Is God a Mathematician?

by Mario Livio

  to help publicize a novel entitled: The novel is Uncle Petros and Goldbach’s Conjecture, by A. K. Doxiadis (Doxiadis 2000).

  innocent-looking example known as Catalan’s conjecture: For a detailed description see Ribenboim 1994.

  Some mathematicians, philosophers, cognitive scientists: I shall discuss these opinions extensively in chapter 9.

  “According to the prophets, the last”: Bell 1940.

  Chapter 3. Magicians: The Master and the Heretic

  “Some existing things are natural”: Aristotle ca. 330 BCa, b; see also Koyré 1978.

  Using a clever thought experiment: Galileo 1589–92.

  virtually complete system of logical inference: This and other logical constructs will be discussed extensively in chapter 7.

  When the historian of mathematics: Bell 1937.

  written by one Heracleides: This is mentioned in commentaries on the Measurement of a Circle by the mathematician Eutocius (ca. 480–540 AD); see Heiberg 1910–15.

  more interested in the military accomplishments: Plutarch ca. 75 AD.

  Archimedes was born in Syracuse: His year of birth has been determined based on the Chiliades, by the twelfth century Byzantine writer Johannes Tzetzes.

  Archimedes spent some time in Alexandria: Evidence discussed in Dijksterhuis 1957.

  This immediately triggered a solution: The Roman architect Marcus Vitruvius Pollio (first century BC) tells us the story in his treatise De Architectura. (See Vitruvius 1st century BC.) He says that Archimedes immersed in water a piece of gold and a piece of silver, both having the same weight as the wreath. He thus found that the wreath displaced more water than the gold but less than the silver. It is easy to show that from the different volumes of water displaced one can calculate the ratio of the weights of the gold and the silver in the wreath. Therefore, contrary to some popular accounts, Archimedes did not need to use the laws of hydrostatics to solve the problem of the wreath.

  has been cited by: In a letter from Thomas Jefferson to M. Correa de Serra in 1814, he wrote: “The good opinion of mankind, like the lever of Archimedes, with the given fulcrum, moves the world.” Lord Byron mentions Archimedes’ statement in Don Juan. JFK used the phrase in a campaign speech, cited in The New York Times, on November 3, 1960. Mark Twain used it in an article entitled “Archimedes” in 1887.

  Archimedes used an assembly of mirrors: A group of MIT students attempted to reproduce the burning of a ship with mirrors in October 2005. Some of them also repeated the experiment for the TV show Myth Busters. The results were somewhat inconclusive in that while the students were able to achieve a burning area that was self-sustaining, they did not produce a large ignition. A similar experiment performed in Germany in September 2002 did manage to ignite the sail of a ship by using 500 mirrors. A discussion of the burning mirrors can be found on a website by Michael Lahanas.

  According to some accounts: Those precise words from Archimedes are mentioned in the Chiliades by Tzetzes; see Dijksterhuis 1957. Plutarch says simply that Archimedes refused to follow the soldier to Marcellus until he had solved the problem in which he was absorbed (Plutarch ca. 75 AD).

  As the British mathematician and philosopher: Whitehead 1911.

  Archimedes’ opus covers an astonishing range: A superb book on Archimedes’ work is The Works of Archimedes (Heath 1897). Other excellent expositions can be found in Dijksterhuis 1957 and Hawking 2005.

  “There are some, king Gelon”: Heath 1897.

  The story of this discovery: For a wonderful description of the history of the Palimpsest Project, see Netz and Noel 2007.

  Sometime in the tenth century: Probably in 975 AD.

  The scribe Ioannes Myronas: Netz and Noel 2007.

  I was fortunate enough to meet: Will Noel, who is the director of the project, arranged for a meeting with William Christens-Barry, Roger Easton, and Keith Knox. This team designed the narrow-band imaging system and invented the algorithm used to reveal some of the text. Image-processing techniques have also been developed by researchers Anna Tonazzini, Luigi Bedini, and Emanuele Salerno.

  “I will send you the proofs”: Dijksterhuis 1957.

  his anticipation of integral and differential calculus: For a beautiful description of the history and meaning of calculus see Berlinski 1996.

  The Greek mathematician Geminus: Heath 1921.

  he requested it be engraved: Plutarch ca. 75 AD.

  Here is Cicero’s rather moving description: Cicero 1st century BC. For a scholarly analysis of Cicero’s text in terms of structure, rhetoric, and symbolic function, see Jaeger 2002.

