Is God a Mathematician?

by Mario Livio

  became known as the problem of the catenary: An excellent description of the problem, and in particular of Huygens’s solution, can be found in Bukowski 2008. The solutions of Bernoulli, Leibniz, and Huygens appear in Truesdell 1960.

  “You say that my brother proposed”: Quoted in Truesdell 1960.

  in his Philosophical Essay on Probabilities: Laplace 1814 (translated by Truscot and Emory in 1902).

  John Graunt (1620–74) was trained: Excellent descriptions of Graunt’s life and work can be found in Hald 1990, Cohen 2006, and Graunt 1662.

  Halley’s paper, which had the rather long title: The paper is reprinted in Newman 1956.

  Here is how Jakob Bernoulli described: Quoted in Newman 1956. His work is summarized in Todhunter 1865.

  Adolphe Quetelet was born: Two excellent books on Quetelet and his work are Hankins 1908 and Lottin 1912. Shorter but also informative pieces can be found in Stigler 1997, Krüger 1987, and Cohen 2006.

  “Chance, that mysterious, much abused word”: Quetelet 1828.

  was in fact a type that nature: Quetelet wrote in his memoir on the propensity to crime: “If the average man were determined for a nation he would represent the type of that nation; if he could be determined from the ensemble of men, he would represent the type of the entire human race.”

  The person who first introduced: For a popular exposition of the work of Galton and Pearson, see Kaplan and Kaplan 2006.

  The serious study of probability: Recently published, entertaining popular accounts of probability, its history, and its uses include Aczel 2004, Kaplan and Kaplan 2006, Connor 2006, Burger and Starbird 2005, and Tabak 2004.

  in a letter dated July 29, 1654: Todhunter 1865, Hald 1990.

  The essence of probability theory: An excellent, popular, brief description of some of the essential principles of probability theory can be found in Kline 1967.

  Probability theory provides us with accurate information: The relevance of probability theory to many real-life situations is beautifully described in Rosenthal 2006.

  The person who brought probability: For an excellent biography, see Orel 1996.

  Mendel published his paper: Mendel 1865. An English translation can be found on the Web page created by R. B. Blumberg at http://www.mendelweb.org.

  While some questions related to the accuracy: See Fisher 1936, for example.

  the influential British statistician: For a brief description of some of his work see Tabak 2004. Fisher wrote an extremely original, nontechnical article about the design of experiments entitled “Mathematics of a Lady Tasting Tea” (see Fisher 1956).

  in his book Ars Conjectandi: For a superb translation see Bernoulli 1713b.

  He then proceeded to explain: Reprinted in Newman 1956.

  Shaw once wrote an insightful article: The article “The Vice of Gambling and the Virtue of Insurance” appears in Newman 1956.

  In a pamphlet entitled The Analyst: The pamphlet was written by George Berkeley in 1734. An edited version by David Wilkins is maintained on the Web; see Berkeley 1734.

  Chapter 6. Geometers: Future Shock

  In his famous book Future Shock: Toffler 1970.

  Hume identified “truths”: Hume 1748.

  Kant asked not what we can know: According to Kant, one of the fundamental philosophical tasks is to account for the possibility of synthetic a priori knowledge of mathematical concepts. Among the many references, I note Höffe 1994 and Kuehn 2001 for the general concepts. A good discussion of the application to mathematics can be found in Trudeau 1987.

  “Space is not an empirical”: Kant 1781.

  The first four Euclidean axioms: For a relatively gentle introduction to Euclidean and non-Euclidean geometries, see Greenberg 1974.

  the proofs of the first twenty-eight: Theorems proven without the fifth postulate are discussed in Trudeau 1987.

  Some of those endeavors started: An excellent description of all the attempts that had eventually led to the development of non-Euclidean geometry can be found in Bonola 1955.

  The first to publish an entire treatise: George Bruce Halsted’s 1891 translation of Lobachevsky’s “Geometrical Researches on the Theory of Parallels” is included in Bonola 1955.

  a young Hungarian mathematician, János Bolyai: For a biography and a description of his work, see Gray 2004. The reason I have not included a picture of János Bolyai is that the picture usually used is of doubtful authenticity. Apparently his only relatively reliable portrait is a relief in the façade of the Palace of Culture in Marosvásárhely.

