Is God a Mathematician? (37 page)
Read Is God a Mathematician? Online
Authors: Mario Livio
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 precisely 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.
to contradict what has become known as the
principle of mediocrity: For an excellent discussion of this topic see Vilenkin 2006.
adopt an intermediate position known as
realism: Putnam 1975.
Let me first briefly review:
There are other opinions that I do not discuss. For instance, Steiner (2005) argues that Wigner does not show that the examples that he gives for “unreasonable effectiveness” have anything to do with the fact that the concepts are mathematical.
Physics Nobel laureate David Gross writes:
Gross 1988. For further discussion of the relationship between mathematics and physics, see Vafa 2000.
Sir Michael Atiyah, whose views on the nature:
Atiyah 1995; see also Atiyah 1993.
mathematician and computer scientist Richard Hamming:
Hamming 1980.
A similar interpretation was proposed:
Weinberg 1993.
Gelfand was once quoted:
In Borovik 2006.
Raskin concluded that:
Raskin 1998.
Hersh proposed that in the spirit:
Excellent article by Hersh in Hersh 2000.
Kepler used a huge body of data:
Kepler’s own books, reprinted as Kepler 1981 and 1997, make for very interesting reading in the history of science. Excellent biographies include Caspar 1993 and Gingerich 1973.
the orbits of the planets may eventually:
For a review, see Lecar et al. 2001.
The answer is actually simpler:
An interesting discussion of the utility of mathematics appears in Raymond 2005. Insightful perspectives on Wigner’s enigma are also found in Wilczek 2006, 2007.
Bertrand Russell in
The Problems of Philosophy: Russell 1912.
Aczel, A. D. 2000.
The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity
(New York: Four Walls Eight Windows).
———. 2004.
Chance: A Guide to Gambling, Love, the Stock Market, and Just about Everything Else
(New York: Thunder’s Mouth Press).
———. 2005.
Descartes’ Secret Notebook
(New York: Broadway Books).
Adam, C., and Tannery, P., eds. 1897–1910.
Oeuvres des Descartes.
Revised edition 1964–76 (Paris: Vrin/CNRS). The most comprehensive translation into English is Cottingham, J., Stoothoff, R., and Murdoch, D., eds. 1985.
The Philosophical Writing of Descartes
(Cambridge: Cambridge University Press).
Adams, C. 1994.
The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
(New York: W. H. Freeman).
Alexander, J. W. 1928.
Transactions of the American Mathematical Society
, 30, 275.
Applegate, D. L., Bixby, R. E., Chvátal, V., and Cook, W. J. 2007.
The Traveling Salesman Problem
(Princeton: Princeton University Press).
Archibald, R. C. 1914.
American Mathematical Society Bulletin
, 20, 409.
Aristotle. Ca. 350 BC.
Metaphysics.
In Barnes, J., ed. 1984.
The Complete Works of Aristotle
(Princeton: Princeton University Press).
———. Ca. 330 BCa.
Physics.
Translated by R. P. Hardie and R. K. Gaye. http://people.bu.edu/wwildman/WeirdWildWeb/courses/wphil/readings/wphil_rdg07_physics_entire.htm (public domain English translation).
———. Ca. 330 BCb.
Physics.
Translated by P. H. Wickstead and F. M. Cornford, 1960 (London: Heinemann).
Aronoff, M., and Rees-Miller, J. 2001.
The Handbook of Linguistics
(Oxford: Blackwell Publishing).
Ashley, C. W. 1944.
The Ashley Book of Knots
(New York: Doubleday).
Atiyah, M. 1989.
Publications Mathématiques de l’Inst. des Hautes Etudes Scientifiques
, Paris, 68, 175.
———. 1990.
The Geometry and Physics of Knots
(Cambridge: Cambridge University Press).
———. 1993.
Proceedings of the American Philosophical Society
, 137(4), 517.
———. 1994.
Supplement to Royal Society News
, 7, (12), (i).
———. 1995.
Times Higher Education Supplement
, 29 September.
Baillet, A. 1691.
La Vie de M. Des-Cartes
(Paris: Daniel Horthemels). Photographic facsimiles were published in 1972 (Hildesheim: Olms) and 1987 (New York: Garner).
Balz, A. G. A. 1952.
Descartes and the Modern Mind
(New Haven: Yale University Press).
Barrow, J. D. 1992.
Pi in the Sky: Counting, Thinking, and Being
(Oxford: Clarendon Press).
———. 2005.
The Infinite Book: A Short Guide to the Boundless, Timeless and Endless
(New York: Pantheon).
Beaney, M. 2003. In Griffin, N., ed.
The Cambridge Companion to Bertrand Russell
(Cambridge: Cambridge University Press).
Bell, E. T. 1937.
Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré
(New York: Touchstone).
———. 1940.
The Development of Mathematics
(New York: McGraw-Hill).
———. 1951.
Mathematics: Queen and Servant of Science
(New York: McGraw-Hill).
Beltrán Mari, A. 1994. “Introduction.” In Galilei, G.
Diálogo Sobre los Dos Máximos Sistemas del Mundo
(Madrid: Alianza Editorial).
