to say about mathematics, science and society and how they do it with an economy that no other form of mass expression can in mathematics. The author says that a great equation tells us something that we did not know before. He goes on to expound that though only three distinct branches of mathematics are apparent, the first is arithmetic or algebra, the science of quantity; the second is geometry, the science of shape and the third is applied mathematics, the science of translating mathematics into solutions to concrete problems of engineering, physics and economics. He adds that there is a fourth wellspring which is the science of the infinite – the analyses of both infinitely large and infinitely small quantities which are essential to understand any process of continuous motion or change. That is why he has included four chapters in this book with each having a theme or ‘storyline’ running throughout, relating to evolution of the four branches over the ages – algebra, geometry, applied mathematics and analysis.
Mathematics may look like a work of alchemy, transforming one quantity into another that at first seem completely different, yet every step can be explained and justified. “The only magic is in the human mind that can discover such connections,” he says. He adds that a great equation has the aesthetics of Japanese calligraphy; it contains nothing but the essentials; it says something simple and powerful. According to him, “The equations that make the deepest impressions are the ones that revolutionise mathematics, change our view of the world, or change the material possibilities of our lives.”
One of the great attractions of mathematics is that an equation proven today will remain true forever. It is not subject to the whims of fashion; it is the same across the globe and it cannot be censored or legislated.
Some of the equations presented in the book are not mathematical theorems, but physical ‘laws’ or theories, for example, Maxwell’s equations. Physical theories are generally confirmed by induction from data, or the ‘scientific method’ rather than by deduction from a set of axioms. Unlike mathematical theorems, they are subject to empirical evidence and statistical testing and occasionally, when more sensitive experiments come along, they are proven wanting.
This book is meant essentially for students of mathematics.
(Princeton University Press, 41, William Street, Princeton, New Jersey – 08540)
Rousseau, the Age of Enlightenment and Their Legacies, Robert Wolker, Bryan Garsten (Ed.), Princeton University Press, Pp 396, £ 30.95
Robert Wolker was one of the world’s leading experts on Rousseau and the Enlightenment, but some of his best work was published in the form of widely scattered and difficult-to-find essays. This book collects for the first time a representative selection of the most important essays on Rousseau and the legacy of enlightenment political thought. The essays concern great themes of the age like liberty, equality and the origins of revolution besides addressing a number of less prominent debates, including those over cosmopolitanism, the nature and social role of music and the origins of the human sciences in the Enlightenment controversy, over the relationship between humans and the great apes. These essays also explore Rousseau’s relationships to Rameau, Pufendorf, Voltaire and Marx; reflect on the work of important earlier scholars of the Enlightenment, including Ernst Cassirer and Isaiah Berlin; and examine the influence of the Enlightenment on the 20th century. One of the central themes of the book is defence of the Enlightenment against the common charge that it bears responsibility for the Terror of the French Revolution, the totalitarian regimes of the 20th century and the Holo-caust.
Rousseau, by focusing upon both his own psychic state and character on the one hand, and the beha-viour of savage peoples on the other, sought to define the inward and outward limits of mankind, not for the sake of ascertaining our origins, but in order to establish the essence of humanity itself within these boundaries. He perceived the polarities between our animal and moral attributes, between our sentimental and rational traits and in general, between our national and cultural patterns of behaviour, along lines which have marked the development of the human sciences ever since.
Robert Wolker was the son of stateless Jews who, to escape the Holocaust, reached to Paris and later to San Francisco, where he grew up to become a talented violinist and graduate from the University of Chicago. He went to England and did D. Phil from Oxford and was appointed lecturer at the University of Manchester while he completed his doctorate. Manchester became his base while he visited other universities to lecture and he did research at Yale. He died of cancer on July 30, 2006, aged 63. Wolker’s essays in this book cover the length of his academic career from the 1970s, beginning with ‘Rousseau or Rameau and Revolution’ and ‘Perfectible Apes in Decadent Cultures’, through some of the last writings that he was able to complete before he died – Rousseau’s Reading of Book of Genesis and Rites of Passage and the Grand Tour.
They span the breadth of his writing for methodological reflections in the Manuscript Authority of Political Thoughts to what might be thought to be journalism, for The Enlightenment Hostilities of Voltaire and Rousseau, including examples of both of his various tributes to his mentors, such as Isaiah Berlin’s Enlightenment and Counter-Enlightenment as well as Projecting the Enlightenment, an essay directed against Alasdair MacIntyre’s influential account of The Enlightenment Project in his After Virtue. The book reveals that Wolker wrote prose that could serve as the vehicle for his formidable erudition, but which was always set out with elegance and a surprising lightness of touch.
(Princeton University Press, 41, William Street, Princeton, New Jersey – 08540.)
