Leibniz's later criticisms of Descartes' work may have been his way of further distancing himself from Descartes' genius lest he be accused of having exploited his ideas. Nothing in Descartes' work led directly to Leibniz's calculus, but Descartes' discoveries in mathematics were certainly the forerunners of the calculus.
We know that in 1661, during his first year of study at Cambridge University, Isaac Newton read books about Descartes' mathematics. Much later, after he had become a famous mathematician and scientist, Newton openly declared: “If I have seen a little farther than others, it is because I have stood on the shoulders of giants”—thus implicitly acknowledging the contributions of Galileo, Kepler, and Descartes. For without Descartes' unification of algebra and geometry it would have been impossible to describe graphs using mathematical equations, and hence, except perhaps as a pure theory, the calculus would be completely devoid of meaning.
Reluctantly, Leibniz returned to Hanover at the end of summer 1676 and spent the remaining years of his life serving the duke of Hanover in various posts. He was an educator, diplomat, counselor, and librarian. He traveled much, to Vienna, Berlin, and Italy. His final task was to write the history of the Brunswick family whom he served. When he died in 1716, that history had still not been completed. Leibniz never married. In his eulogy of Leibniz, Bernard de Fontenelle recounted that when Leibniz was fifty years old, he proposed to a lady, but that she took so much time to consider his offer that he finally withdrew it. When he died, the only heir to his considerable fortune was his sister's son. When this nephew's wife heard how much money she and her husband had inherited, she suffered a shock and died.
A Twenty first-Century epilogue
DESCARTES COULD BE VIEWED AS an early cosmologist—a scientist working to unveil the secrets of the universe. As such, he was the forerunner of Einstein, who in the fall of 1919—exactly three hundred years after Descartes' rapturous moment of discovery in November 1619— became the celebrity we know when his general theory of relativity was confirmed through measurements of the bending of starlight around the sun made by Arthur Eddington during a total solar eclipse a few months earlier that year. Descartes' spirit of discovery is further carried forward today by Stephen Hawking, Roger Penrose, and Alan Guth—the leading cosmologists of our own generation, expanding our horizons as we learn more about the workings of the universe.
In its essence, Descartes' work consisted of placing physics and cosmology on a firm mathematical foundation using Euclid's geometry as its base. Anyone who reads the works of modern cosmologists will be struck by the extent of the use of geometry in constructing models of the universe. The difference between the work of present-day scientists and that of Descartes is that modern cosmology is based on more advanced, specialized geometries such as the non-Euclidean geometries developed in the nineteenth century and used extensively by Einstein. Such geometries abandon Euclid's assumption that space is characterized by straight lines, and allow for a much more general structure of space, in which curves of various kinds are the basic elements.
But amazingly, the methods used by modern-day cosmologists are fundamentally extensions of those pioneered by Descartes. Physical space is so complex that in order to study its essential properties, cosmologists must rely on purely algebraic methods. They study the geometry of space by analyzing the properties of groups. A group is an abstract collection of elements with certain mathematical properties, a concept that is a direct result of the algebra studied by Descartes. And the connection between geometry and algebra, the very tool that allows modern cosmologists to carry out their advanced analysis, was, as we know, established by Descartes. But do the regular solids of ancient Greece— the elements of Descartes' most prized secret—have anything to do with cosmology?
On the eve of the transit of Venus across the face of the sun on June 8, 2004, a roughly twice-in-a-century event that was about to be observed by astronomers at the Observatory of the Aristotle University of Thessaloniki, in Greece, the American astronomer Jay M. Pasachoff delivered a lecture on the history of our understanding of the solar system. Referring to Kepler's cosmological model based on the five Platonic solids, Pasachoff said: “It was a beautiful theoretical model of the universe. Unfortunately, it was completely wrong.”
