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Curriculum Vitae


The Mathematical Experience (National Book Award, w. Phil Davis)         (Birkhauser)

Descartes’ Dream (w. Phil Davis) (Dover)

What Is Mathematics, Really? (Oxford)

18 Unconventional Essays on the Nature of Mathematics (edited)  (Springer)

Loving and Hating Mathematics (w. Vera John-Steiner)  (Princeton, 2011)



 Mathematical practice and life

“Mathematicians’ Proof,  Established Mathematics, and Warranted Assertibility”,  to appear in “From a Heuristic Point of View,” ed. E. Ippoliti, Cambridge Scholars

“Paul Cohen in 1963,”  The Mathematical Intelligencer, 24 (3) 38-40, 2011

“Dialogo fra un matematico de succeso e uno studente testardo”,  Archimede (Firenze, Italia)   ottobre-dicembre 2010   4/2010, 202--205

“Alvin White,”  Journal of Humanistic Mathematics, 2011

“Under-represented Then over-represented:  A Memoir of Jews in American    Mathematics,”    College Mathematics Journal, January 2010, pp. 2-9

“Refuge from Misery and Suffering,”  Mathematical Intelligencer, Winter, 2008, 30 (1)  22-26

“Clarence Stephens and the Potsdam Model,” Math Horizons, February 2010, pp 18-21, 29

Introduction to Indiscrete thoughts by Gian-Carlo Rota, Birkhauser, 1997

“Report on programs for minorities in mathematics” , Newsletter of SW region of MAA, 1988; Focus

Obituary of Nicholas Kazarinoff, SIAM News

“Proof is convincing and explaining,” Ed Stud in Math vol 24 pp 389-399 1993

“What is humanistic mathematics?”,  Mathematics in College, 1990 49-50

“Humanistic mathematics and the real world,” in Essays in Humanistic Mathematics,   MAA 1993, ed. A. White, pp. 15-18

“Szeged in 1934,” (edited from manuscripts of Edgar Lorch) Amer Math Monthly 100(3)          Winner of Ford Prize

“Math lingo vs. plain English: double entendre”, Amer Math Monthly January 1997

Biographies of David Blackwell and Solomon Lefschetz, Dictionary of Mathematicians

 “Mathematical menopause, or, a young man’s game?” The Mathematical Intelligencer   23(3) 2001 52-60

“Form and function in mathematical modeling,” talk at Budapest meeting of Santa Fe      Institute

“The Origin of Chaos,” talk at Santa Fe Institute

“Mathematics and Ethics,” The Mathematical Intelligencer 12(3) 1990  12-15

“Ethics for Mathematicians,” in Philosophy of Math Education, P. Ernest ed., 2007


Partial differential equations

“Mixed problems in several variables,” J Math Mech 12(3) 1963

“Boundary conditions for equations of evolution,” Arch Rat Mech Anal 16(4) 1964

“On surface waves with finite and infinite speed of propagation,” Arch Rat Mech Anal   19(4) 1965

“On vibration, diffusion or equilibrium across a plane interface,” Arch Rat Mech Anal     21(5)1966

“The three-dimensional wave equation in a characteristic quarter-space,” with Y W Chen,  J Math and Mech, November 1967

“Finite difference approximations for mixed initial-boundary value problems of general    type,” SIAM J on Numer Anal 5(2) June 1968

“On the general theory of mixed problems,” Proc Battelle Rencontres 1968, 85-96

“Direct solution of general one-dimensional parabolic equations via an abstract   Plancherel formula,” Proc Nat Acad Sci July 1969

“The method of reflection for two-sided problems of general type,” in Studies and Essays  Presented to Yu-Why Chen, Math Res Center, Taipei, October 1970, pp 209-222

“A perturbation series for Cauchy’s problem for higher-order abstract parabolic equations” (with J Donaldson) Proc Nat Acad Sci vol 67 September 1970, pp        41-44

“How to classify differential polynomials,” Amer Math Monthly June-July 1973 pp  641-654

“The method of transmutations,”  Lecture Notes in Mathematics,  Partial Differential Equations and Related Topics, Springer-Verlag 1975, no 446, pp 264-282    

“Stochastic solutions of hyperbolic equations”, ibid, pp 283-301

“Hyperbolic equations with coefficients in an enveloping algebra” (with S Steinberg) J Diff Eq 34(3) December 1979 pp 405-426


Random evolutions

“Random semigroups, Markov chains, and systems of partial differential equations” (with Richard J Griego) Proc Nat Acad Sci February 1969

“Theory of random evolutions with applications to partial differential equations” (with R J Griego) Trans Amer Math Soc vol 156 May 1971 pp 405-418

“Random evolutions are asymptotically Gaussian” (with M Pinsky) Comm Pure & Appl Math vol 25 1972 pp 33-44

