Is God a Mathematician_ - Mario Livio [120]
to describe all the symmetries of the world: A popular description of symmetry, group theory, and their intertwined history is given in The Equation That Couldn’t Be Solved (Livio 2005), Stewart 2007, Ronan 2006, and Du Sautoy 2008.
He noticed that a sequence of numbers: A wonderful popular description of the emergence of chaos theory can be found in Gleick 1987.
Black-Scholes option pricing formula: Black and Scholes 1973.
The traveling salesman problem was solved: A superb but technical description of the problem and its solutions can be found in Applegate et al. 2007.
expressed his views very clearly: Changeux and Connes 1995.
He once wittily remarked: Gardner 2003.
While reviewing a book: Atiyah 1995.
In the words of the French neuroscientist: Changeux and Connes 1995.
In one place she complains: A brief biography of Marjory Fleming can be found, for instance, at Wallechinsky and Wallace 1975–81.
author Ian Stewart once put it: Stewart 2004.
Chapter 2. Mystics: The Numerologist and the Philosopher
Descartes was one of the principal architects: A more detailed description of Descartes’ contributions is presented in chapter 4.
“I recognize no matter”: Descartes 1644.
credited with introducing the words: Iamblichus ca. 300 ADa, b; discussed in Guthrie 1987.
biographies of Pythagoras from the third century: Laertius ca. 250 AD; Porphyry ca. 270 AD; Iamblichus ca. 300 ADa, b.
finds it difficult to identify: Aristotle ca. 350 BC; discussed in Burkert 1972.
The Greek historian Herodotus: Herodotus 440 BC.
Empedocles (ca. 492–432 BC) added in admiration: Porphyry ca. 270 AD.
For instance, the monad: A clear discussion of the Pythagorean perspective can be found in Strohmeier and Westbrook 1999.
The English historian of philosophy: Stanley 1687.
The fact that someone would find numbers: For a fascinating compilation of properties of numbers see Wells 1986.
Pythagoras asks someone to count: Cited in Heath 1921.
“I swear by the discoverer”: Iamblichus ca. 300 ADa; discussed in Guthrie 1987.
When two similar strings: Strohmeier and Westbrook 1999; Stanley 1687.
The word “gnomon” (a “marker”): T. L. Heath gives a detailed discussion of the term and what it meant at different times (Heath 1921). The mathematician Theon of Smyrna (ca. 70–135 AD) used the term in relation to the figurative expression of numbers described in the text in Mathematics, Useful for Understanding Plato (Theon of Smyrna ca. 130 AD).
“If we listen to those who wish”: You will notice that in his comment Proclus does not state specifically what he himself believes with respect to the question of whether Pythagoras was the first to formulate the theorem. The story about the ox appears in the writings of Laertius, Porphyry, and the historian Plutarch (ca. 46–120 AD). It is based on verses by Apollodorus. However, the verses only talk about “that famous proposition” without stating which proposition this was. See Laertius ca. 250 AD, Plutarch ca. 75 AD.
These constructions were clearly known: Renon and Felliozat 1947, van der Waerden 1983.
The basic philosophy expressed by the table: This cosmology was based on the notion that reality emerges from the fact that Matter (considered indefinite) is shaped by Form (considered the limit).
The book Philosophy for Dummies: Morris 1999.
The oldest surviving story: Joost-Gaugier 2006.
From the perspective of the questions: Good discussions of the Pythagorean contributions and their influence can be found in Huffman 1999, Riedweg 2005, Joost-Gaugier 2006, and Huffman 2006 in the Stanford Encyclopedia of Philosophy.
One of the Pythagoreans: Fritz 1945.
the recognition of the existence of “countable”: I do not discuss topics such as transfinite numbers and the works of Cantor and Dedekind in the present book. Excellent popular accounts can be found in Aczel 2000, Barrow 2005, Devlin 2000,