51社区

Education

Ph.D. University of Illinois, Chicago

Research

Areas of Specialization or Competence

• Logic
• History and Philosophy of Mathematics
• Philosophy of Science
• Philosophy of Physics

In his paper Recent Work On The Principles of Mathematics, which appeared in 1901, Bertrand Russell reported that the three central problems of traditional mathematical philosophy--the nature of the infinite, the nature of the infinitesimal, and the nature of the continuum--had all been 鈥渃ompletely solved鈥� [1901, p. 89]. Indeed, as Russell went on to add: 鈥淭he solutions, for those acquainted with mathematics, are so clear as to leave no longer the slightest doubt or difficulty鈥� [1901, p. 89].

According to Russell, the structure of the infinite and the continuum were completely revealed by Cantor and Dedekind, and the concept of an infinitesimal had been found to be incoherent and was 鈥渂anish[ed] from mathematics鈥� through the work of Weierstrass and others [1901, pp. 88, 90]. These themes were reiterated in Russell鈥檚 often reprinted Mathematics and the Metaphysician [1918], and further developed in both editions of Russell鈥檚 The Principles of Mathematics [1903; 1937], the works which perhaps more than any other helped to promulgate these ideas among historians and philosophers of mathematics.

Having been persuaded that infinitesimals had indeed been 鈥渂anished鈥� from mathematics and that the problems of the infinite and the continuum had been completely solved, Russell and most other analytic philosophers of mathematics after him turned their attention to finding a secure foundation for the newly developed theories of the infinite and the continuum and for mathematics, more generally.

More than twenty years ago, however, I started to realize that the historical picture painted by Russell and others was not only historically inaccurate, but that the work done by Dedekind, while revolutionary, only revealed a glimpse of a far richer theory of continua that not only allows for infinitesimals but leads to a vast generalization of portions Cantor鈥檚 theory of the infinite, a generalization that also provides a setting for Abraham Robinson鈥檚 infinitesimal approach to analysis [1961; 1966] as well as for the profound and all too often overlooked non-Cantorian theories of the infinite (and infinitesimal) pioneered by Giuseppe Veronese [1891; 1894], Tullio Levi-Civita [1892; 1898], David Hilbert [1899] and Hans Hahn [1907] in connection with their work on non-Archimedean ordered algebraic and geometric systems and by Paul du Bois-Reymond (cf. [1870-71;1875; 1877; 1882]), Otto Stolz [1883; 1885], Felix Hausdorff [1907; 1909] and G. H. Hardy [1910; 1912] in connection with their work on the rate of growth of real functions. Central to the theory is J.H. Conway鈥檚 theory of surreal numbers [1976; 2001], and the present author鈥檚 amplifications and generalizations thereof and other contributions thereto.

Since that time, the bulk of my research has been devoted to developing the theory, rewriting the related history, and working out the implications of this work for the philosophy of geometry, the philosophy of number, the philosophy of the infinite and the infinitesimal, the theory of measurement and the philosophy of space and time.

Awards

51社区 Presidential Research Scholar in Arts and Humanities (2002-2007)

National Science Foundation Scholars Award (# SBR-0724700) (2007-11)

National Science Foundation Scholars Award (#SBR-9602154)(1996-99)

National Science Foundation Scholars Award (#SBR-9223839)(1993-95)

51社区 Professional Development Award (Fall 1999)

51社区 Professional Development Award (Spring 1998)

51社区 Professional Development Award (Fall 1996)

Honors

Associate of Center for Philosophy of Science, University of Pittsburgh (1999-)

Fellowships

Visiting Fellow, Center for the Philosophy of Science (Winter, 2002), University of Pittsburgh.

Research Fellow, Center for the Philosophy and History of Science (1992-1993), Boston University, Boston, MA.

Publications

Articles

鈥淭he Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small鈥�, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45.

鈥淐onway Names, the Simplicity Hierarchy and the Surreal Number Tree鈥�, The Journal of Logic and Analysis 3 (2011) no. 1, pp. 1-26.

鈥淭he Absolute Arithmetic Continuum and its Peircean Counterpart,鈥� in New Essays on Peirce鈥檚 Mathematical Philosophy, edited by Matthew Moore, Open Court Press, 2010.

鈥淭he Absolute Arithmetic Continuum,鈥� Synthese (forthcoming).

鈥淛.L. Bell, The Continuous and the Infinitesimal in Mathematics and Philosophy,鈥� (Book Review) The Bulletin of Symbolic Logic 13 (2007), no. 3, pp. 361-363.

鈥淭he Rise of non-Archimedean Mathematics and the Roots of a Misconception I: the Emergence of non-Archimedean Systems of Magnitudes,鈥� Archive for History of Exact Sciences 60 (2006), pp. 1-121.

鈥淐ontinuity,鈥� in Encyclopedia of Philosophy, Second Edition, Donald M. Borchart, Editor in Chief, Macmillan Reference USA, 2005, Volume 2, pp. 489-518.

鈥淎rthur Fine,鈥� entry in The Dictionary of Modern American Philosophers, General Editor, John R. Shook, Bristol: Thoemmes Press, 2005.

鈥淪urreal Numbers: An Alternative Construction,鈥� The Bulletin of Symbolic Logic 8 (2002), no. 3, p. 448.

鈥淣umber Systems with Simplicity Hierarchies: A Generalization of Conway鈥檚 Theory of Surreal Numbers,鈥� The Journal of Symbolic Logic 66 (2001), no. 3, pp. 1231-1258. Errata.

