From the Armillary Sphere to the Computer

Professor Michael S. Mahoney

Meeting Time:

The seminar will meet Tuesday afternoons, 1:30-4:30, in Dickinson 230

Topics and Readings

Week 1 (2/2)
Introduction: The Issues
Paul Cilliers, Complexity and Postmodernism: Understanding Complex Systems (London/Ny, 1998)

Background and Overview
The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (Dordrecht, 1961) [8105.492]

Georges Canguilhem (ed.), La mathématisation des doctrines informes (Paris, 1972)

Goldstein, Jerome & Joella Yoder (eds.), Mathematics Applied to Science: In Memoriam Edward D. Conway [SM 101.M39 1988]

Shea, William R. (ed.), Nature Mathematized: Historical and Philosophical Case Studies in Classical Modern Natural Philosophy (3rd Pisa Conference on History and Philosophy of Science, Montreal, 1980) [Q174.I563 1980a]

Enrico Bellone, A World on Paper: Studies on the Second Scientific Revolution (Cambridge/London, 1980); cf. his earlier study, I modelli e la concezione del mondo nella fisica moderna da Laplace a Bohr (Milan, 1973)

John L. Casti, Alternate Realities: Mathematical Models of Nature and Man

Week 2 (2/9)
Modeling the Heavens

Reporter: Ole Molvig
Nicolaus Copernicus, De revolutionibus orbium coelestium, Preface and Book I (trans. Rosen); for assistance, cf. N.M. Swerdlow and O. Neugebauer
Theorica planetarum, in Source Book in Medieval Science, ed. E. Grant
Morris Cohen and I.E. Drabkin (eds.), Source Book in Greek Science, 90-91 (Simplicius on astronomy and physics)

Gregory Vlastos, Plato's Universe
G.J. Toomer, "Hipparchus", Dictionary of Scientific Biography
G.E.R. Lloyd, Greek Science After Aristotle, Chap. 5
Alexander Jones, "The Adaptation of Babylonian Methods in Greek Numerical Astronomy", ISIS 82(1991), 441-54
M.S. Mahoney, "Ptolemaic Astronomy in the Middle Ages" (preprint of article since published in Dictionary of the Middle Ages, ed. J.R. Strayer, s.v. "Astronomy, Ptolemaic")

Otto Neugebauer, The Exact Sciences in Antiquity and History of Ancient Mathematical Astronomy; cf. J. Needham, Science and Civilization in China, vol. 3, for similar form of astronomy based on arithmetical relations
B.L. van der Waerden, Greek Astronomy
Arpad Szabó & Erkka Maula, Les débuts de l'astronomie, de la géographie et de la trigonometrie chez les Grecs (Paris, 1986) [QB21.S9314]
M.S. Mahoney, "Babylonian Algebra: Form vs. Content" (Essay review of O. Neugebauer, Vorgriechische Mathematik, 2nd unrev. ed., Berlin, 1969), Studies in History and Philosophy of Science 1(1970), 369-380
Ido Yavetz, "On the Homocentric Spheres of Eudoxus", Archive for History of Exact Sciences 52(1998), 221-78

Jerome R. Ravetz, "The Origins of the Copernican Revolution", Scientific American (Oct. 1966), 88-98
Noel Swerdlow, "The Derivation and First Draft of Copernicus's Planetary Theory:  A Translation of the Commentariolus with Commentary", Proceedings of the American Philosophical Society, 117(1973), 423-512

Week 3 (2/16)
Real and Heuristic Models: Descartes' Optics
René Descartes, The World, or Treatise on Light
----, Optics (Dioptrique) and Meteorology (Météores)

William R. Shea, The Magic of Numbers and Motion, Chaps. 7, 10-12
Nancy L. Maull, "Cartesian Optics and the Geometrization of Nature", in Descartes: Philosophy, Mathematics & Physics, ed. Stephen Gaukroger
Neil M. Ribe, "Cartesian Optics and the Mastery of Nature", ISIS 88(1997), 42-61

Daniel Garber, Descartes' Mathematical Physics
Emily R. Grosholz, Cartesian Method and the Problem of Reduction

Week 4 (2/23)
The Mathematization of Motion I
Reporter: Rebecca Press
Galileo Galilei, The Two New Sciences, Day III
Christiaan Huygens, [Determination of the period of a simple pendulum and of the cycloidal pendulum], (trans. M.S. Mahoney)

