Ivar Ekeland has a Norwegian name and teaches at the University of British Columbia in Canada, but the style and spirit of his book are unmistakably French. The book is a rapid run through the history of the last four hundred years, seen through the eyes of a French mathematician. Mathematics appears as a unifying principle for history. Ekeland moves easily from mathematics to physics, biology, ethics, and philosophy. The central figure of his narrative is the French savant Pierre de Maupertuis (1698–1759), a man of many talents, who formulated the principle of least action in 1745 in a memoir with the title The Laws of Motion and Rest Deduced from a Metaphysical Principle. The principle of least action says that nature arranges all processes so as to minimize a quantity called action, which is a measure of the effort required to bring the processes to completion. The action of any mechanical motion is defined as the moving mass multiplied by the velocity and by the distance moved. Maupertuis was able to demonstrate mathematically that if a collection of objects moves in such a way as to make the total action as small as possible, then the movement obeys Newton’s laws of motion. Thus the whole science of Newtonian mechanics follows from the principle of least action.
Maupertuis was dazzled by the beauty of his discovery. “How satisfying for the human spirit,” he wrote, “to contemplate these laws, so beautiful and simple, which may be the only ones that the Creator and Ordainer of things has established in matter to sustain all phenomena of this visible world.” He went on to identify action with evil, so that the principle of least action became a principle of maximum goodness. He concluded that God has ordered the universe so as to maximize goodness. The world that we live in is the best of all the possible worlds that God might have created. This simple principle unites science with history and morality. Mathematics is the key to the understanding of human destiny.
One of the contemporaries of Maupertuis was Voltaire, the great skeptic, who demolished Maupertuis’s optimistic philosophy in a book with the title Story of Doctor Akakia and the Native of Saint-Malo. Akakia is Greek for “absence of evil,” and the native of Saint-Malo is Maupertuis. “The native of Saint-Malo,” Voltaire writes, “had long fallen a prey to a chronic sickness, which some call philotimia [Greek for love of honors] and others philocratia [Greek for love of power].” Voltaire’s book sold well and Maupertuis’s day of glory ended. After Maupertuis died, Voltaire made him posthumously ridiculous by writing the novel Candide, in which Maupertuis appears as the optimistic philosopher Pangloss, wandering from one disaster to another but unshaken in his belief that “all is well that ends well in the best of all possible worlds.”
Maupertuis was in fact no Pangloss. He spent only a small part of his time as an optimistic philosopher. He was also a brilliant scientist and a capable administrator. He became famous as a young man for leading an expedition to Lapland to measure the shape of the earth at high latitude. His measurements were accurate enough to prove that the earth is not a perfect sphere but an ellipsoid, flattened at the poles as Newton predicted as a consequence of its rotation. This confirmation of Newton’s theory was historically important, since up to that time Newtonian physics was not widely known or accepted in France. Maupertuis also learned to travel on skis in Lapland, and brought home with him the first pair of skis that had ever been seen in France. For many years after the Lapland expedition, he was one of the most active members of the French Academy of Sciences. When King Frederick the Great of Prussia founded his own Academy of Sciences in Berlin, he invited Maupertuis to be the first president. Maupertuis spent the rest of his life in Berlin, successfully launching and running the Prussian Academy. Voltaire hated King Frederick, and Maupertuis’s friendship with the King gave Voltaire another reason to hate and belittle Maupertuis.
Ekeland’s sketch of history is divided into two parts, before Maupertuis and after Maupertuis. Before Maupertuis, the two chief characters are Galileo and Descartes. Galileo started modern science by using the pendulum as a tool to make accurate measurements of time. Ancient Greek science was based on geometry, measuring space but not time. Archimedes understood statics but did not understand dynamics. Galileo with his pendulum and his falling weights made the decisive step from a static to a dynamic view of nature. He introduced time as a quantity accessible to mathematical analysis. He said, “Nature’s great book is written in mathematical symbols.” That remark of Galileo was the lever that moved the world into the modern era of scientific understanding.
After Galileo came René Descartes, a great mathematician and a great philosopher but not yet a great scientist. Descartes took to heart Galileo’s insight that mathematics is the language that nature speaks. He tried to deduce the laws of nature from the laws of mathematics by pure reason alone. He did not listen to another statement of Galileo, that nature answers questions that we ask by doing experiments. Descartes held experimental results in low esteem, thinking them less trustworthy than logic. His was a normative science, telling nature what it was supposed to do, and not an experimental science, investigating what nature was actually doing. Descartes published in 1637 his great work, A Discourse on the Method of Rightly Conducting the Reason and of Seeking Truth in the Sciences. He describes a scientific method that is broad enough to deal with moral as well as with physical problems. “I showed what the laws of nature were,” he wrote,
and without basing my arguments on any principle other than the infinite perfections of God, I tried to demonstrate all those laws about which we could have any doubt, and to show that they are such that, even if God created many worlds, there could not be any in which they failed to be observed.