鈥淔ields of Surreal Numbers and Exponentiation,鈥� (co-authored with Lou van den Dries), Fundamenta Mathematicae 167 (2001), no. 2, pp. 173-188. Erratum: Fundamenta Mathematicae 168 (2001), no. 2, pp. 295-297.

鈥淔rom Completeness to Archimedean Completeness: An Essay in the Foundations of Euclidean Geometry,鈥� in A Symposium on David Hilbert edited by Alfred Tauber and Akihiro Kanamori, Synthese 110 (1997), pp. 57-76.

鈥淒edekind Cuts of Archimedean Complete Ordered Abelian Groups,鈥� Algebra Universalis 37 (1997), pp. 223-234.

鈥淗ahn鈥檚 鈥溍渂er die nichtarchimedischen Gr枚ssensysteme鈥� and the Origins of the Modern Theory of Magnitudes and Numbers to Measure Them,鈥� in From Dedekind to G枚del: Essays on the Development of the Foundations of Mathematics, edited by Jaakko Hintikka, Kluwer Academic Publishers, 1995, pp. 165-213.

鈥淎ll Numbers Great and Small,鈥� in Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich, Kluwer Academic Publishers, 1994, pp. 239-258.

鈥淯niversally Extending Arithmetic Continua,鈥� in Le Labyrinthe du Continu, Colloque de Cerisy, edited by H. Sinaceur and J.M. Salanskis, Springer-Verlag France, Paris, 1992, pp. 168-178.

鈥淯niversally Extended Continua,鈥� Abstracts of Papers Presented to the American Mathematical Society, 10 (January, 1989), p. 15.

鈥淎bsolutely Saturated Models,鈥� Fundamenta Mathematica 133 (1989), pp. 39-46.

鈥淎n Alternative Construction of Conway鈥檚 Ordered Field No,鈥� Algebra Universalis 25 (1988), pp. 7-16. Errata, Ibid. 25, p. 233.

鈥淭he Absolute Arithmetic and Geometric Continua,鈥� PSA 1986, Volume 2, edited by Arthur Fine and Peter Machamer, Philosophy of Science Association, Lansing, MI (1987), pp. 237-247.

鈥淎n Alternative Construction of Conway鈥檚 Surreal Numbers,鈥� (co-authored with Norman Alling), Comptes Rendus Mathematiques De L鈥橝cademie Des Sciences, Canada VIII (1986), pp. 241-46. Reprinted in Collected Papers of Norman Alling, edited by Paulo Ribenboim, Queen鈥檚 Papers in Pure and Applied Mathematics, Volume 107, 1998, Kingston, Ontario, Canada.

鈥淎n Abstract Characterization of a Full Class of Surreal Numbers,鈥� (co-authored with Norman Alling), Comptes Rendus Mathematiques De L鈥橝cademie Des Sciences, Canada VIII (1986), pp. 303-8. Reprinted in Collected Papers of Norman Alling, edited by Paulo Ribenboim, Queen鈥檚 Papers in Pure and Applied Mathematics, Volume 107, 1998, Kingston, Ontario, Canada.

鈥淣egative, Infinite and Hotter than Infinite Temperatures,鈥� Synthese 50 (1982), pp. 233-77. Reprinted in Philosophical Problems of Modern Physics, edited by Hans S. Plendl, Reidel Publishing Co., Boston (1982).

鈥淭he Concept of Temperature and its Dependence on the Laws of Thermodynamics,鈥� The American Journal of Physics 49 (1981), pp. 622-32.

Edited Books

Real Numbers, Generalizations of the Reals, and Theories of Continua, edited with a General Introduction by Philip Ehrlich, Kluwer Academic Publishers, 1994. The contemporary contributors are Douglas S. Bridges, J. H. Conway, Gordon Fisher, Hourya Sinaceur, H. J. Keisler, Philip Ehrlich, Dieter Klaua, and Mathieu Marion; there are also little-known classical contributions by E. W. Hobson, Henri Poincar茅, and Giuseppe Veronese.

Philosophical and Foundational Issues in Measurement Theory, (co-edited with C. Wade Savage) Lawrence Erlbaum Associates, Inc., Publishers, 365 Broadway, Hillsdale, NJ 07642, 1990. The contributors are Patrick Suppes, Mario Zanotti, Ernest Adams, Karel Berka, Zolton Domotor, Brian Ellis, Arnold Koslow, Henry Kyburg, Louis Narens, John Burgess, Wolfgang Balzer, and R.D. Luce.

Portions or Chapters of Books

鈥淕eneral Introduction鈥�, in Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich, Kluwer Academic Publishers, 1994, pp. vii-xxxii.

鈥淓ditorial Notes鈥� to 鈥淥n Non-Archimedean Geometry: Invited Address to the International Congress of Mathematics, Rome, April 1908, by Giuseppe Veronese鈥�, translated by Mathieu Marion (with editorial notes by Philip Ehrlich), in Real Numbers, Generalizations of the Reals, and Theories of Continua, edited by Philip Ehrlich, Kluwer Academic Publishers, 1994, pp. (for notes) 182-187.

鈥淎 Brief Introduction to Measurement Theory,鈥� (co-authored with C. Wade Salvage), in Philosophical and Foundational Issues in Measurement Theory, (co-edited with C. Wade Savage) Lawrence Erlbaum Associates, Inc., Publishers, 365 Broadway, Hillsdale, NJ 07642, 1990, pp. 1-14.

Sections 4.02 and 4.03 of Norman Alling鈥檚 Foundations of Analysis Over Surreal Number Fields, North-Holland Publishing Co., Amsterdam, (1987).