Cells to civil: Creativity, ability to plan make humans unique
Cells To Civilisations: The Principles of Change that Shape Life, Enrico Coen, Princeton University Press, Pp 322, £ 19.95
When tracing the story of humanity, we naturally ask when it began and what makes human beings unique. We learn about this on reading this book that it was their qualities like language, imitation, the ability to plan ahead, the human soul or creativity, which imparted them greatness. No obvious beginnings account for evolution, development, or learning; rather the transformations arise through a constellation of ingredients coming together and interacting with each other. Instead of clear starting points, we find loops and interactions.
The same applies to cultural change. Our outlook today cannot be traced to a single cause but depends on a recipe with many interacting ingredients. Variations in human beings are a result of differences in birth and upbringing. The brain we are born with is influenced by the particular genes we inherit and then modify our development in the womb. After birth, individual characteristics are acquired as we encounter various situations and learn from them. The variations that allow cultural change in human population are due to the interplay between evolution, development and learning.
Similar considerations apply in cultural development. Transmission of knowledge and expertise from one person to another depends on our ability to communicate. The skill in turn stems from how each of us develops and learns. Our brain enables us to learn languages, gestures and actions and thus communicate effectively with others. This ability to communicate also plays a key role in cultural reinforcement. Here the neural connections of our brain come into play.
The historical relationship is seen during our evolution, then development, then learning and finally cultural change.
The author highlights seven key principles that underlie processes ranging from the evolution of bacteria to the working of our brain. “These seven principles or ‘ingredients’ and the way they work together define what I call life’s creative recipe,” says the author. It is his recipe that lies at the root of how life transforms itself.
(Princeton University Press, 41, William Street, Princeton, New Jersey 08540; press.princeton.edu)
Slaves in ancient Greece: A scintillating narrative
Slaves Tell Tales, Sara Forsdyke, Princeton University Press, Pp 275, $27.95
in this stimulating work, Sara Forsdyke tries to understand the ways that ordinary farmers, craftsmen and slaves in ancient Greece made sense of their world and their place in it. Unlike the wealthy elites, who produced written texts illustrating their world views, the ordinary people of ancient Greece left little or nothing from which their experiences and perspectives could be recovered. Similar to the “trace of the ancient plough”, the culture of these groups has been largely lost since it existed in “living” forms, such as festivals and oral storytelling.
Here Sara argues that diverse forms of popular culture, such as festival revellry, oral storytelling and the spontaneous collective punishment of social offenders were crucial aspects of ancient politics. By looking at popular culture throughout Greece, she puts Athens in perspective and shows, contrary to much modern scholarship, that Athens had more in common with other city-states (e.g. Sparta) and indeed pre-modern and modern peasant societies in general, than is currently recognised. If seen from the right angle, the surviving evidence can reveal traces of this lost culture.
Most studies of ancient Greek politics focus on formal institutions, such as the political assembly and the law courts, overlooking the role that informal social parties placed in the regulation of the political order. So the author attempts to excavate popular forms of culture that lie barely discernible beneath the surface of ancient Greek literature. Also this is not simply an antiquarian inquiry into cultural relics by Sara, but an attempt to show that forms of popular culture were vital to the practice of politics in ancient Greek communities.
By looking beyond institutional contexts, Sara recovers the ways the groups that were excluded from the formal political sphere – especially women and slaves – participated in the process by which society was ordered.
Giving an overview chapterwise, Sara begins each chapter with an apparently marginal incident in Greek history – the worship of a dead slave by masters on Chios, the naming of Sicyon’s civic divisions after lowly animals, such as pigs and asses, and the riding of an adultress on a donkey through the streets of Cyme – to show how these episodes demonstrate the significance of informal social practices and discourses in the regulation and reproduction of the social order.
In ancient Greece, rural labourers, “whether free or unfree, citizen or metic”, had more in common with one another than with urban artisans, bankers and traders. It was not necessary that slaves aligned themselves ideologically against their masters or that citizens put up a united front to keep the slaves in check. In contrast to the American South, slaves were distinctly different from the white men, who formed militias to hunt down runaway slaves. In ancient Greece, the lines between free men and slaves were often blurred and could be overridden.
The most important distinction to complicate class and status divisions was that of gender. All women were treated like non-citizens and slaves were denied full political rights. Yet, despite their dominant ideology, women created a culture of their own and asserted their opinions.
Sara thus presents the overview of ancient Greek social structure in which existed a myriad of overlapping social groups and dividing lines as were evident between males and females, citizens and free non-citizens and metics and women. Yet these three groups were similarly privileged in contrast to slaves who had no legal rights at all. In this way Sara presents a new perspective on politics and popular culture in ancient Greece.
(Princeton University Press, 41, William Street, Princeton, New Jersey 08540; press.princeton.edu)