Saturday, June 13, 2009

Some Mathematical History

The Fjordman Report

The noted blogger Fjordman is filing this report via Gates of Vienna.
For a complete Fjordman blogography, see The Fjordman Files. There is also a multi-index listing here.

An important insight in the history of science was formulated by the French mathematician, biologist and astronomer Pierre-Louis de Maupertuis (1698-1759). Maupertuis had studied in Paris and in Basel, Switzerland, with Johann Bernoulli. He became a leading member of the Berlin Academy of Sciences in 1741 and in 1744 enunciated the principle of least action. He hoped that the principle might unify the laws of the universe and prove the existence of God.

As John Gribbin writes, “De Maupertuis had been a soldier before turning to science; his big idea is known as the principle of least action. ‘Action’ is the name given by physicists to a property of a body which is measured in terms of the changing position of an object and its momentum (that is, it relates mass, velocity and distance travelled by a particle). The principle of least action says that nature always operates to keep this quantity to a minimum (in other words, nature is lazy). This turned out to be hugely important in quantum mechanics, but the simplest example of the principle of least action at work is that light always travels in straight lines.” Euler, too, developed the idea of the principle of least action, and “This pointed the way for the work of Joseph Lagrange (1736-1813), which in turn provided the basis for a mathematical description of the quantum world in the twentieth century.”

Lagrange was born Giuseppe Lodovico Lagrangia in Turin, Italy, where he lived during the early years of his life. He replaced Euler when the great Swiss scholar left Berlin for Saint Petersburg, Russia, in 1766 and spent twenty productive years in that city. By 1786 he moved to France, where he became known as Joseph-Louis Lagrange. According to O’Connor and Robertson he “excelled in all fields of analysis and number theory and analytical and celestial mechanics.” Lagrange was a better mathematician than de Maupertuis and provided the concept of least action with a more thorough mathematical foundation.

The Swiss mathematician Leonhard Euler (1707-1783) is widely recognized as one of the greatest mathematicians of all time. He was certainly the most productive measured in published pages, although the eccentric and prolific Hungarian Jewish mathematician Paul Erdos (1913-1996) beat him in the number of published papers. Erdos, too, was interested in combinatorics, graph theory and number theory and the “contributions which Erdös made to mathematics were numerous and broad. However, basically Erdös was a solver of problems, not a builder of theories.” The mathematical community was obviously much smaller in the eighteenth century than it is today; it would be difficult for even the most gifted scholar to have such a wide-ranging influence and dominate the entire, now very diverse field of mathematics as Euler did in his time, but his achievement is nevertheless impressive.
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Euler was born in Basel, Switzerland, and his career was connected to that of the Bernoulli family. He graduated with honors from the University of Basel and got a post in the newly formed St. Petersburg Academy of Sciences, created by Tsar Peter the Great as part of his modernization efforts of the Russian state. He accepted the invitation of Frederick II of Prussia to join the Berlin Academy of Sciences, founded on the advice of Leibniz. Euler returned to Russia in 1766 at the invitation of Empress Catherine the Great, whose succession to the throne marked a return to the Westernizing policies. He made contributions in most fields of mathematics which existed in his day, including hydraulics, ship construction and artillery, but he was especially devoted to astronomy. The tremendous prestige of his textbooks settled forever many questions of notation on calculus and algebra. Lagrange, Laplace and Gauss followed Euler in their works, and Riemann knew his work well. Euler was able to dictate his articles and letters to his sons and others virtually until the day of his sudden death in 1783. As The Oxford Guide to the History of Physics and Astronomy states:

“Yet productivity was perhaps the least important of Euler’s claims to mathematical distinction. One of his great contributions was his clarity….He contributed to every branch of mathematics of his day except probability. He achieved much in the realm of number theory. He arguably founded graph theory and combinatorics when he solved the Königsberg Bridge problem in 1736….in addition Euler contributed to ordinary and partial differential equations, the calculus of variations, and differential geometry….Euler made major contributions to every branch of mechanics. The motion of mass points, celestial mechanics, the mechanics of continuous media (mechanics of solids and nonviscous fluids, theories of materials, hydrodynamics, hydraulics, elasticity theory, the motion of a vibrating string, and rigid-body kinematics and dynamics), ballistics, acoustics, vibration theory, optics, and ship theory all received something important from him. If Beethoven did not need to hear to compose music, Euler did not need to see to create mathematics. He began to go blind in one eye in 1738 and became totally blind thirty years later. This only increased his productivity, since total blindness relieved him of academic chores like proofreading and eliminated unwanted visual distractions. Euler did not miss eyes for another reason; he had a prodigious memory.”

