While reading this account, I became more and more intrigued by the question how a Norwegian working in Canada acquired a view of mathematics and of history that is so quintessentially French. The characters in his story are mostly French, and the dominant role of mathematics in their thinking is a hallmark of French culture. Nowhere else except in France do mathematicians command such respect. As soon as I consulted Google, I found the solution to the mystery. In spite of his Norwegian name, Ekeland is French. Born in Paris, educated at the historic École Normale Supérieure, professor at the University of Paris–Dauphine, and subsequently president of the university, he is a charter member of the French academic establishment. His books were mostly written in French before being published in other languages. This book is a translation of a book with the same title published in French in the year 2000, revised and brought up to date for English-speaking readers. It gives us a vivid picture of human history and destiny as seen through the eyes of a senior academic trained in the French educational system.
There is at least one Frenchman who does not share Ekeland’s view of the world. Pierre de Gennes is a brilliant French physicist who won a Nobel Prize in 1991 for understanding the behavior of squishy materials on the borderland between liquid and solid. He called the things that he studied “soft matter.” After the Nobel Prize made him a French national hero, he was inundated with invitations to visit high schools and inspire the students to follow in his footsteps. He accepted the invitations and spent a year and a half as a traveling guru, explaining science to the kids. He enjoyed the contact with young people so much that he turned his talks into a book, Fragile Objects: Soft Matter, Hard Science, and the Thrill of Discovery. The book was translated into English and published by Springer-Verlag in 1996. It describes in simple words how the science of soft matter explains the behavior of ordinary materials such as soap, glue, ink, rubber, and flesh and blood that children encounter in their everyday lives. De Gennes’s talks were aimed at the average child, not at the talented few who might become professional scientists. His book is well pitched to give average readers a practical understanding of how science works.
At the end of his book, de Gennes adds a few chapters aimed not at the children but at their teachers. One of these chapters, with the title “The Imperialism of Mathematics,” is a diatribe against the dominance of mathematics in the French educational system. He writes:
Whenever an entrance examination is instituted in a scientific discipline, it invariably becomes an exercise in mathematics…. Why is there such a focus on mathematics? In reality, the trend toward mathematization turns our graduates, our future engineers, into hemiplegics…. They may have learned to master certain tools, to prepare reports, but they will suffer from crippling weaknesses in observation, manual skills, common sense, and sociability.
De Gennes is not a typical French intellectual. He mixes theory with experiment, and prefers concrete objects to abstract ideas. In his research and in his teaching, he fights against the imperialism of mathematics.
In America we have the opposite situation. Our children study a variety of subjects without much formal discipline, and most of them remain mathematically illiterate. It is good for us to be reminded that different countries have profoundly different cultures and different virtues and vices. The imperialism of mathematics is difficult for Americans to imagine, but for France it is a real problem. If American children could learn more mathematics and French children less, both countries would benefit. Americans should not be misled by de Gennes’s diatribe into thinking that we have nothing to learn from France. He describes eloquently the vices of the French educational establishment. He does not emphasize its virtues. The most important virtue of the French system is the strict discipline that it imposes. Every child and every student must meet rigid standards of knowledge and skill. De Gennes takes for granted the fact that the children he is talking to are literate and have a firm grasp of elementary mathematics. Americans should ask themselves why such a standard of literary and mathematical competence cannot be taken for granted in America.
Ekeland does not entirely exclude people who were not French from his narrative. He recognizes the great contributions of Galileo, Newton, Euler, and Darwin to the development of modern science, and the great contributions of the historians Thucydides and Guicciardini to the understanding of human destiny. Some of the most illuminating passages in the book are quotations from Thucydides and Guicciardini, both of them generals who fought on the losing side in catastrophic wars and then wrote their histories to teach whatever bitter lessons posterity might learn from their defeat. Both of them saw tragedy arising not from implacable fate but from human folly and unlucky accidents. With wiser leaders, mistakes might have been avoided and tragedy averted. The worst mistakes are mistakes of overconfidence, made by arrogant leaders who do not respect the skill of their enemies or the vagaries of chance. For the American edition of his book, Ekeland has inserted some acid remarks about arrogance and overconfidence displayed in recent actions of the American government.
A different book about the cultural history of the last four hundred years might have been written by a different Ekeland who was educated in the Anglo-American tradition instead of the French. I call the imaginary Ekeland Akeland, and I assume that Akeland is as strongly biased toward English as Ekeland is biased toward French. For Akeland, modern science still begins with Galileo, but then continues with Francis Bacon instead of René Descartes. Bacon was three years older than Galileo and thirty-five years older than Descartes. Bacon pushed English science as strongly in the direction of experiment as Descartes pushed French science in the direction of theory. Bacon had a low opinion of theory. He wrote: “The logic now in use serves rather to fix and give stability to the errors which have their foundation in commonly received notions than to help the search after truth.” Bacon preached humility toward nature as the only way to arrive at truth: “Man, being the servant and interpreter of Nature, can do and understand so much and so much only as he has observed in fact or in thought of the course of Nature; beyond this he neither knows anything nor can do anything.” He had a grand vision of the future of science but a modest view of the science of his own time: “For though it be true that I am principally in pursuit of works and the active department of the sciences, yet I wait for harvest-time, and do not attempt to mow the moss or to reap the green corn.” He did not live to see the harvest of discoveries that began thirty-four years after his death when the Royal Society of London was founded. He died while the corn was still green and Descartes had not yet started to mow the moss.