Bennett, D. 2004.
Logic Made Easy: How to Know When Language Deceives You
(New York: W. W. Norton).
Berkeley, G. 1734. “The Analyst: Or a Discourse Addressed to an Infidel Mathematician, D. R. Wilkins, ed. http:///www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.html.
Berlinski, D. 1996.
A Tour of the Calculus
(New York: Pantheon Books).
Bernoulli, J. 1713a.
The Art of Conjecturing
[
Ars Conjectandi
]. Translated by E.D. Sylla, with introduction and notes, 2006 (Baltimore: Johns Hopkins University Press).
———. 1713b.
Ars Conjectandi
(Basel: Tharnisiorum).
Beyssade, M. 1993. “The Cogito.” In Voss, S., ed.
Essays on the Philosophy and Science of René Descartes
(Oxford: Oxford University Press).
Black, F., and Scholes, M. 1973.
Journal of Political Economy
, 81(3), 637.
Bodanis, D. 2000.
E = mc
2
: A Biography of the World’s Most Famous Equation
(New York: Walker).
Bonola, R. 1955.
Non-Euclidean Geometry.
Translated by H. S. Carshaw. (New York: Dover Publications). This is a republication of the 1912 translation (Chicago: Open Court Publishing Company).
Boole, G. 1847.
The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning.
In Ewald, W. 1996.
From
Kant to Hilbert: A Source Book in the Foundations of Mathematics
(Oxford: Clarendon Press).
———. 1854.
An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities
(London: Macmillan). Reprinted 1958 (Mineola, N.Y.: Dover Publications).
Boolos, G. 1985.
Mind
, 94, 331.
———. 1999.
Logic, Logic, Logic
(Cambridge, Mass.: Harvard University Press).
Borovik, A. 2006.
Mathematics under the Microscope.
http://www.maths.manchester.ac.uk/%7Eavb/micromathematics/downloads.
Brewster, D. 1831.
The Life of Sir Isaac Newton
(London: John Murray, Albemarle Street).
Bukowski, J. 2008.
The College Mathematics Journal
, 39(1), 2.
Burger, E. B., and Starbird, M. 2005.
Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas
(New York: W. W. Norton).
Burkert, W. 1972.
Lore and Science in Ancient Pythagoreanism
(Cambridge, Mass.: Harvard University Press).
Cajori, F. 1926.
The American Mathematical Monthly
, 33(8), 397.
———. 1928. In The History of Science Society.
Sir Isaac Newton 1727–1927: A Bicentenary Evaluation of His Work
(Baltimore: The Williams & Wilkins Company).
Cardano, G. 1545.
Artis Magnae, sive de regulis algebraices.
Published in 1968 under the title
The Great Art or the Rules of Algebra
, translated and edited by T. R. Witmer (Cambridge, Mass.: MIT Press).
Caspar, M. 1993.
Kepler.
Translated by C. D. Hellman (Mineola, N.Y.: Dover Publications).
Chandrasekhar, S. 1995.
Newton’s “Principia” for the Common Reader
(Oxford: Clarendon Press).
Changeux, J.-P., and Connes, A. 1995.
Conversations on Mind, Matter, and Mathematics
(Princeton: Princeton University Press).
Cherniss, H. 1945.
The Riddle of the Early Academy
(Berkeley: University of California Press). Reprinted 1980 (New York: Garland).
———. 1951.
Review of Metaphysics
, 4, 395.
Chomsky, N. 1957.
Syntactic Structures
(The Hague: Mouton & Co.).
Cicero. 1st century BC.
Discussion at Tusculam
[sometimes translated as
Tusculan Disputations
]. In Grant, M., trans. 1971.
Cicero: On the Good Life
(London: Penguin Classics).
Clark, M. 2002.
Paradoxes from A to Z
(London: Routledge).
Clarke, D. M. 1992. In Cottingham, J., ed.
The Cambridge Companion to Descartes
(Cambridge: Cambridge University Press).
Cohen, I. B. 1982. In Bechler, Z., ed.
Contemporary Newtonian Research
(Dordrecht: Reidel).
———. 2006.
The Triumph of Numbers
(New York: W. W. Norton & Company).
Cohen, P. J. 1966.
Set Theory and the Continuum Hypothesis
(New York: W. A. Benjamin).
Cole, J. R. 1992.
The Olympian Dreams and Youthful Rebellion of René Descartes
(Champaign: University of Illinois Press).
Connor, J. A. 2006.
Pascal’s Wager: The Man Who Played Dice with God
(New York: HarperCollins).
Conway, J. H. 1970. In Leech, J., ed.
Computational Problems in Abstract Algebra
(Oxford: Pergamon Press).
Coresio, G. 1612.
Operetta intorno al galleggiare de’ corpi solidi.
Reprinted in Favaro, A. 1968.
Le Opere di Galileo Galilei.
Edizione Nazionale (Florence: Barbera).
Cottingham, J. 1986.
Descartes
(Oxford: Blackwell).
Craig, Sir J. 1946.
Newton at the Mint
(Cambridge: Cambridge University Press).