  Galileo Galilei was born in Pisa: An authoritative modern biography is S. Drake’s Galileo at Work (Drake 1978). A more popular account is J. Reston’s Galileo: A Life (Reston 1994). See also Van Helden and Burr 1995. The complete works of Galileo appear (in Italian) in Favaro 1890–1909.

  “Those who read his works”: In The Little Balance, Galilei 1586.

  “wood moves more swiftly”: Galileo 1589–92 (Galilei 1600a and Galilei 1600b). C. B. Schmitt suggests (Schmitt 1969, after D. A. Maklich) that Galileo’s statement may be the result of the hand holding a lead ball being more tired than the hand holding a wooden ball, and consequently that the release of the wooden ball is more prompt. An excellent presentation of Galileo’s correct ideas on falling bodies can be found in Frova and Marenzana 1998 (McManus’s 2006 translation). A superb discussion of Galileo’s physics can be found in Koyré 1978.

  Viviani created the popular image: A thorough discussion of Galileo’s methods and thought process can be found in Shea 1972, and in Machamer 1998.

  “was ignorant not only”: Galileo 1589–92. Galileo profusely criticizes Aristotle in De Motu. See Galilei 1600a, b.

  Virginia, Livia, and Vincenzio: The life story of Virginia, later known as Sister Maria Celeste, is beautifully told in Dava Sobel’s Galileo’s Daughter (Sobel 1999).

  “About 10 months ago”: Galilei 1610a, b. An excellent description of the work that led to the telescope can be found in is Reeves 2008.

  As the historian of science Noel Swerdlow: Swerdlow 1998. For a detailed description of Galileo’s discoveries with the telescope, see Shea 1972, Drake 1990.

  Turning his telescope to the Moon: A more popular and very engaging description of Galileo’s discoveries, as well as a general history of the telescope, can be found in Panek 1998.

  The importance of the discovery: Galileo’s Copernicanism is discussed extensively by Shea 1998 and Swerdlow 1998.

  a playful Galileo sent Kepler: The letter itself was written to the Tuscan ambassador to Prague, but Galileo enclosed the anagram for Kepler.

  Kepler tried unsuccessfully to decipher: In fact, he wrote to Galileo: “I abjure you not to leave us long in doubt of the meaning. For you see you are dealing with real Germans. Think in what distress you place me by your silence.” Quoted in Caspar 1993.

  Scheiner argued that it was impossible: The entire episode is discussed in detail in Shea 1972.

  The Scottish poet Thomas Seggett: The epigram was in Latin. Seggett (1570–1627) had been a pupil with Galileo in Padua. The epigram appears in Favaro’s Le Opere. A beautiful discussion of poetry related to telescopes can be found in Nicolson’s Modern Philology (Nicolson 1935).

  Sir Henry Wotton, an English diplomat: Curzon 2004.

  Here is the Aristotelian Giorgio Coresio: Coresio 1612. Also cited in Shea 1972.

  the Pisan philosopher Vincenzo di Grazia: Appears in di Grazia’s Considerazioni (1612), which is reprinted in Favaro’s Opere di Galileo, vol. 4, p. 385.

  In the draft of his treatise: Cited in Shea 1972.

  His premise of celestial immutability: The entire story of the controversy over the nature of sunspots is described beautifully in Van Helden 1996 and in Swerdlow 1998. See also Shea 1972.

  The entire story of The Assayer: Galilei 1623.

  were delivered by Galileo’s disciple: Antonio Favaro, who edited all of Galileo’s works, found that large par
ts of Guiducci’s manuscript (containing the lectures) were written in Galileo’s handwriting.

  Discourses and Mathematical Demonstrations: Galilei 1638.

  what was truly at the heart: Excellent discussions of Galileo’s opinions on the relation between science and scripture can be found in Feldberg 1995 and in McMullin 1998.

  In a long letter to Castelli: Appears also in von Gebler 1879.

  was clearly at odds with that: Theologian Melchor Cano stated in 1585 that “not only the words but even every comma [in the scripture] has been supplied by the Holy Spirit.” Cited in Vawter 1972.

  Galileo’s further attempts to rely: An extensive description can be found in Redondi 1998.

  Dialogue Concerning the Two Chief: Galilei 1632.

  “We condemn you to the formal”: de Santillana 1955.

