Is God a Mathematician_ - Mario Livio [128]
logic dealt with the relationships: An extremely accessible introduction to logic can be found in Bennett 2004. More technical, but brilliant, is Quine 1982. 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