Philip Ehrlich
Professor
Department of Philosophy Ohio University Ellis Hall, 220 C Athens, Ohio 45701 (740)5934595
office (740)5934597 fax 

Education
Ph.D. University of Illinois, Chicago
Areas of Specialization or Competence
Logic, History and Philosophy of Mathematics,
Philosophy of Science, Philosophy of Physics
Current Research
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
philosophythe nature of the infinite, the nature of the infinitesimal, and
the nature of the continuumhad all been Òcompletely solvedÓ [1901, p. 89]. Indeed,
as Russell went on to add: ÒThe solutions, for those acquainted with
mathematics, are so clear as to leave no longer the slightest doubt or
difficultyÓ [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 Òbanish[ed] from mathematicsÓ through the work of Weierstrass and others [1901, pp. 88, 90]. These themes
were reiterated in RussellÕs often reprinted Mathematics and the
Metaphysician [1918], and further developed in both editions of RussellÕs 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 ÒbanishedÓ 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 CantorÕs theory of the infinite, a
generalization that also provides a setting for Abraham RobinsonÕs
infinitesimal approach to analysis [1961; 1966] as well as for the profound and
all too often overlooked nonCantorian theories of
the infinite (and infinitesimal) pioneered by Giuseppe Veronese [1891; 1894], Tullio LeviCivita [1892; 1898],
David Hilbert [1899] and Hans Hahn [1907] in connection with their work on
nonArchimedean ordered algebraic and geometric systems and by Paul du BoisReymond (cf. [187071;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. ConwayÕs theory of surreal numbers [1976; 2001], and
the present authorÕs 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 (20022007)
National
Science Foundation Scholars Award (# SBR0724700) (200711)
National Science Foundation Scholars Award (#SBR9602154)(199699)
National Science Foundation Scholars Award (#SBR9223839)(199395)
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
(19921993), Boston University, Boston, MA.
Articles
ÒThe Absolute
Arithmetic Continuum,Ó Synthese (forthcoming).
ÒArthur Fine,Ó entry in The Dictionary of
Modern American Philosophers, General Editor, John R. Shook, Bristol: Thoemmes Press, 2005.
ÒSurreal Numbers: An Alternative Construction,Ó The
Bulletin of Symbolic Logic 8 (2002), no. 3, p. 448.
ÒNumber Systems
with Simplicity Hierarchies: A Generalization of ConwayÕs Theory of Surreal
Numbers,Ó The Journal of Symbolic Logic 66 (2001), no. 3, pp.
12311258. Errata.
ÒFields of Surreal Numbers and Exponentiation,Ó (coauthored with Lou van den Dries), Fundamenta
Mathematicae 167 (2001), no. 2, pp. 173188.
Erratum: Fundamenta Mathematicae
168 (2001), no. 2, pp. 295297.
ÒHahnÕs Ò†ber die nichtarchimedischen GršssensystemeÓ and the Origins of the Modern Theory of
Magnitudes and Numbers to Measure Them,Ó in From Dedekind to Gšdel:
Essays on the Development of the Foundations of Mathematics, edited by Jaakko Hintikka, Kluwer Academic
Publishers, 1995, pp. 165213.
ÒAll Numbers Great and
Small,Ó in Real Numbers, Generalizations of the Reals,
and Theories of Continua, edited by Philip Ehrlich, Kluwer Academic
Publishers, 1994, pp. 239258.
ÒUniversally Extended Continua,Ó Abstracts of
Papers Presented to the American Mathematical Society, 10 (January, 1989), p.
15.
ÒAbsolutely
Saturated Models,Ó Fundamenta Mathematica
133 (1989), pp. 3946.
ÒAn Alternative
Construction of ConwayÕs Ordered Field No,Ó Algebra Universalis
25 (1988), pp. 716. Errata, Ibid. 25, p. 233.
ÒThe Absolute Arithmetic and
Geometric Continua,Ó PSA 1986, Volume 2, edited by Arthur Fine and
Peter Machamer, Philosophy of Science Association,
Lansing, MI (1987), pp. 237247.
ÒAn Alternative
Construction of ConwayÕs Surreal Numbers,Ó (coauthored with Norman Alling), Comptes Rendus Mathematiques De LÕAcademie Des Sciences, Canada VIII (1986), pp.
24146. Reprinted in Collected
Papers of Norman Alling, edited by Paulo Ribenboim, QueenÕs Papers in Pure and Applied
Mathematics, Volume 107, 1998, Kingston, Ontario, Canada.
ÒAn Abstract
Characterization of a Full Class of Surreal Numbers,Ó (coauthored with Norman Alling), Comptes Rendus Mathematiques De LÕAcademie Des Sciences, Canada VIII (1986), pp. 3038. Reprinted in Collected Papers of Norman Alling,
edited by Paulo Ribenboim, QueenÕs Papers in Pure
and Applied Mathematics, Volume 107, 1998, Kingston, Ontario, Canada.
ÒNegative, Infinite and Hotter than Infinite
Temperatures,Ó Synthese 50 (1982), pp. 23377.
Reprinted in Philosophical Problems of Modern Physics, edited by Hans S.
Plendl, Reidel Publishing
Co., Boston (1982).
ÒThe Concept of Temperature and its Dependence
on the Laws of Thermodynamics,Ó The American Journal of Physics 49
(1981), pp. 62232.
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 littleknown classical contributions by E. W. Hobson, Henri PoincarŽ, and Giuseppe Veronese.
Philosophical and
Foundational Issues in Measurement Theory,
(coedited 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
ÒGeneral IntroductionÓ, in Real Numbers,
Generalizations of the Reals, and Theories of
Continua, edited by Philip Ehrlich, Kluwer Academic Publishers, 1994, pp.
viixxxii.
ÒEditorial NotesÓ to ÒOn NonArchimedean
Geometry: Invited Address to the International Congress of Mathematics, Rome,
April 1908, by Giuseppe VeroneseÓ, 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) 182187.
ÒA Brief Introduction to Measurement Theory,Ó
(coauthored with C. Wade Salvage), in Philosophical and Foundational Issues
in Measurement Theory, (coedited with C. Wade Savage) Lawrence Erlbaum
Associates, Inc., Publishers, 365 Broadway, Hillsdale, NJ 07642, 1990, pp.
114.
Sections 4.02 and 4.03 of Norman AllingÕs Foundations of Analysis Over Surreal Number
Fields, NorthHolland Publishing Co., Amsterdam, (1987).