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 “completely solved” [1901, p. 89]. Indeed, as Russell went on to add: “The solutions, for those acquainted with mathematics, are so clear as to leave no longer the slightest doubt or difficulty” [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 “banish[ed] from mathematics” through the work of Weierstrass and others [1901, pp. 88, 90]. These themes were reiterated in Russell’s often reprinted Mathematics and the Metaphysician [1918], and further developed in both editions of Russell’s 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 “banished” 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’s theory of the infinite, a generalization that also provides a setting for Abraham Robinson’s 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’s theory of surreal numbers [1976; 2001], and the present author’s 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
Ohio University 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)
Ohio University Professional Development Award (Fall 1999)
Ohio University Professional Development Award (Spring 1998)
Ohio University 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
“The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45.
“Conway Names, the Simplicity Hierarchy and the Surreal Number Tree”, The Journal of Logic and Analysis 3 (2011) no. 1, pp. 1-26.
“The Absolute Arithmetic Continuum and its Peircean Counterpart,” in New Essays on Peirce’s Mathematical Philosophy, edited by Matthew Moore, Open Court Press, 2010.
“The Absolute Arithmetic Continuum,” Synthese (forthcoming).
“J.L. Bell, The Continuous and the Infinitesimal in Mathematics and Philosophy,” (Book Review) The Bulletin of Symbolic Logic 13 (2007), no. 3, pp. 361-363.
“The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: the Emergence of non-Archimedean Systems of Magnitudes,” Archive for History of Exact Sciences 60 (2006), pp. 1-121.
“Continuity,” in Encyclopedia of Philosophy, Second Edition, Donald M. Borchart, Editor in Chief, Macmillan Reference USA, 2005, Volume 2, pp. 489-518.
“Arthur Fine,” entry in The Dictionary of Modern American Philosophers , General Editor, John R. Shook, Bristol: Thoemmes Press, 2005.
“Surreal Numbers: An Alternative Construction,” The Bulletin of Symbolic Logic 8 (2002), no. 3, p. 448.
“Number Systems with Simplicity Hierarchies: A Generalization of Conway’s Theory of Surreal Numbers,” The Journal of Symbolic Logic 66 (2001), no. 3, pp. 1231-1258. Errata.
“Fields of Surreal Numbers and Exponentiation,” (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.
“From Completeness to Archimedean Completeness: An Essay in the Foundations of Euclidean Geometry,” in A Symposium on David Hilbert edited by Alfred Tauber and Akihiro Kanamori, Synthese 110 (1997), pp. 57-76.
“Dedekind Cuts of Archimedean Complete Ordered Abelian Groups,” Algebra Universalis 37 (1997), pp. 223-234.
“Hahn’s “Über die nichtarchimedischen Grössensysteme” and the Origins of the Modern Theory of Magnitudes and Numbers to Measure Them,” in From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics , edited by Jaakko Hintikka, Kluwer Academic Publishers, 1995, pp. 165-213.
“All Numbers Great and Small,” in Real Numbers, Generalizations of the Reals, and Theories of Continua , edited by Philip Ehrlich, Kluwer Academic Publishers, 1994, pp. 239-258.
“Universally Extending Arithmetic Continua,” in Le Labyrinthe du Continu, Colloque de Cerisy , edited by H. Sinaceur and J.M. Salanskis, Springer-Verlag France, Paris, 1992, pp. 168-178.
“Universally Extended Continua,” Abstracts of Papers Presented to the American Mathematical Society , 10 (January, 1989), p. 15.
“Absolutely Saturated Models,” Fundamenta Mathematica 133 (1989), pp. 39-46.
“An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25 (1988), pp. 7-16. Errata, Ibid. 25, p. 233.
“The Absolute Arithmetic and Geometric Continua,” PSA 1986, Volume 2 , edited by Arthur Fine and Peter Machamer, Philosophy of Science Association, Lansing, MI (1987), pp. 237-247.
“An Alternative Construction of Conway’s Surreal Numbers,” (co-authored with Norman Alling), Comptes Rendus Mathematiques De L’Academie Des Sciences, Canada VIII (1986), pp. 241-46. Reprinted in Collected Papers of Norman Alling, edited by Paulo Ribenboim, Queen’s Papers in Pure and Applied Mathematics , Volume 107, 1998, Kingston, Ontario, Canada.
“An Abstract Characterization of a Full Class of Surreal Numbers,” (co-authored with Norman Alling), Comptes Rendus Mathematiques De L’Academie Des Sciences, Canada VIII (1986), pp. 303-8. Reprinted in Collected Papers of Norman Alling, edited by Paulo Ribenboim , Queen’s Papers in Pure and Applied Mathematics , Volume 107, 1998, Kingston, Ontario, Canada.
“Negative, Infinite and Hotter than Infinite Temperatures,” Synthese 50 (1982), pp. 233-77. Reprinted in Philosophical Problems of Modern Physics , edited by Hans S. Plendl, Reidel Publishing Co., Boston (1982).
“The Concept of Temperature and its Dependence on the Laws of Thermodynamics,” 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 Poincaré, 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
“General Introduction”, in Real Numbers, Generalizations of the Reals, and Theories of Continua , edited by Philip Ehrlich, Kluwer Academic Publishers, 1994, pp. vii-xxxii.
“Editorial Notes” to “On Non-Archimedean Geometry: Invited Address to the International Congress of Mathematics, Rome, April 1908, by Giuseppe Veronese”, 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.
“A Brief Introduction to Measurement Theory,” (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’s Foundations of Analysis Over Surreal Number Fields , North-Holland Publishing Co., Amsterdam, (1987).