  “Therefore, desiring to remove from”: de Santillana 1955.

  “The fact that the Pope”: Beltrán Mari 1994. See also discussion in Frova and Marenzana 1998.

  Chapter 4. Magicians: The Skeptic and the Giant

  “the greatest single step ever made”: Cited in Sedgwick and Tyler 1917.

  René Descartes was born on March 31: There are numerous biographies of Descartes. The classic is Baillet 1691. Other books I found helpful were Vrooman 1970 and the relatively recent Rodis-Lewis 1998. Bell 1937 gives a brief but beautiful summary. Very interesting also are Finkel 1898, Watson 2002, and Grayling 2005.

  Descartes asked the first passer-by: While there is no doubt that Descartes did indeed meet Beeckman on that day, Beeckman never mentions any problem on a billboard in his journal. Beeckman rather says that Descartes “made every effort to prove that in reality the angle does not exist.”

  whose influence on Descartes’ physico-mathematical: See Gaukroger 2002 for a description.

  Descartes experienced three dreams: Most biographers locate this night as occurring in the town of Ulm in the state of Neuburg. Descartes himself told the story in a notebook that was seen by his early biographers. Only a few transcribed passages have survived. Descartes repeated impressions of these dreams in his Discourse (Adam and Tannery 1897–1910). A quite comprehensive description of the dreams and their possible interpretations can be found in Grayling 2005 and Cole 1992.

  as Descartes wrote to his friend: Letter to Pierre Chanut, France’s ambassador to Sweden, who was also an amateur philosopher. Adam and Tannery 1897–1910.

  Descartes was buried in Sweden: Originally he was buried in the cemetery of Nord-Malmoe. When the remains were transferred to France, there were rumors (Adam and Tannery 1897–1910) that part of them, the skull in particular, remained in Sweden. In France, the remains were first buried in the Abbey of Sainte-Geneviève, then in the convent of the Petits-Augustines. Finally, the remains were put in the Saint-Germain-des-Prés Cathedral, in what is today the Saint-Benoit Chapel. I had a hard time finding it, because I couldn’t believe that Descartes was not buried all by himself. In fact, in the same chapel are buried the two Benedictines Mabillon and Montfaucon, and there is only the bust of Mabillon.

  What makes Descartes a true modern: For one perspective see Balz 1952.

  Descartes recognized that the methods: The standard, authoritative compilation of Descartes’ works is that by Adam and Tannery 1897–1910. Most of my quotes come from this source. Many translations exist of a number of individual works, such as Veitch’s 1901 The Philosophy of Descartes, which contains Discourse on Method, the Meditations, and the Principles of Philosophy. On Descartes’ philosophy of science see also Clarke 1992.

  this deluge of troubling doubts: An excellent introduction to Descartes’ philosophy in general can be found in Cottingham 1986. For a discussion of the Cartesian doubt and the ensuing Cogito see Wolterstorff 1999, Ricoeur 1996, Sorell 2005, Curley 1993, and Beyssade 1993.

  He outlined it in a 106-page appendix: Descartes 1637. One of the translations of the entire book is P. J. Olscamp’s 1965 edition (Descartes 1637a). A beautiful translation of The Geometry, which also includes a facsimile of the first edition, is The Geometry of René Descartes (translated by D. E. Smith and M. L. Latham; Descartes 1637b).

  Descartes discovered a way to represent: Descartes’ mathematical achievements are nicely summarized in Rouse Ball 1908. A beautiful popular description of Descartes’ life and work can be found in Aczel 2005. The level of abstraction exhibited in Descartes’ algebra is analyzed in Gaukroger 1992.

  writing his treatise on cosmology and physics: Descartes’ conviction in the existence of “laws of nature” can be gleaned from a letter he wrote to Mersenne in May 1632: “Now I have become bold enough to seek the cause of the position of each fixed star. For although their distribution seems irregular, in various parts of the universe, I have no doubt that there is between them a natural order which is regular and determinate.”

  Two of his laws closely resemble: Adam and Tannery 1897–1910. See also Miller and Miller 1983. A good discussion of Descartes’ physics can be found in Garber 1992. A more general description of Descartes’ natural philosophy appears in Keeling 1968.

  Newton’s tomb inside Westminster Abbey: The monument was erected in 1731. It was commissioned from William Kent and the Flemish sculptor Michael Rysbrack. In addition to Newton’s figure, whose elbow rests on some of his works, the sculpture shows youths carrying emblems of Newton’s main discoveries. Behind the sarcophagus there is a pyramid, from the middle of which rises a globe on which several constellations are drawn, as well as the path of the comet of 1681.