It would have seemed, therefore, that nothing about the Platonic solids had anything to do with the structure of the universe. And certainly Descartes' obsession with hiding his discovery about the nature of these solids was completely unnecessary, since Kepler's idea was not valid. The Platonic solids held no secret of the structure of the universe, and therefore did not constitute a real challenge to the earth-centered doctrine held by the church. But new research described in an article published in a mathematics journal in June 2004 may have changed everything.
On June 30, 2001, NASA launched the Wilkinson Microwave Anisotropy Probe (WMAP), a satellite designed to study minute fluctuations in the microwave background radiation that permeates all space as an echo of the Big Bang that created our universe. The fluctuations that the satellite was to study are believed to contain essential information on the geometry of the universe as a whole.
On August 10, 2001, the WMAP satellite reached its orbit far above the earth and directed its microwave antennas away from the earth into deep space. The stream of data the satellite has been producing ever since has been studied by scientists around the world.
But an enigma about the data puzzled the scientists. If the universe indeed had the infinite, flat geometry that scientists had assumed it to have, all fluctuation frequencies should have been present in the data. But surprisingly, certain fluctuations were not there. The absence of particular frequencies in the data implied to the scientists that the size of the universe was the culprit. The frequencies of the microwave background radiation that permeates space are similar in their essence to the frequencies of sound. And in the same way that the vibrations of a bell cannot be larger than the bell itself, the radiation frequencies in space are limited by the size of space itself. So cosmologists needed to look for new models of the structure of the universe: ones that would agree with the data from the satellite. These models should thus forbid the occurrence of radiation frequencies that were not found.
Complicated mathematical analysis was carried out in an effort to solve this mystery. And the answer they obtained surprised the scientists: the large-scale geometry of our universe that can answer the discrepancies in the data is a geometry based on some of the Platonic solids. It seems that while the orbits of the planets in our solar system do not follow the structure of the ancient Greek solids, models of the geometry of the entire universe do. In particular, cosmologist and MacArthur fellow Jeffrey Weeks exposed in an article in the Notices of the American Mathematical Society a theory that showed that tetrahedral, octahedral, and dodecahedral models of the geometry of the universe agree very well with the new findings—they completely solve the mystery of the missing fluctuations.
One new model of the geometry of the universe is thus a gigantic octahedron that is “folded onto itself in every direction. It is an octahedron in which opposite faces are identified with each other. This means that if a spaceship travels outward from the inside of the octahedron toward one of the faces, and then passes through that face, it will arrive—speeding back inward into the octahedron—from exactly the opposite face of the octahedron. Another model is that of a huge icosa-hedron, again with opposite faces identified with each other. And a third possible model that satisfies the data is a giant dodecahedron with opposite faces identified. These models give us a universe that is closed and yet has no boundaries. Traveling (in three dimensions) within such a universe is similar to traveling on the (two-dimensional) surface of the earth: if you keep going constantly east, for example, you will travel around the world and arrive back at your starting point. You will never hit a “boundary,” and you will arrive back home from “the opposite direction.” Applying this principle to traveling in the “folded onto i
tself dodecahedron, you will arrive back—in three-dimensional space—from the other side, that is, from the face opposite to the one you went through.
Since imagining such geometries is difficult, and since to a mathematician, one dodecahedron is exactly the same as another dodecahedron of the same size, a way of understanding the new proposed geometries of the universe is by visualizing a repeating pattern of such dodecahedrons (or octahedrons or icosahedrons). Space could thus be seen as a three-dimensional array of connected octahedrons, icosahedrons, or dodecahedrons, infinitely extended in every direction. These possible geometries of our universe are shown below.
If this theory holds under the scrutiny of other experts, and survives the test of time, Kepler will have been proven right in assuming a connection between the Platonic solids and cosmology—albeit one he could not have envisioned. And Descartes may well have been correct in believing that the objects of his great mathematical discovery held profound cosmological relevance.
Cosmological models of the universe based on Platonic solids
INTRODUCTION
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5 “Who are we as minds in relation to our bodies?”: Roger Ariew and Marjorie Grene, eds. Descartes and His Contemporaries, 1.
