Author's note: Much of the following is excerpted from Chapter 5 of my book in progress, The Anti-Copernican Revolution. The book is concerned with the relationship between philosophy and physics, and it spans the period from Copernicus to the present.
Like the lives of individuals, history has its key moments. In Ayn Rand's novel The Fountainhead, the hero, Howard Roark, is asked: “When you look back, does it seem to you that all your days rolled forward evenly, like a sort of typing exercise, all alike? Or were there stops—points reached—and then the typing rolled on again?” Roark answers: “There were stops.”1 Even on the grand scale of history we can see such stops, that is, points at which one period ends, the direction changes, and another period begins.
One historian, Alan Cromer, refers to the extraordinary transition that took place at the end of the 18th century and writes: “In many ways, the decade of the 1780s divides the past from the present.”2 That is a profound truth, in a much deeper sense than Cromer explains. I will argue that this was the point at which the Enlightenment reached its zenith—and at which it was brought down by its one failure.
The Enlightenment is the century between two major figures: Isaac Newton and Immanuel Kant. In The Ominous Parallels, Leonard Peikoff wrote that this period is the only time in modern history that “an authentic respect for reason became the hallmark of an entire culture.”3 Man's nature, most thinkers of the period agreed, was clear from his accomplishments, particularly those in science. He is not the helpless animal described by the skeptics or the depraved animal described by the mystics; he is, as Aristotle said long ago, the rational animal. As such, he is a being with unlimited potential for gaining knowledge and acting to achieve his spiritual and physical well-being. In 1750, French economist and statesman Anne Robert Jacques Turgot expressed the attitude that dominated the era: “At last all clouds are dissipated. What a glorious light is cast on all sides! What a crowd of great men on all paths of knowledge! What perfection of human reason!”4
In physical science, the crowd of great men knew who had cleared their path. “The physics of the eighteenth century,” writes one historian, “provides an example of the profound influence exerted by the work of a single man, Isaac Newton, to a degree that is unique in the development of modern science.”5 Newton had blazed the trail with two works of genius. In the Principia, he had presented the universal laws of motion and gravitation and thus opened men's eyes to the extraordinary power of mathematics. In his Optics, he had provided a tour de force demonstration of how to ask questions of nature and obtain the answers by systematic experimentation. The Enlightenment made the most of both lessons.
By far the greatest contribution to mathematical physics was made by Leonhard Euler. As one physicist puts it: “All branches of mathematics abound with Euler's theorems, Euler's coefficients, Euler's methods, Euler's proofs, Euler's constant, Euler's integrals, Euler's functions, and Euler's everything else.”6 He wrote exhaustive treatises on differential and integral calculus, and he presented the first modern treatment of methods for solving differential equations. He made fundamental contributions to several new areas of mathematics, including the theory of functions of complex variables, the theory of special functions, and the calculus of variations. Furthermore, he invented a great deal of the notation used today in mathematics, including the ubiquitous modern symbols for summations, finite differences, pi, the square root of minus one, the base of natural logarithms, and trigonometric functions.
Above all, Euler did for calculus what Euclid did for geometry. Neither man was the original discoverer, but each made an enormous contribution to his respective field and then presented the theory systematically. The comparison also highlights an interesting difference between the two men. Euclid was more the theorist concerned with logical foundations, whereas Euler was more the “practical” thinker so typical of the Enlightenment. Euler never lost sight of applications to the physical world, and such applications motivated his major innovations in mathematics.
Both logical rigor and applications are crucial. Without the first, we cannot be certain that our statements are true; without the second, it does not matter whether or not they are true. Throughout most of history, however, mathematicians have committed the Platonic error of denigrating applications. Newton's influence caused a profound (and unfortunately temporary) change of attitude. During the Enlightenment, mathematicians celebrated the indispensable role of their science in understanding the physical world. . . .
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Acknowledgment: The author wishes to acknowledge the generous support of the Ayn Rand Institute.
1. Ayn Rand, The Fountainhead (New York: Signet, 1993), pp. 542–543.
2. Alan Cromer, Uncommon Sense: The Heretical Nature of Science (New York: Oxford University Press, 1993), p. 5.
3. Leonard Peikoff, The Ominous Parallels (New York: Stein and Day, 1982), p. 102.
4. Richard Panek, Seeing and Believing (New York: Viking Penguin, 1998), pp. 102–103.
5. I. Bernard Cohen, Franklin and Newton (Baltimore: J. H. Furst Company, 1956), quoted from Preface, vii.
6. Petr Beckmann, A History of Pi (New York: St. Martin's Press, 1971), p. 148.
7. Thomas L. Hankins, Science and the Enlightenment (Cambridge University Press, 1985), p. 21.
8. St. Augustine, The Essential Augustine, edited by Vernon Bourke (Indianapolis: Hackett Publishing Company, 1974), p. 114.