“Non-commuting random evolutions, and an operator-valued Feynman-Kac formula”  (with G Papanicolaou) Comm Pure & Appl Math vol 25 1972 pp 337-367

“Two limit theorems for random differential equations” (with R Cogburn) Indiana U Math J  vol 22 1973 pp 1067-1089

“Random evolutions, a survey of results and problems,” Rocky Mt J Math, Summer 1974 pp 443-477

“Weyl’s theorem for certain operator-valued potentials” (with R. Griego) Indiana Univ Math J 1978 27(2) pp 195-209

“The birth of random evolutions,” The Mathematical Intelligencer 25(1) 2003 53-60


“A class of central limit theorems for convolution products of generalized functions,”  Trans Amer Math Soc June 1969

“Some limit theorems for stochastic equations  and applications” (with G Papanicolaou) Indiana U Math J vol 21 1972 pp 815-840

“Maxwell’s coefficients are conditional probabilities” Proc Amer Math Soc June 1974 pp 449-453

“Introduction to a special issue on stochastic differential equations,” Rocky Mt J Math Summer 1974

“Quasi-standard random variables and stochastic differentials” (with P. Greenwood) Stochastic differential equations Springer Lecture Series no 451 pp 35-62

Brownian motion and nonstandard analysis, University of New Mexico Technical Report No. 277, May 1973.

Linear operator equations

“Explicit solution of a class of abstract Cauchy problems,” J Diff Eqs vol 8 1970  570-  579

“Perturbation and approximation theory for higher-order abstract Cauchy problems”  (with L Bobisud) Rocky Mt J Math vol 2 1972 pp 57-73

“On the invariance principle of scattering theory” (with J A Donaldson and A G Gibson)  J Func Anal October 1973 14(2) pp131-145

“High-accuracy stable difference schemes for well-posed initial-value problems” (with T.Kato) SIAM J Numer Anal 1979 pp670-682

Pedagogy and philosophy

“Why Faulhaber Polynomials are Powers of (n + 1/2)”, College Math J., 2011

“From counting to quaternions”, Journal of Humanistic Mathematics, Vol 1, No. 1, 2011

“Mathematical Intuition:  Poincare, Polya, Dewey”,  in Knowledge and Logic” ,  Ed. C Cellucci et al., Cambridge Scholars Publishing,  Newcastle on Tyne,  pp.  223-242  (also at Rio de Janeiro, CLF Conference,  May 2010)

 “Mathematical discourse:  the link from pre-mathematical to fully mathematical thinking,” (with KristinUmland) in Philosophy and Education 19         “Mathematical practice as a scientific problem,” in  Proof & Other Dilemmas:  Mathematics and Philosophy, ed B Gold, MAA 2008

 “Definition of mathematics” in Dictionary of Scientific Terms, ed by G. Sica

“On Platonism,” Newsletter of the European Mathematical Society, June, 2008

“On von Glasersfeld’s Radical Constructivism,” Constructivist Notes, 2008

“What I would like my freshman students to know about proof”, in Teaching and Learning Proof Across the Grades, Ed. D. A. Stylianou et al., Routledge and National Council of Teachers of Mathematics,  2009

“Wings, not foundations!”, in G. Sica ed.,Essays on the foundations of mathematics and logic, Polimetrica, 2005

Foreword to New Mexico Math Contest Problem Book by L S Hahn, UNM Press 2005

 “On the interdisciplinary study of mathematical practice,” in B. v. Kerkhove & J.P.v.Bendegem, Perspectives on mathematical practices, Springer, 2007

“Independent Thinking”  College Math J. 34 (2) 2003

“Sines and cosines are just exponentials,” Math Rev (England) 2(4) April 1992 22-25

“The classroom encounter,” Humanistic Math Newsletter

“Reply to Martin Gardner,” The Mathematical Intelligencer 23 (1) 2001, 3-5

“The origin of geometry,” College Math J 33(3) May 2002 207-211

“A nifty derivation of Heron’s area formula by 11th grade algebra,” Focus 22(8)  2002 p 22

“Heron’s area formula:  what about a tetrahedron?”  College Math J 35(2) March 2004 112-114

“A simple derivation of Heron’s area formula,” Wiskunde & Onderweis 2003

 “Some proposals for reviving the philosophy of mathematics,” Advance in Mathematics 31(1) January 1979 pp 31-50

“Introducing Imre Lakatos,” Mathematical Intelligencer 1(3) 1978 pp 148-151

“Rhetoric and mathematics” (with P J Davis) in The rhetoric of the human sciences  University of Wisconsin Press 1987 53-68

“Mathematics has a front and a back,” Eureka 1988, Synthese 88(2) August 1991

“Introduction to special issue,” Synthese 88(2) August 1991

“A mathematician’s perspective,” in P. Connolly and T. Vilardi, Writing to learn mathematics and science, TC Press, 1989