Michel Blay, Reasoning with the Infinite
M.S. Mahoney, "The Mathematical Realm of Nature", in The Cambridge History of Seventeenth Century Philosophy
M.S. Mahoney, "Huygens and the Pendulum: From Device to Mathematical Relation", in The Growth of Mathematical Knowledge

M.S. Mahoney, "Christiaan Huygens: The measure of time and longitude at sea", in H.J.M. Bos et al. (eds.), Studies on Christiaan Huygens; cf. other related articles by Mahoney
Joella G. Yoder, Unrolling Time: Christiaan Huygens and the Mathematization of Nature

Week 5 (3/9)
The Mathematization of Motion II
Reporter: Rebecca Press
Isaac Newton, Philosophiae naturalis principia mathematica (The Mathematical Principles of Natural Philosophy), Book I, Sects.2, 3, 7 and 8 (Props. 32-41), Book III (read for structure of argument)
Pierre Varignon, "Du mouvement en générale par toutes sortes de courbes; & des forces centrales, tant centrifuges que centripètes, nécessaires aux corps qui les décrivent", Mémoires de l'Académie Royale des Sciences (MARS), 1700, 83-101 (xerox)
----, "Des forces centrales, ou des pesanteurs necessaires aux Planètes pour leur faire décrire des orbes qu'on leur a supposés jusqu'ici", MARS, 1700, 218-237 xerox), Lemma II and Problem I

D.T. Whiteside, "The Mathematical Principles Underlying Newton's Principia mathematica", J. Hist. Astronomy 1(1970), 116-138
François de Gandt, "Le style mathématique des Principia de Newton", Revue d'histoire des sciences 39,3(1986): 195-222
Michael S. Mahoney, "Algebraic vs. Geometric Techniques in Newton's Determination of Planetary Orbits", in Paul Theerman and Adele F. Seeff (eds.), Action and Reaction: Proceedings of a Symposium to Commemorate the Tercentenary of Newton's Principia , 183-205
M.S. Mahoney, "Pierre Varignon and the Calculus of Motion" (draft article)
Michel Blay, La naissance de la mécanique analytique

John W. Herivel, The Background to Newton's 'Principia'
Richard S. Westfall, Force in Newton's Physics: The Science of Dynamics in the Seventeenth Century (1971) Cf. essay review by E.J. Aiton, "The Concept of Force", History of Science 10(1971), 88-102

Week 6 (3/23)
Laplacian Physics
The major original sources for this week are both obvious and forbidding:
  • Leonhard Euler, Mechanica sive motus scientia analytice exposita (2 vols, St. Petersburg, 1736)
  • Joseph-Louis Lagrange, Mécanique analytique (Paris, 1788)
  • Pierre-Simon de Laplace, Mécanique céleste (5 vols., Paris, 1799-1825); Engl. trans. by Nathaniel Bowditch

Nonetheless, you may want to page through them, using the above accounts as a guide. Don't overlook the DSB for an overview of the three men's careers; the long article on Laplace is particularly valuable, not only in summarizing the then current scholarship but in extending it.

Clifford Truesdell, "A Program toward rediscovering the rational mechanics of the Age of Reason", Archive for History of Exact Sciences 1,1(1960): 1-36
Craig Fraser, "J.L. Lagrange's Early Contributions to the Principles and Methods of Mechanics", Archive for History of Exact Sciences 28,3(1983): 197-241
John L. Heilbron, "Laplace's school", in Weighing Imponderables and Other Quantitative Science Around 1800 (Historical Studies in Physical and Biological Sciences [HSPS], Suppl. to 24(1993), Pt. 1)
Robert Fox, "The rise and fall of Laplacian physics", HSPS 4(1974): 89-136; cf. his "Laplacian physics", in Companion to the History of Modern Science (ed. R.C. Olby et al.), Chap. 18

Enrico Bellone, A World on Paper: Studies on the Second Scientific Revolution (Cambridge/London, 1980); cf. his earlier study, I modelli e la concezione del mondo nella fisica moderna da Laplace a Bohr (Milan, 1973)

Week 7 (3/30)
Mathematical Physics in the 19th Century
Reporter: Ole Molvig

Joseph Fourier, Théorie analytique de la chaleur [1812, 1822]; cf. original versions, "Mémoire sur la propagation de la chaleur avec notes séparées sur cette propagation ... ", in Ivor Grattan-Guiness and J.R. Ravetz, Joseph Fourier 1768-1830 (Cambridge, MA, 1972)
A.-M. Ampère, Théorie mathématique des phénomènes électro-dynamiques [1826]