9. Duane Roller, The Development of the Concept of Electric Charge (Cambridge: Harvard University Press, 1954), p. 52.
10. Ibid., p. 51.
11. Jean-Jacques Rousseau, The Essential Rousseau (New York: Penguin Books, 1983), p. 210.
12. Ibid., p. 214.
13. Ibid., p. 215.
14. Ibid., pp. 258–259.
15. Bertrand Russell, A History of Western Philosophy (New York: Simon & Schuster, 1945), p. 688.
16. Thomas L. Hankins, Jean d'Alembert: Science and the Enlightenment (London: Oxford University Press, 1970), p. 70.
17. Ibid., p. 88.
18. Ibid., p. 3.
19. Ibid., p. 165.
20. Ibid., p. 153.
21. Ibid., p. 185.
22. Ibid., p. 197.
23. Ibid., p. 129.
24. Thomas L. Hankins, Science and the Enlightenment (New York: Cambridge University Press, 1985), p. 112.
25. The Beginnings of Modern Science, edited by Holmes Boynton (New York: Walter J. Black, Inc., 1948), p. 615.
26. Thomas L. Hankins, Science and the Enlightenment (New York: Cambridge University Press, 1985), p. 109.
27. E. T. Bell, Men of Mathematics (New York: Simon & Schuster, 1986) p. 181.
28. Richard Panek, Seeing and Believing (New York: Viking Penguin, 1998), p. 115.
29. Alan Cromer, Uncommon Sense: The Heretical Nature of Science (New York: Oxford University Press, 1993), p. 4.
30. Immanuel Kant, Critique of Pure Reason, translated by Norman Kemp Smith (New York: St Martin's Press, 1965), p. 286.
31. Ibid., p. 22.
32. Ibid., p. 29.
33. Immanuel Kant, Religion within the Limits of Reason Alone, translated by Theodore M. Greene and Hoyt H. Hudson (New York: Harper & Row, 1960), p. 9.
34. Immanuel Kant, Kant's Philosophy of Material Nature, translated by James W. Ellington (Indianapolis: Hackett Publishing Company, 1985), p. 55.
35. Immanuel Kant, Critique of Pure Reason, translated by Norman Kemp Smith (New York: St Martin's Press, 1965), p. 284.
36. Ibid., p. 385.
37. Ibid., p. 280.
38. Ibid., p. 449.
39. Immanuel Kant, Kant's Philosophy of Material Nature, translated by James W. Ellington (Indianapolis: Hackett Publishing Company, 1985), p. 39
40. Ibid., p. 61.
41. Ibid., pp. 55–56.
42. Ibid., pp. 114–115.
43. Ibid., p. 112.
44. Ibid., p. 73.
45. Ibid., p. 217.
46. Ibid., p. 19.
47. Ibid., pp. 13–14.
48. Ibid., p. 78.
49. Ibid., pp. 62–63.
50. Isaac Newton, Newton's Philosophy of Nature: Selections from His Writings, edited by H. S. Thayer (New York: Hafner Publishing Company, 1953), p. 54.
51. Immanuel Kant, Kant's Philosophy of Material Nature, translated by James W. Ellington (Indianapolis: Hackett Publishing Company, 1985), p. 93.
52. The Encyclopedia of Philosophy, edited by Paul Edwards (New York: Macmillan Publishing Company, 1967), vol. 3, p. 300.
53. H. G. Schenk, The Mind of the European Romantics (New York: Anchor Books, 1969), p. 4.
54. Jonathon Norton Leonard, Crusaders of Chemistry (New York: Doubleday, Doran & Company, 1930), p. 263.
55. Ibid., p. 298.
56. Douglas McKie, Antoine Lavoisier: Scientist, Economist and Social Reformer (New York: Henry Schuman, 1952), p. 406.
57. Johann Gottlieb Fichte, Science of Knowledge, edited and translated by Peter Heath and John Lachs (Meredith Corporation, 1970), pp. 129–130.
58. Novalis, Hymns to the Night and Other Selected Writings, translated by Charles E. Passage (New York: Bobbs-Merrill Company, 1960), p. 66.
59. Ibid., p. 15.
60. Ibid., p. 46.
61. Romanticism and the Sciences, edited by Andrew Cunningham and Nicholas Jardine (Cambridge University Press, 1990), p. 200.
62. F. W. J. Schelling, Ideas for a Philosophy of Nature, translated by Errol Harris and Peter Heath (New York: Cambridge University Press, 1988), p. 177.
63. Ibid., p. 175.
64. Hegel's Philosophy of Nature, edited and translated by M. J. Petry (London: Unwin Brothers Limited, 1970), p. 262.
65. Ibid., p. 246.
66. Goethe's Color Theory, edited by Rupprecht Matthaei (New York: Van Nostrand Reinhold Company, 1971), p. 212.
67. Selected Writings of Hermann von Helmholtz, translated and edited by Russell Kahl (Connecticut: Wesleyan University Press, 1971), p. 65.
68. Werner Heisenberg, Across the Frontiers, edited by Ruth Anshen and translated by Peter Heath (New York: Harper & Row Publishers, 1974), pp. 126–127.
69. Charles Coulston Gillispie, The Edge of Objectivity (New Jersey: Princeton University Press, 1960), p. 195.
70. Werner Heisenberg, Philosophical Problems of Quantum Physics, translated by F. C. Hayes (Connecticut: Ox Bow Press, 1979), p. 71.
71. Ibid., p. 37.
72. Neil Ribe and Friedrich Steinle, “Exploratory Experimentation: Goethe, Land, and Color Theory,” in Physics Today, July 2002, p. 48.
73. Arthur Schopenhauer, On the Will in Nature, translated by E. F. J. Payne and edited by David E. Cartwright (Oxford: Berg Publishers, 1992), pp. 110–111.