  The manuscript was entitled The Science Absolute of Space: A facsimile of the original (in Latin) and the translation into English by George Bruce Halsted appear in Gray 2004.

  There is very little doubt, however: An excellent description of the entire episode, from the perspective of Gauss’s life and work, can be found in Dunnington 1955. A concise but accurate summary of the claims of Lobachevsky and Bolyai for priority is given in Kline 1972. ome of Gauss’s correspondence on non-Euclidean geometry is presented in Ewald 1996.

  In a brilliant lecture delivered in Göttingen: An English translation of the lecture, as well as other seminal papers on non-Euclidean geometries, together with illuminating notes, can be found in Pesic 2007.

  Poincaré’s views were inspired: Poincaré 1891.

  in the first chapter of the Ars Magna: Cardano 1545.

  In another important book, Treatise of Algebra: Wallis 1685. A concise summary of Wallis’s biography and work can be found in Rouse Ball 1908.

  Opinions eventually started to change: A brief summary of the history is given in Cajori 1926.

  In an article entitled “Dimension”: This article appeared in Diderot’s Encyclopédie. Quoted in Archibald 1914.

  stating more assertively in 1797: Lagrange 1797.

  Grassmann, one of twelve children: An excellent biography and description of Grassmann’s work (in German) can be found in Petsche 2006. A good brief summary can be found in O’Connor and Robertson 2005.

  It is fascinating to follow: Relatively accessible (but still technical) descriptions of his work in linear algebra can be found in Fearnley-Sander 1979 and 1982.

  By the 1860s n-dimensional geometry: A good introductory text is Sommerville 1929.

  by the following “declaration of independence”: The text appears in Ewald 1996.

  To which algebraist Richard Dedekind: The text appears in Ewald 1996.

  Here is how the French mathematician: Stieltjer’s first letter to Hermite was dated November 8, 1882. The correspondence between the two mathematicians consists of 432 letters. The full correspondence appears in Hermite 1905. I translated the text that appears here.

  “Mathematicians have constructed a very large”: The lecture can be found in O’Connor and Robertson 2007.

  Chapter 7. Logicians: Thinking About Reasoning

  The sign outside a barber shop: The paradox of the village barber is discussed in many books. See Quine 1966, Rescher 2001, and Sorensen 2003, for example.

  Here is how Russell himself described: Russell 1919. This was Russell’s more popular exposition of his ideas in logic.

  For completeness, I should note: Brouwer’s intuitionist program is summarized nicely by van Stegt 1998. An excellent popular exposition is by Barrow 1992. The debate between formalism and intuitionism is popularly described in Hellman 2006.

  “the meaning of a mathematical statement”: Dummett adds that “an individual cannot communicate what he cannot be observed to communicate: if an individual associated with a mathematical symbol or formula some mental content, where the association did not lie in the use he made of the symbol or formula, then he could not convey that content by means of the symbol or formula, for his audience would be unaware of the association and would have no means of becoming aware of it.” Dummett 1978.

  logic dealt with the relationships: An extremely accessible introduction to logic can be found in Bennett 2004. More technical, but brilliant, is Quine 198
2. A nice summary of the history of logic, by Czeslaw Lejewski, appears in the Encyclopaedia Britannica, 15th edition.

  De Morgan was an incredibly prolific: A concise but insightful description of his life and work is given in Ewald 1996.

  recounted in The Mathematical Analysis of Logic: Boole 1847.

  George Boole was born: For a full-length biography see MacHale 1985.

  “The design of the following treatise”: Boole 1854.

  In spite of the soundness of Boole’s conclusion: Boole concluded that when it comes to the belief in God’s existence, the faith-based, non-logical “feeble steps of an understanding limited in its faculties and its materials of knowledge, are of more avail than the ambitious attempt to arrive at a certainty unattainable on the ground of natural religion.”

  Frege managed to publish his first revolutionary work: Frege 1879. his is one of the most important works in the history of logic.

  In his Basic Laws of Arithmetic: Frege 1893, 1903.

  Frege’s logical axioms generally: For a general discussion of Frege’s ideas and formalism see Resnik 1980, Demopoulos and Clark 2005, Zalta 2005 and 2007, and Boolos 1985. For an excellent general discussion of mathematical logic, see DeLong 1970.

  He then went on to define all the natural numbers: Frege 1884.

