Is God a Mathematician? (34 page)

Thus, to sum up our discussion of the value of philosophy; Philosophy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good.

Chapter 1. A Mystery

As the British physicist James Jeans:
Jeans 1930.

Einstein once wondered:
Einstein 1934.

he singled out geometry as the paradigm:
Hobbes 1651.

Penrose identifies three different:
Penrose beautifully discusses these “three worlds” in
Emperor’s New Mind
and
Road to Reality.

Physics Nobel laureate Eugene Wigner:
Wigner 1960. We shall return to this article many times in this book.

that he emphatically declared:
Hardy 1940.

One of his works was reincarnated:
For a discussion of the Hardy-Weinberg law in context see for example Hedrick 2004.

the British mathematician Clifford Cocks:
Cocks invented in 1973 what has become known as the RSA encryption algorithm, but at the time it was classified. The algorithm was independently invented a few years later by R. Rivest, A. Shamir, and L. Adleman at MIT. See Rivest, Shamir, and Adleman 1978.

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, Rucker 1995, and Wallace 2003.

the philosopher Iamblichus reports:
Iamblichus ca. 300 ADa, b.

to the Pythagoreans, God was not:
See discussion in Netz 2005.

“the safest generalization that can be made”:
Whitehead 1929.

Who was this relentless seeker:
The titles of texts about Plato and his ideas can, of course, by themselves fill an entire volume. Here are just a few texts that I found to be very helpful. On Plato in general: Hamilton and Cairns 1961, Havelock 1963, Gosling 1973, Ross 1951, Kraut 1992. On mathematics: Heath 1921, Cherniss 1951, Mueller 1991, Fowler 1999, Herz-Fischler 1998.

According to an oration by the fourth century:
The oration was written in 362 AD, but it did not give any details on the contents of the inscription. The words of the inscription come from a marginal note in a manuscript of Aelius Aristides. The note may have been written by the fourth century orator Sopatros, and it reads (in a translation by Andrew Barker): “There had been inscribed at the front of the School of Plato, ‘Let no one who is not a geometer enter.’ [That is] in place of ‘unfair’ or ‘unjust’: for geometry pursues fairness and justice.” The note seems to imply that Plato’s inscription replaced “unfair or unjust person” in a sign that was common in sacred places (“Let no unfair or unjust person enter”) with the phrase “one who is not a geometer.” This story was later repeated by no fewer than five sixth century Alexandrian philosophers, and it eventually made its way into the book
Chiliades,
by the twelfth century polymath Johannes Tzetzes (ca. 1110–80). For a detailed discussion see Fowler 1999.

I was disappointed to discover:
A summary of many unsuccessful archaeological attempts can be found in Glucker 1978.

The first century philosopher and historian:
Discussed in Cherniss 1945, Mekler 1902.

To which the Neoplatonic philosopher:
Cherniss 1945, Proclus ca. 450.

“What we require is that those who take”:
Plato ca. 360 BC.

“The science of figures, to a certain degree”:
Washington 1788.

is no more real than shadows projected:
An interesting discussion of the allegory can be found in Stewart 1905.

Plato’s views formed the basis:
For interesting discussions of Platonism and its place in the philosophy of mathematics, see Tiles 1996, Mueller 1992, White 1992, Russell 1945, Tait 1996. For excellent presentations in popular texts, see Davis and Hersh 1981, Barrow 1992.

mathematics becomes closely associated with the divine:
For a discussion of this topic see Mueller 2005.

He argued that in true astronomy:
Plato’s comments on astronomy and planetary motion appear in the
Republic
(Plato ca. 360 BC), in Timaeus, and in
Laws.
G. Vlostos and I. Mueller discuss the implications of Plato’s position (Vlostos 1975, Mueller 1992).

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.