James Clerk Maxwell, "On Faraday's Lines of Force" (1855/56), in The Scientific Papers of James Clerk Maxwell (ed. W.D. Niven), I: 155-229
---, "On Physical Lines of Force", ibid., I:451-513
---, "A Dynamical Theory of the Electromagnetic Field" (1864), ibid., I:526-597
---, "On the Proof of the Equations of Motion of a Connected System" (1876), ibid., II:308-309
---, "On Action at a Distance", ibid., II:311-323
---, "[Review of Thompson & Tait, Elements of Natural Philosophy]", ibid., II:324-328

Clearly, you cannot read all or even most of the above. I have listed them in chronological order, but the last three are the most discursive and might be the place to start. You should approach the first three with an eye toward seeing how Maxwell set up his subject and go as far as your math, physics, and tolerance for perplexity take you. Most secondary sources on the subject, preeminently those of Jed Buchwald and Norton Wise, not to mention E.T. Whittaker, are as technical as Maxwell himself, so for our purposes it is just as well to go to the originals. Two articles that may help are:

Jed Z. Buchwald, "William Thompson and the mathematization of Faraday's Electrostatics", Historical Studies in the Physical Sciences 8(1977): 101-136
M. Norton Wise, "The mutual embrace of electricity and magnetism", Science 203 (1979), 1310-18

Eugene Frankel, "J.B. Biot and the mathematization of experimental physics in Napoleonic France", Historical Studies in the Physical Sciences 8(1977): 33-72
Ivor Grattan-Guiness, "Mathematical physics in France, 1800-1840: Knowledge, activity and historiography", in J. Dauben (ed.), Mathematical Perspectives: Essays on the History of Mathematics in Honor of Kurt.-R. Biermann
--- , "Mathematical physics 1800-1835: Genesis in France and development in Germany", in H.N. Jahnke and M. Otte (eds.), Epistemological and Social Problems of the Sciences in the Early Nineteenth Century
--- , "Mathematics and mathematical physics in Cambridge, 1815-1840: A survey of their achievements and of the French influences", in P.M. Harman (ed.), Wranglers and Physicists: Studies on Cambridge Physics in the Nineteenth Century
Daniel Siegel, "Mechanical Image and Reality in Maxwell's Electromagnetic Theory", in Harman, Wranglers and Physicists

Week 8 (4/6)
Modeling populations
Reporters: Tapio Schneider and Alex Chekovich
Vito Volterra, Leçons sur la théorie mathématique de la lutte pour la vie (1931)

Stephen Brush, Statistical Physics and the Atomic Theory of Matter From Boyle and Newton to Landau and Onsager, Chaps. 1-2
Lorenz Krüger, "Reduction as a Problem: Some Remarks on the History of Statistical Mechanics from a Philosophical Point of View", in J. Hintikka et al. (eds.), Probabilistic Thinking, Thermodynamics, and the Interaction of History and Philosophy of Science, Vol II, 147-74
William B. Provine, The Origins of Theoretical Population Genetics
Sharon E. Kingsland, Modeling nature : episodes in the history of population ecology

Lorraine J. Daston, Classical Probability in the Enlightenment
Theodore M. Porter, The Rise of Statistical Thinking, 1830-1900
Stephen M. Stigler, The History of Statistics: The Measurement of Uncertainty Before 1900

Week 9 (4/13)
Modeling Behavior
John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (1944), Chapter I
Herbert A. Simon, "Some strategic considerations in the construction of social science models", in Paul F. Lazarsfeld, Mathematical Thinking in the Social Sciences (1954); cf. Simon's Sciences of the Artificial (1969)
<sources cited in the secondary readings, as fancy strikes and time permits>

Margaret Schabas, "Alfred W. Marshall, W. Stanley Jevons, and the mathematization of economics", ISIS 80(1989): 60-73; cf. her A World Ruled by Number: William Stanley Jevons and the Rise of Mathematical Economics (1990)
E. Roy Weintraub and Philip Mirowski, "The Pure and the Applied: Bourbakism Comes to Mathematical Economics", Science in Context 7,2(1994), 243-72
Robert J. Leonard, "Creating a context for game theory", in E. Roy Weintraub (ed.), Toward a History of Game Theory, 29-76
Philip Mirowski, "What were von Neuman and Morgenstern trying to accomplish?", ibid., 113-147