  Actually, Newton may have written that phrase: It is impossible to know for sure whether Newton meant this as an insult or not. R.K. Merton did find “on the shoulders of giants” to be a fairly common phrase by Newton’s time (Merton 1993).

  In his reply to Hooke’s letter: Impressively, Newton’s entire correspondence has been collected in Turnbull, Scott, Hall, and Tilling 1959–77.

  The feud between the two scientists: The feud is described in great detail in a few excellent biographies of Newton, including Westfall 1983, Hall 1992, and Gleick 2003.

  looked like nothing but a collection: In an essay published in 1674, Hooke wrote about gravity that its “attractive powers are so much more powerful in operating, by how much nearer the body wrought upon is to their own centers.” Hence, while he had the correct intuition, he failed to describe it mathematically.

  “We offer this work as the mathematical principles”: There are a number of excellent translations of Newton’s Principia, including Motte 1729 and Cohen and Whitman 1999 (see Newton 1729). The most accessible with helpful notes is Chandrasekhar’s 1995 edited version. The general concept of a law of gravity and its history is discussed extensively in Girifalco 2008, Greene 2004, Hawking 2007, and Penrose 2004.

  even in his more experimentally based book on light: Newton 1730.

  In his Memoirs of Sir Isaac Newton’s Life: Stukeley 1752. In addition to the full biographies, there are small books describing certain episodes in the life of Newton or his relatives. Among these I note De Morgan 1885 and Craig 1946.

  Irrespective of whether the mythical event: In his biography of Newton, David Brewster wrote in 1831: “The celebrated apple tree, the fall of one of the apples of which is said to have turned the attention of Newton to the subject of gravity, was destroyed by wind about four years ago; but Mr. Turnor [the proprietor of Newton’s house in Woolsthorpe] has preserved it in the form of a chair.” Brewster 1831.

  It may have all started in Newton’s youth: A good description of Newton’s studies of mathematics is given in Hall 1992.

  “And the same year [1666] I began”: The memorandum is in the Portsmouth Collection. There are other documents suggesting that Newton did indeed think of the inverse square law of gravity during the plague years. See Whiston 1753, for example.

  For reasons that are not entirely clear: For a general discussion of the reasons for the delay in Newton’s announcement of the law of gravitation see Cajori 1928 and Cohen 1982. In the
next section I summarize what I regard as the two most convincing suggestions as to what the reasons might have been.

  “In 1684 Dr Halley came”: De Moivre was recalling here what Newton had described to him.

  Some even speculate that he: This is suggested by Cohen 1982, to name just one source.

  In his address at the bicentenary: Glaisher 1888.

  For Newton, the world’s very existence: In Principia he says about God: “He is omnipresent not only virtually but also substantially…He is all eye, all ear, all brain, all arm, all force of sensing, of understanding, and of acting.” In a manuscript from the early 1700s, purchased at Sotheby’s in 1936 and exhibited in Jerusalem in 2007, Newton used the biblical book of Daniel to calculate the date for the apocalypse. In case you are worried, he reached the conclusion that he saw no reason for “its [the world] ending sooner” than 2060.

  The validity of the cosmological, teleological: For excellent recent discussions of the history of these arguments and an assessment of their logical soundness, see Dennett 2006, Dawkins 2006, and Paulos 2008.

  This type of logical maneuvering: See Dennett 2006, Dawkins 2006, Paulos 2008.

  Chapter 5. Statisticians and Probabilists: The Science of Uncertainty

  The branch of mathematics called calculus: Extremely accessible descriptions of calculus and its applications can be found in Berlinski 1996, Kline 1967, and Bell 1951. Somewhat more technical, but truly excellent is Kline 1972.

  were members of the legendary Bernoulli family: For some of the achievements of this remarkable family, see Maor 1994, Dunham 1994. See also the “Bernoulli-Edition” (in German) on the University of Basel Web page (http://www.ub.unibas.ch/spez/bernoulli.htm). Information on the project in English can be found at http://www.springer.com/cda/content/document/cda_downloaddocument/bernoulli2005web.Pdf?SGWID=0-0-45-169442-0.

  known for their bitter intrafamily feuds: Described in Hellman 2006.