In Akeland’s version of history, the scientist who personifies eighteenth-century enlightenment is Benjamin Franklin rather than the Marquis de Maupertuis. Instead of the mathematicians Lagrange and Poincaré, the scientists who bring us into the modern world are the nineteenth-century British physicists Michael Faraday and James Clerk Maxwell, who set out the basic laws of electricity and magnetism. Bacon, Franklin, Faraday, and Maxwell, the chief characters in Akeland’s narrative, are nowhere mentioned by Ekeland. Likewise, Akeland fails to mention Descartes, Maupertuis, Lagrange, and Poincaré. His main theme is the emergence of electricity in the eighteenth century as the growing point of science. Electricity was a product of purely Baconian science, emerging from unexpected observations of nature rather than from mathematical deduction.
Ekeland’s book puts mathematical optimization at the focus of history. Optimization means choosing the best out of a set of alternatives. Mathematical optimization means using mathematics to make the choice. Maupertuis is the central character of the history because he claimed that the universe is mathematically optimized. Akeland’s book has the opposite emphasis. For Akeland, things are more important than theorems. Experiments are more important than mathematics. The great scientific achievement of the Age of Enlightenment was the experimental study of electricity. Electricity was the driving force of science for two hundred years, from the death of Newton to the rise of molecular biology. Electricity also enlarged the scope of science, moving out from the logical and mechanical universe of Newton into the color and variety of the modern world. The biologist Stephen Jay Gould formulated the philosophical principle that Akeland borrows for the title of his book: “We are the offspring of history and must establish our own paths in this most diverse and interesting of conceivable universes.” Instead of mathematical optimization, Akeland postulates maximum diversity as the governing principle of the universe. His title is The Most Interesting of All Possible Worlds: Electricity and Destiny.
Franklin had no theoretical understanding of electricity. Electricity was outside the Newtonian domain of mechanics and gravitation that constituted the theoretical science of his time. Franklin explored electricity because it was a part of nature that nobody understood. Without pretending to understand electricity, he learned how to control it. His invention of the lightning conductor made him world-famous and earned him a warm welcome when he came to live in France. He came to France too late to meet with Maupertuis. If they had met, they would have found that they had much in common. Franklin was only eight years younger than Maupertuis. Both were good organizers as well as good scientists. Franklin was organizing the American Philosophical Society in Philadelphia while Maupertuis was organizing the Prussian Academy in Berlin. Both were gentlemen of the Enlightenment, adventurers and travelers in an age when travel was slow and arduous. Both were by temperament optimists, but neither was a Pangloss. The only serious difference between them was that Maupertuis was a mathematician and Franklin was an experimenter.
The next pair of characters in Ekeland and Akeland’s histories were Lagrange in France and Faraday in England. They lived in different centuries and had less in common than Maupertuis and Franklin. They were extreme examples of Cartesian and Baconian scientists. Faraday explored the new worlds of electricity and magnetism, chemistry and metallurgy, pushing into unknown territory far ahead of any theoretical understanding. Lagrange (1736–1813) created the science of analytical mechanics, an abstract mathematical framework that included all the results of Newtonian dynamics as special cases. Each was master of his trade, but theirs were very different trades. By unifying Newton’s ideas into a single scheme, Lagrange left the world simpler than he found it. By discovering a host of unexpected new phenomena, Faraday (1791–1867) left the world more complicated than he found it. Lagrange was a unifier; Faraday was a diversifier. Although Lagrange’s great work was published three years before Faraday was born, Faraday never read it and never felt a need for it. All the mathematics that Faraday needed was elementary arithmetic and a little algebra.
The histories of Ekeland and Akeland begin to diverge with Maupertuis and Franklin and reach a point of maximum divergence with Lagrange and Faraday. With the last pair of characters, Poincaré and Maxwell, the histories converge. Poincaré (1854–1912) was a mathematician with a taste for diversity. He was interested in the new science of electromagnetism as well as the old science of mechanics, and he discovered in the dynamics of stars and planets a variety of chaotic motions that Lagrange never dreamed of. Maxwell (1831–1879) was a physicist with a passion for unification. Starting from the observations of Faraday, he discovered the equations that unify the theories of electricity and magnetism and light into a mathematical structure as elegant as Lagrange’s mechanics. The convergence of Ekeland and Akeland became complete when Poincaré explored the group of symmetries of the Maxwell equations, the group that is now known to physicists as the Poincaré Group. Maxwell and Poincaré together prepared the way that led Einstein to the new world of relativity.
The real Ekeland and the fictitious Akeland are teaching us a simple lesson. Each of them gives us a slanted and partial view of history. The true history of modern science must include both of them. Modern science started its rapid growth in the seventeenth century, taking its aims and methods not from Descartes alone and not from Bacon alone but from the cross-fertilization of Cartesian and Baconian ideas. Isaac Newton, the greatest figure in the history of the physical sciences, was an intimate mixture of Descartes and Bacon. He was Baconian in his study of optics, when he separated white light into its colored components and invented his reflecting telescope. He was Cartesian when he wrote his Principia Mathematica, deducing the system of the world from a logical sequence of mathematical propositions. He cleverly used a Cartesian style of argument, together with a Baconian knowledge of planetary motions, to demolish Descartes’s cosmology of vortices in space.
In a true history of science, mathematics and electricity make equal contributions to human destiny. Our world may be the best of all possible worlds and may be the most interesting. Both possibilities are open. Our destiny depends on choices that we have not yet made, probably concerned more with biology—and particularly with our incipient understanding of the human brain—than with mathematics or electricity.