“Let’s teach philosophy of math!” College Math J 21(2) March 1990 105-111

“Inner knowledge, outer truth,” in Mathematics and Science, ed. R. Mickens, World   Scientific, Singapore, 1990

“A visit to Hungarian mathematics,” (with V. John-Steiner) The Mathematical Intelligencer Spring 1993

“Fresh breezes in the philosophy of mathematics,” NY Acad of Sci December 2, 1993; in Amer Math Monthly and in Mathematics, Education and Philosophy ed P Ernest   London 1994

Popular articles

Non-Cantorian set theory (with Paul Cohen) Scientific American December 1967

Brownian motion and potential theory (with Richard Griego) Scientific American March 1969

Nonstandard analysis (with M Davis) Scientific American June 1972 pp 78-86

Hilbert’s tenth problem (with M Davis) Scientific American November 1973 pp 84-91 Winner of the Chauvenet Prize

Brownian motion, Encyclopedia Britannica 15th edition

Infinity.  Encyclopedia Britannica


“Freud on math,” The Mathematical Intelligencer 1995

“How to do and write math research”, The Mathematical Intelligencer September 1997

“Follow-up on math lingo,” MAA Online

“Independent thinking,” College Math J 34(2) March 2003  112-115

“The wedding,” The Mathematical Intelligencer, 2006

Book reviews

Kees van Deemter,  “Not Exactly…in praise of vagueness”  SIAM Review

Peter Lax, “Selected Papers,” AMS Bulletin

William Byers, “How mathematicians think,” Notices of the AMS,   54 (11)  December,   2007

Neal Koblitz, “Random Curves,”  College Math J., 40 (2), March, 2009

Alain Badiou, “Number and numbers,”  Nordisk Matematisk Tidskrift, 2009

George Szpiro, “Poincare’s Prize,” College Math J., September, 2009

Alain Badiou, “Number and numbers,” The Mathematical Intelligencer, 2009, #3

Masha Gessen, “Perfect Rigor,”  The Mathematical Intelligencer, 2010

Roland Omnes, “Converging Realities,” American Scientist

Jonathan Borwein and David Bailey, “Mathematics by experiment” and “Experimentation in mathematics,” SIAM Review (with R. Frye)

Harold Edwards, “Essays in constructive mathematics,” Springer Verlag

Stanislaw Ulam,”Sets, numbers and universes,” “Analogies between analogies,”  “Science, computers and people”  The Mathematical Intelligencer 14 (4) 1992,      71-73

R. Utton,  “The Social Life of Numbers”  J Anthropological Res.

Paul Hoffman, “Archimedes Revenge”, American Scientist 77(5) sept-oct 1989, 500-501

Mark Kac, “Enigmas of chance,” Advances in Mathematics September 1986 p 305

Gian-Carlo Rota, “Indiscrete Thoughts,” College Math Journal

Zalman Usiskin et al., “Mathematics for high school teachers,” Prentice Hall

Alberto Martinez, “Negative Math,” Princeton U. Press

Morris Kline, “Mathematics and the search for knowledge,” American Mathematical Monthly 94 (3) March 1987, 314-315

Imre Lakatos, “Proofs and Refutations,” Advances in mathematics July 1978, 131-133

Philip Kitcher & W. Asprey, “History and philosophy of modern mathematics Vol XI Minnesota Studies in the Philosophy of Science,” Historia Mathematica vol.18,         288-291

John Casti, “Alternate realities—mathematical models of nature and man,” UMAP  Journal, November 4, 359-361

Paul Ernest, “Philosophy of  Mathematics Education,” Philosophia Mathematica, May1994, 172--176

L. Salem et al. “The most beautiful mathematical formulas,” UMAP Journal

Marcia Ascher, “Ethnomathematics,” SIAM Review 35 (1) March 1993, 151-153

Clifford A. Pickover, “Mazes for the Mind,” IEEE Spectrum, April 1993, pp 7. 12

David Darling, “Equations of Infinity,” IEEE Spectrum

Mark Kac, G.-C. Rota and J. Schwartz, “Discrete Thoughts,” Review of Mathematics Books and Software

Jacot & Hudson, “The way men think,” w. V. John-Steiner, Journal of Gender Studies

Keith Devlin, “The math gene,” The Mathematical Intelligencer

Stanislas Dehaene, “The Number Sense,” American Mathematical Monthly

Marcus du Sautoy, “The music of the primes,”   London Times

John Derbyshire, “Prime obsession,”  London Times

Keith Ball, “Strange curves, counting rabbits…”   London Times

Thomas A. Garrity, “All the mathematics you missed,” UMAP Journal

Amir Aczel, “Geometrical landscapes,” American Scientist

David Corfield, “Towards a philosophy of real mathematics,” SIAM Review

Donald Estep, “Practical analysis in one variable,” SIAM Review


What is mathematics, really? Edge website, Circuit (Canberra Math Assoc March 1999)