Curley, E. 1993. In Voss, S., ed.
Essays on the Philosophy and Science of René Descartes
(Oxford: Oxford University Press).
Curzon, G. 2004.
Wotton and His Words: Spying, Science and Venetian Intrigues
(Philadelphia: Xlibris Corporation).
Davies, P. 2001.
How to Build a Time Machine
(New York: Allen Lane).
Davis, P. J., and Hersh, R. 1981.
The Mathematical Experience
(Boston: Birkhaüser). Revised edition 1998 (Boston: Mariner Books).
Dawkins, R. 2006.
The God Delusion
(New York: Houghton Mifflin Company).
Dawson, J. 1997.
Logical Dilemmas: The Life and Work of Kurt Gödel
(Natick, Mass.: A. K. Peters).
Dehaene, S. 1997.
The Number Sense
(Oxford: Oxford University Press).
Dehaene, S., Izard, V., Pica, P., and Spelke, E. 2006.
Science
, 311, 381.
DeLong, H. 1970.
A Profile of Mathematical Logic
(Reading, Mass.: Addison-Wesley). Republished 2004 (Mineola, N.Y.: Dover Publications).
Demopoulos, W., and Clark, P. 2005. In Shapiro, S., ed.
The Oxford Handbook of Philosophy of Mathematics and Logic
(Oxford: Oxford University Press).
De Morgan, A. 1885.
Newton: His Friend: and His Niece
(London: Elliot Stock).
Dennett, D. C. 2006.
Breaking the Spell: Religion as a Natural Phenomenon
(New York: Viking).
De Santillana, G. 1955.
The Crime of Galileo
(Chicago: University of Chicago Press).
Descartes, R. 1637a.
Discourse on Method, Optics, Geometry, and Meteorology.
Translated by P. J. Olscamp, 1965 (Indianapolis: The Bobbs-Merrill Company).
———. 1637b.
The Geometry of René Descartes.
Translated by D. E. Smith and M. L. Latham, 1954 (Mineola, N.Y.: Dover Publications).
———. 1644.
Principles of Philosophy
, II:64. In Cottingham, J., Stoothoff, R., and Murdoch, D., eds. 1985.
Philosophical Works of Descartes
(Cambridge: Cambridge University Press).
———. 1637–1644.
The Philosophy of Descartes: Containing the Method, Meditations, and Other Works.
Translated by J. Veitch, 1901 (New York: Tudor Publishing).
Detlefsen, M. 2005. In Shapiro, S., ed.
The Oxford Handbook of Philosophy of Mathematics and Logic
(Oxford: Oxford University Press).
Deutsch, D. 1997.
The Fabric of Reality
(New York: Allen Lane).
Devlin, K. 1993.
The Joy of Sets: Fundamentals of Contemporary Set Theory
, 2nd ed. (New York: Springer-Verlag).
———. 2000.
The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are like Gossip
(New York: Basic Books).
Dijksterhuis, E. J. 1957.
Archimedes
(New York: The Humanities Press).
Doxiadis, A. K. 2000.
Uncle Petros and Goldbach’s Conjecture
(New York: Bloomsbury).
Drake, S. 1978.
Galileo at Work: His Scientific Biography
(Chicago: University of Chicago Press).
———. 1990.
Galileo: Pioneer Scientist
(Toronto: University of Toronto Press).
Dummett, M. 1978.
Truth and Other Enigmas
(Cambridge, Mass.: Harvard University Press).
Dunham, W. 1994.
The Mathematical Universe: An Alphabetical Journey through the Great Proofs, Problems and Personalities
(New York: John Wiley & Sons).
Dunnington, G. W. 1955.
Carl Friedrich Gauss: Titan of Science
(New York: Hafner Publishing).
Du Sautoy, M. 2008.
Symmetry: A Journey into the Patterns of Nature
(New York: Harper Collins).
Einstein, A. 1934. “Geometrie und Erfuhrung.” In
Mein Weltbild
(Frankfurt am Main: Ullstein Materialien).
Ewald, W. 1996.
From Kant to Hilbert: A Source Book in the Foundations of Mathematics
(Oxford: Clarendon Press).
Favaro, A., ed. 1890–1909.
Le Opere di Galileo Galilei, Edizione Nationale
(Florence: Barbera). There have been a number of reprints, the most recent 1964–66. This text is searchable online at http://www.imss.fi.it/istituto/index.html.
Fearnley-Sander, D. 1979.
The American Mathematical Monthly
, 86(10), 809.
———. 1982.
The American Mathematical Monthly
, 89(3), 161.
Feldberg, R. 1995.
Galileo and the Church: Political Inquisition or Critical Dialogue
(Cambridge: Cambridge University Press).
Ferris, T. 1997.
The Whole Shebang
(New York: Simon & Schuster).
Finkel, B. F. 1898. “Biography: René Descartes.”
American Mathematical Monthly
, 5(8–9), 191.
Fisher, R. A. 1936.
Annals of Science
, 1, 115.
———. 1956. In Newman, J. R., ed.
The World of Mathematics
(New York: Simon & Schuster).