Ekeland concludes that Descartes’s method “has been used in science with tremendous success, and there is no reason why it should not be as useful in philosophy, or in trying to establish some principles by which to guide our collective and individual lives.” Unfortunately, the Cartesian way of doing science with minimum recourse to experimentation led him into bad mistakes. From his philosophical principle that nature abhors a vacuum, he was led to deduce that the space around the planets is filled with enormous vortices, or whirling masses, and that the pressure of the vortices confines the planets to their orbits and pushes them on their way. This theory of planetary motions was generally accepted in France as a preferable alternative to Newton’s theory of universal gravitation. Descartes also deduced that the rotating earth creates another enormous vortex that squeezes the earth into the shape of an American or rugby football. According to Descartes, the earth should be an ellipsoid elongated at the poles, instead of being flattened as predicted by Newton. Maupertuis’s measurements in Lapland proved Newton right and Descartes wrong.
Ekeland’s history continues after Maupertuis with a couple of great mathematicians—Joseph-Louis Lagrange and Henri Poincaré, who used the ideas of Maupertuis to build a grand edifice of classical dynamics. Poincaré, in the late nineteenth century, discovered chaos, a general property of dynamical systems that makes their behavior unpredictable over long times. He discovered that almost all complicated dynamical systems are chaotic. In particular, the orbital motions of planetary systems with more than two planets, and the fluid motions of atmospheres or oceans, are likely to be chaotic. The discovery of chaos opened a new chapter in the history of astronomy and meteorology, as well as in the history of mathematics.
After his discussion of Poincaré, Ekeland devotes chapters to biology and ethics, with backward glances to establish connections with Maupertuis. In biology, the guiding principle of evolution is the survival of the fittest. Darwin’s notion of nature selecting a population with maximum fitness resembles Maupertuis’s notion of God selecting a universe with maximum goodness. Darwin himself understood that fitness is not the same as goodness, but other evolutionary thinkers such as Herbert Spencer allowed the distinction between fitness and goodness to be blurred. Darwin rarely used the word “evolution,” which Spencer introduced into biology. Darwin preferred to speak of “descent with variation,” emphasizing the fact that variations are random and not usually progressive.
In ethics, the problem of optimization is even more tricky. Ekeland begins his discussion of ethics with Jean-Jacques Rousseau, the philosopher of the French Enlightenment, whose ideas prepared the way for the revolution of 1789. Rousseau believed that human beings were naturally virtuous and wise. They needed only to be set free from tyrannical governments, and then they would order their affairs harmoniously. A democratic government, responsive to the will of a free people, would make sure that everyone was treated fairly. Before the revolution could put these ideas to a practical test, some theoretical difficulties were raised by the Marquis de Condorcet, who for the first time used mathematics to model human behavior. The marquis discovered a logical inconsistency known as Condorcet’s paradox, which demonstrates that an assembly ruling by majority vote may make decisions that are logically incompatible. For example, if three candidates A, B, C are running for a job to be filled by majority vote, it is possible that a majority prefers A to B, another majority prefers B to C, and a third majority prefers C to A. Then the result of the election will depend on the order in which the votes are taken. Another learned academician, the Chevalier de Borda, devised a system of preferential voting for election of members to the French Academy of Sciences. The de Borda scheme avoided the Condorcet paradox, but led to another paradox that could be exploited by unscrupulous politicians to win elections. It turned out that no system of voting is free from mathematical paradoxes. And the revolution, when it came, brought a quarter-century of death and destruction instead of the peace and harmony that Rousseau had promised.
To sum up the lessons to be learned from history, Ekeland writes:
We have now reached the end of our journey. It started in the world of the Renaissance, impregnated with Christian values…. The laws of nature then are simply the rules God followed when creating the world, and the purpose of science is to recover them from observations. There is then also a deeper science, which is to seek the purpose God himself had in creating the world. This is what Maupertuis, in a glorious moment, thought he had achieved, thereby reconciling forever science and religion, both being the quest for God’s will, in the physical world and in the moral one. Our journey ends in a world where God has receded, leaving humankind alone in a world not of its choosing.