5-6 which incorporated into philosophy the elements of modern psychology: Victor Cousin, Histoire generale de la philosophic depuis les temps les plus anciens, 359.
7 between the rue du Roi de Sidle and the rue des Bhncs-Manteaux: Baillet, whose 1691 biography of Descartes is still the most comprehensive after more than three hundred years, does not mention today's rue des Rosiers, which lies between the two cross streets.
PROLOGUE: LEIBNIZ'S SEARCH IN PARIS
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11 found it difficult to reassemble the manuscripts: Charles Adam and Paul Tannery, Oeuvres de Descartes (1974), hxviii, after Baillet. Adam and Tannery, who quote this story from Baillet, note that Baillet would well have known exactly what happened to the papers, since he collaborated in his own biography of Descartes with Father Legrand, who received the papers from Clerselier in 1684.
14 eagerly asked him if there was anything else: The dates—June 1, 1676, for starting the copying and June 5, 1676, for copying the secret notebook—are mentioned in Henri Gouhier, Les premieres pensees de Descartes, 14, and are based on dates inserted by Leibniz in his manuscript copy, later recopied in the nineteenth century by Foucher de Careil.
14 imposed tight restrictions on the access to this notebook: Pierre Costabel, ed. Rene Descartes: Exercises pour les elements des solides, ix.
16 Part of Leibniz's copy of Descartes' secret notebook: Reproduced by permission from the Gottfried Wilhelm Leibniz Library in Hanover, Germany. I am indebted to Birgit Zimny for the reproduction and a copy of the entire Leibniz manuscript.
CHAPTER 1: THE GARDENS OF TOURAINE
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18 Joachim Descartes was the councillor of the Parliament of Brittany: Clearly, pre-Revolution France was not a democracy, so we cannot interpret the function of a regional Parlement as that of, for example, the British Parliament of today. This institution had legislative and judicial roles, but they were subjected to royal authority, and these roles were more akin to those of a high court. Indeed, some of Descartes' biographers have translated the name of the institution at which Descartes' father worked as the “High Court of Rennes.”
20 there were only seventy-two Protestant baptisms in La Haye: From information kept by the Descartes Museum in Descartes, Touraine.
21 His baptismal certificate reads: I am indebted to Ms. Daisy Esposito of the Descartes Museum in Descartes, Touraine, for showing me a copy of the “Acte de bapteme de Rene Descartes” and allowing me to translate it.
21 Her brother's wife was Jeanne (Jehanne) Proust: This information is based on the genealogy drawn by Alfred Barbier in 1898. I am indebted to Ms. Daisy Esposito for providing me with the family tree produced by Mr. Barbier.
22 “the land of the bears, between rocks and ice?”: Descartes to Princess Elizabeth, April 23, 1649, quoted in Jean-Marc Varaut, Descartes: On cavalier francais, 256.
24 dust rising up from the earth as it was being plowed: Varaut, 44. See also Descartes, Discours de la Methode, edited by Etienne Gilson, 108.
25 eighteen years after Rene's death: Genevieve Rodis-Lewis, Descartes (Paris, 1995), 18.
25 the three-generations requirement, years after Rene's death: Baillet's errors, however, are few and far between, and modern scholars have found it difficult to disprove facts in his biography of Descartes. Generally, he seems to have done a very good job, and his biography serves as the primary source of information about the life of the philosopher-mathematician, along with the many surviving letters Descartes had written.
CHAPTER 2: JESUIT MATHEMATICS AND THE PLEASURES OF THE CAPITAL
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28 right after Easter 1607: Varaut, 48.
30 they attended a spiritual lecture: Varaut, 49.
30 as well as logic, physics, and metaphysics: Vittorio Boria, “Marin Mersenne: Educator of Scientists,” 12-30. 32 “that has thus hid them out”: Descartes, Oiscours de h methode, Gallimard ed.,
83-84, author's translation. 34 forever to remain in the church: BaiUet (1691), 1:22.