  “all of those classes that are not members”: Russell’s paradox and its implications and possible remedies are discussed, for instance, in Boolos 1999, Clark 2002, Sainsbury 1988, and Irvine 2003.

  the landmark three-volume Principia Mathematica: Whitehead and Russell 1910. For a popular but illuminating description of Principia’s contents, see Russell 1919.

  In the Principia, Russell and Whitehead: For the interaction between Russell’s and Frege’s ideas, see Beaney 2003. For Russell’s logicism, see Shapiro 2000 and Godwyn and Irvine 2003.

  Russell proposed a theory of types: An excellent discussion can be found in Urquhart 2003.

  Russell’s theory of types was viewed: The theory of types has indeed fallen out of favor with most mathematicians. However, a similar construct has found new applications in computer programming. See Mitchell 1990, for example.

  the German mathematician Ernst Zermelo: See Ewald 1996 for a description of his contributions.

  Zermelo’s scheme was further augmented: Translations of the original papers by Zermelo, Fraenkel, and logician Thoralf Skolem can be found in van Heijenoort 1967. For a relatively gentle introduction to sets and the Zermelo-Fraenkel axioms, see Devlin 1993.

  the axiom of choice states: A very detailed discussion of the axiom can be found in Moore 1982.

  known as the continuum hypothesis: Cantor devised a method to compare the cardinality of infinite sets. In particular, he proved that the cardinality of the set of real numbers is larger than the cardinality of the set of the integers. He then formulated the continuum hypothesis, which stated that there is no set with a cardinality that is strictly between those of the integers and the real numbers. When David Hilbert posed his famous problems in mathematics in 1900, the question of whether the continuum hypothesis held true was his first problem. For a relatively recent discussion of this problem, see Woodin 2001a, b.

  by the American mathematician Paul Cohen: He described his work in Cohen 1966.

  mathematics proper consisted simply of a collection: A good description of the Hilbert program can be found in Sieg 1988. An excellent, updated review of the philosophy of mathematics, and a clear summary of the tensions among logicism, formalism, and intuitionism are presented in Shapiro 2000.

  “My investigations in the new grounding”: Hilbert delivered this lecture in Leipzig in September 1922. The text can be found in Ewald 1996.

  to his formalist followers: For a good discussion on formalism see Detlefsen 2005.

  considered by some to be the greatest: R. Monk presents a wonderful biography (Monk 1990).

  “If I am unclear about the nature”: In Waismann 1979.

  Kurt Gödel was born: A recent biography is Goldstein 2005. The standard biography has been Dawson 1997.

  he published his incompleteness theorems: Excellent books on Gödel’s theorems, their meaning, and their connection with other branches of knowledge include Hofstadter 1979, Nagel and Newman 1959, and Franzén 2005.

  “But, despite their remoteness”: Gödel 1947.

  Gödel the man was every bit: A comprehensive description of Gödel’s philosophical views and how he related philosophical ideas to the foundations of mathematics can be found in Wang 1996.

  “It was in 1946 that Gödel”: Morgenstern 1971.

  Now all that existed was an incomplete: This is clearly a huge over-simplification, allowed only in a popular text. In fact, serious attempts in logicism continue even today. These typically assume that many mathematical truths are knowable a priori. See Wright 1997 and Tennant 1997, for example.

  Chapter 8. Unreasonable Effectiveness?

  Various knots were even given: An interesting book on making knots is Ashley 1944.

  The mathematical theory of knots: Vandermonde 1771. An excellent review of the history of knot theory can be found in Przytycki 1992. lively introduction to the theory itself is presented in Adams 1994. popular account is given in Neuwirth 1979, Peterson 1988, and Menasco and Rudolph 1995.

  Thomson’s efforts concentrated on formulating: Excellent descriptions are presented by Sossinsky 2002 and Atiyah 1990.

  Tait started his classification: Tait 1898, Sossinsky 2002. A brief, well-written biography of Tait can be found in O’Connor and Robertson 2003.

  Maxwell offered the following rhyme: Knott 1911.

  University of Nebraska professor: Little 1899.

  Topology—the rubber-sheet geometry: A technical but still elementary introduction to topology is provided in Messer and Straffin 2006.

  the New York lawyer: Perko 1974.

  A breakthrough in knot theory came: Alexander 1928.

  the prolific English-American mathematician: Conway 1970.