Mathematics has historically been overwhelmingly created by men, often very young men. The French mathematician Évariste Galois (1811-1832) died from wounds suffered in a duel while still in his early twenties, yet he had already managed to leave his mark on the history of mathematics. In 1830 he developed group theory, which was to prove of great importance for the development of quantum mechanics a century later. The Norwegian mathematical pioneer Niels Henrik Abel (1802-1829), too, died while still in his twenties. He suffered from tuberculosis, but continued to develop high quality mathematics as his health deteriorated. Dirk J. Struik writes in A Concise History of Mathematics, Fourth Revised edition:

“Abel proved the impossibility of solving the general quintic equation by means of radicals — a problem which had puzzled mathematicians from the time of Bombelli and Viète (a proof of 1799 by the Italian Paolo Ruffini was considered by Poisson and other mathematicians as too vague). Abel now obtained a stipend which enabled him to travel to Berlin, Italy, and France. But, tortured by poverty most of his life and unable to get a position worthy of his talents, Abel established few personal mathematical contacts and died (1829) soon after his return to his native land….Abel’s investigations on elliptic functions were conducted in a short but exciting competition with Jacobi. Gauss in his private notes had already found that the inversion of elliptic integrals leads to single-valued, doubly periodic functions, but he never published his ideas. Legendre, who had spent so much effort on elliptic integrals, had missed this point entirely and was deeply impressed when, as an old man, he read Abel’s discoveries.”

The Frenchman Adrien-Marie Legendre (1752-1833) was one of the leading mathematicians in Europe at the turn of the nineteenth century and made many personal contributions to the field, but some of his work was perfected by others, among them Abel, Jacobi and Galois. Carl Gustav Jacobi (1804-1851) was a German (Prussian) mathematician, born of Jewish parents, who studied in Potsdam and at the University of Berlin to be able to teach mathematics, Greek and Latin. In 1829 Jacobi met Legendre and other French mathematicians such as Fourier when he made a visit to Paris, and he visited Gauss in Göttingen. He became an influential and inspiring teacher and made contributions to the theory of elliptic functions.

The mathematician and astronomer Sir William Rowan Hamilton (1805-1865), who lived his whole life in Dublin, Ireland, could read Hebrew, Latin, and Greek at the age of five and learned many other languages during his lifetime. By 1822 his mathematical abilities had advanced to such an extent that he discovered a significant error in Laplace’s treatise Celestial Mechanics. In 1843 he introduced quaternions, algebra with hyper-complex numbers and “ the first noncommutative algebra to be studied.” He made major contributions to optics, and Hamiltonian mechanics helped shape quantum mechanics in the twentieth century.

According to scholar Alan Gabbey, “Pierre de Maupertuis’s principle of least action had sounded a new note. Reflecting on the controversy of the 1660s over Fermat’s least-time optical principle, Maupertuis argued (1744) that in all bodily changes, the ‘action’…is the least possible, a principle that for Maupertuis and Euler — though not for d’Alembert and Lagrange — pointed to the governance of all things by a Supreme Being….Maupertuis’s variational principle enjoyed an improved mathematical treatment by William Rowan Hamilton (1834, 1835), whose transformation of Lagrange’s equations was modified and generalized by Carl Gustav Jacobi in the form now known as the Hamilton-Jacobi Equation (1837). In turn, the Hamilton-Jacobi Equation found fruitful application in the establishment of the quantum mechanics of Louis de Broglie (1923) and Erwin Schrödinger (1926).”