34 he moved to Paris: BaiUet found Descartes' year in Poitiers so uninteresting that he didn't even mention it in his biography. 36 characterized his early days in Paris: BaiUet (1691), 1:37.
38 His friends were close to giving him up as lost: BaiUet (1691), 1:36.
39 true judgment on the evaluation of all things: F. Alquie, ed., Descartes: Oeuvres philosophiques, 1:46-47.
CHAPTER 3: THE DUTCH PUZZLE
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41 assistant principal of the Latin School of Utrecht: Adam and Tannery (1986), X:22.
42 “And I suppose you willgive me the solution, once you have solved this problem?”: Adam and Tannery (1986), X:46-47.
45 “Hence, there is no such thing as an angle”: Adam and Tannery, (1986), X:46-47; Beeckman's Journal, 1:23 7, translated in Cole, Olympian Dreams, 80.
46 “at the beginning of Lent”: Descartes to Beeckman, January 24, 1619, in F. Alquie, ed., Descartes: Oeuvres philosophiques, 1:35.
48 “and so on for another twenty hours”: Descartes to Beeckman, April 29, 1619, in Alquie, Oeuvres philosophiques, 1:42—43. 48 new concepts could be derived: Frances A. Yates, The Art of Memory, 180-84.
48 would likely never be met: BaiUet (1692), 28.
51 Prince Frederick of the Pahtinate: Frances A. Yates, The Rosicrucian Enlightenment, 23.
51 “and are you still concerned with getting married?”: In Alquie, Oeuvres philosophiques, 1:45.
52 the two friends met almost every day: Adam and Tannery (1986), X:25. 52 “nor where I shall stop along the way”: Adam and Tannery (1986), X:162.
52 “honor you as the promoter of this work”: Adam and Tannery (1986), X:162. This letter was found together with Beeckman's journal in 1905.
52 Descartes was present at this magnificent ceremony: Some modern researchers have questioned Descartes' extensive itinerary, saying it would have taken too long, given where we know he finally arrived. But Baillet is generally correct, and perhaps there is no justification for underestimating the speed at which travel could be achieved in the seventeenth century. At any rate, from Descartes' own writings in the Discourse on the Method (second part; Gallimard ed., 84), we know that Descartes was certainly at the coronation of the emperor.
53 Baillet tells us: Baillet (1691), 1:63.
54 would not carry a musket, only his sword: Baillet (1692), 30.
CHAPTER 4: THREE DREAMS IN AN OVEN BY THE DANUBE
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56 “to entertain myself with my own thoughts”: Descartes, Discourse on the Method, second part, in Alquie, Oeuvres philosophiques, 1:578. 56 used both for cooking and for heating in winter: Varaut, 69.
58 “an evil spirit that wanted to seduce him”: Baillet (1691), 1:81.
58 tried to take hold of the Cor
pus poetarum, it disappeared: John R. Cole, The Olympian Dreams and Youthful Rebellion of Rene Descartes, 228 n. 14, identi fies the two editions of the anthology Corpus omnium veterum poetarum latinorum that were available during Descartes' early years, and notes that in these editions, the two poems “Quod vitae sectabor iterl” and “Est et Non” ap pear either on the same page or on facing pages. It seems that Descartes' memory of reading these poems served him even in his dream.
59 “truth and falsehood in the secuhr sciences”: Baillet (1691), I;82. 61 “Anno 1619 Kalendis Januarii”: Adam and Tannery (1986), X:7.
61 who discovered the laws of planetary motion: Edouard Mehl, Descartes en Allemagne, 17.
61 has conjectured that such a meeting indeed took place: L. Gabe, “Cartelius oder Cartesius: Eine Korrectur zu meinem Buche iiber Descartes Selbstkritik, Hamburg, 1972,” Archiv fur Geschichte der Philosophy 58 (1976), 58-59.
62 “truly worthy of your consideration”: Quoted in Mehl, 189. See also William R. Shea, The Magic of Numbers and Motion: The Scientific Career of Rene Descartes, 105.
Descartes's Secret Notebook Page 21