Philip Mirowski, More heat than light: Economics as social physics, Physics as nature's economics (1989) [a controversial book; cf. "Review Symposium on Philip Mirowski's 'More Heat than Light ...", Philosophy of the Social Sciences 22(1992), 77-141, esp. the critique of M. Norton Wise]
Bruna Ingrao and Giorgio Israel, The invisible hand: Economic equilibrium in the history of science (1990)
Mary S. Morgan, The history of econometric ideas

Week 10 (4/20)
Computers and Automata
John von Neumann, "First Draft of a Report on the Edvac" (1945) and "General and Logical Theory of Automata" (1954), in Cerebral Mechanisms in Behavior: The Hixon Symposium, ed. L.A. Jeffries (New York: Wiley, 1951), 1-31; repr. in Papers of John von Neumann on Computing and Computer Theory, ed. William Aspray and Arthur Burks (MIT, 1987), 391-431
Arthur W. Burks, "Introduction", in Burks (ed.) Essays on Cellular Automata
Christopher G. Langton, "Self-Reproduction in Cellular Automata", Cellular Automata, ed. D. Farmer et al. (North-Holland), 145-156

M.S. Mahoney, "Computer Science: The Search for a Mathematical Theory", in John Krige and Dominique Pestre (eds.), Science in the Twentieth Century, Chap. 31
William Aspray, John von Neumann and the Origins of Modern Computing, Chap. 8, "A Theory of Information Processing"

Week 11 (4/27)
Automata, Languages, Development
Stanislaw Ulam, "On some mathematical properties connected with patterns of growth of figures", Proceedings of Symposia on Applied Mathematics 14 (American Mathematical society, 1962), 215-224
Noam Chomsky, "Three models of language", IRE Transactions in Information Theory 2,3(1956), 113-24, and "On certain formal properties of grammars", Information and Control 2,2(1959), 137-167
Aristide Lindenmayer, "Mathematical models for cellular interactions in development", J. Theor. Biol. 18(1968), 280-99, 300-15 Cf. his L-systems (An L-system Tutorial)
Stephen Wolfram, "Computation Theory of Cellular Automata" Communications in Mathematical Physics 96(1984), 15-57; repr. in Wolfram, Theory and Application of Cellular Automata, 189-231

Noam Chomsky, The Logical Structure of Linguistic Theory, 1-53 ("Introduction 1973")
Robert Friedin, "Conceptual Shifts in the Science of Grammar: 1951-1992", in Carlos P. Otero (ed.), Noam Chomsky: Critical Assessments
Sheila A. Greibach, "Formal languages: Origins and directions", Annals of the History of Computing 3,1(1981), 14-41
Michael S. Mahoney, The Structures of Computation, Chapter 4 (draft)

Przemyslaw Prusinkiewicz, Mark Hall, and Radomir Mech, Visual Models of Morphogenesis: A Guided Tour (online text with illustrations)
Noam Chomsky and George A. Miller, "Introduction to the Formal Analysis of Natural Languages", in Handbook of Mathematical Psychology [1963-5], ed. R.D. Luce, R.R. Bush, E. Galanter, Vol. 2, Chap. 11

Week 12 (5/4)
Synthetic Biology
Christopher Langton, "Artificial Life", in Margaret A. Boden (ed.) The Philosophy of Artificial Life, Chap. 1
Thomas S. Ray, "An Approach to the Synthesis of Life", ibid., Chap. 3; see the Tierra home page and get a copy of the software
John Maynard Smith, "Evolution -- Natural and Artificial" ibid., Chap. 5
Elliott Sober, "Learning from Functionalism -- Prospects for Strong Artificial Life", ibid., Chap. 14
John L. Casti, "Newton, Aristotle and the Modeling of Living Systems", in John Casti and Anders Karlqvist (eds.), Newton to Aristotle: Toward a Theory of Models for Living Systems, 47-89
Walter Fontana, Günter Wagner, Leo W. Buss, "Beyond Digital Naturalism", in Christoper G. Langton (ed.), Artificial Life: An Overview, 211-228 (online version)
Robert Rosen, "Church's Thesis and Its Relation to the Concept of Realizability in Biology and Physics", Bull. Math. Biophysics 24(1962), 375-393

Claus Emmeche, The Garden in the Machine: The Emerging Science of Artificial Life

John H. Holland, Hidden Order: How Adaptation Builds Complexity
Richard Doyle, On Beyond Living: Rhetorical Transformations of the Life Sciences, Chap. 7, "Emergent Power: Vitality and Theology in Artificial Life"
Stefan Helmreich, Silicon Second Nature: Culturing Artificial Life in a Digital World
John L. Casti, Alternate Realities: Mathematical Models of Nature and Man, Chap. 2