  An examination of that relation eventually revealed: Jones 1985.

  in a wide range of sciences: For instance, mathematician Louis Kauffman has demonstrated a relationship between the Jones polynomial and statistical physics. An excellent but technical book on physics applications is Kauffman 2001.

  The agents that take care: An excellent description of knot theory and the action of enzymes is given in Summers 1995. See also Wasserman and Cozzarelli 1986.

  String theory appears to be: For wonderful popular accounts of string theory, its successes and problems, see Greene 1999, Randall 2005, Krauss 2005, and Smolin 2006. For a technical introduction, see Zweibach 2004.

  string theorists Hirosi Ooguri and Cumrun Vafa: Ooguri and Vafa 2000.

  created an unexpected relation: Witten 1989.

  rethought from a purely mathematical perspective: Atiyah 1989; see Atiyah 1990 for a broader perspective.

  Eric Adelberger, Daniel Kapner, and their collaborators: Kapner et al. 2007.

  Einstein had a very strong reason: There are many excellent expositions of the ideas of special and general relativity. I’ll mention here only a few that I have particularly liked: Davies 2001, Deutsch 1997, Ferris 1997, Gott 2001, Greene 2004, Hawking and Penrose 1996, Kaku 2004, Penrose 2004, Rees 1997, and Smolin 2001. A recent, wonderful description of Einstein the man and his ideas is given in Isaacson 2007. Previous superb depictions of Einstein and his world include Bodanis 2000, Lightman 1993, Overbye 2000, and Pais 1982. or a nice collection of original papers, see Hawking 2007.

  The most recent test was the result: Kramer et al. 2006.

  a group of physicists at Harvard University: Odom et al. 2006.

  In the late 1960s, physicists: An excellent description can be found in Weinberg 1993.

  Chapter 9. On the Human Mind, Mathematics, and the Universe

  Here is how mathematicians: Davis and Hersh 1981.

  representing the Platonic point of view: Hardy 1940.

  expressed prec
isely the opposite perspective: Kasner and Newman 1989.

  Those who believe that mathematics exists: One of the best popular discussions of the nature of mathematics can be found in Barrow 1992. slightly more technical but still very accessible review of some of the major ideas is given in Kline 1972.

  Since I have already discussed pure Platonism: For another excellent discussion of many of the topics in the present book, see Barrow 1992.

  Tegmark argues that: Tegmark 2007a, b.

  in response to a similar assertion: Changeux and Connes 1995.

  concluded in his 1997 book: Dehaene 1997.

  Dehaene and his collaborators: Dehaene et al. 2006.

  Not all cognitive scientists agree: See Holden 2006, for example.

  provided the following observation: Changeux and Connes 1995.

  the most categorical statement: Lakoff and Núñez 2000.

  Neuroscientists have also identified: See Ramachandran and Blakeslee 1999, for instance.

  cognitive neuroscientist Rosemary Varley: Varley et al. 2005; Klessinger et al. 2007.

  Here, again, is how Atiyah argues: Atiyah 1995.

  Since the nineteenth century: For a very detailed description of the Golden Ratio, its history and properties, see Livio 2002, and also Herz-Fischler 1998.

  Prime numbers as a concept: A good discussion of these ideas is provided in an article by Yehuda Rav in Hersh 2000.

  Anthropologist Leslie A. White: White 1947.

  drew attention in the 1960s to the fact: For a popularized description see Hockett 1960.

  The former property represents the ability: For a readable discussion of language and the brain see Obler and Gjerlow 1999.

  are also characteristic of mathematics: The similarities between language and mathematics are also discussed by Sarrukai 2005 and Atiyah 1994.

  Noam Chomsky published his revolutionary work: Chomsky 1957. or more on linguistics, an excellent review can be found in Aronoff and Rees-Miller 2001. A popularized, very interesting perspective is given in Pinker 1994.

  Computer scientist Stephen Wolfram argued: Wolfram 2002.

  Astrophysicist Max Tegmark argues: Tegmark identified four distinct types of parallel universes. In “Level I,” there are universes with the same laws of physics but different initial conditions. In “Level II,” there are universes with the same equations of physics but perhaps different constants of nature. “Level III” employs the “many worlds” interpretation of quantum mechanics, and in “Level IV,” there are different mathematical structures. Tegmark 2004, 2007b.