Education:                             Born: January 9,1930, New York City

                                                B.S.: C.C.N.Y., 1948

                                                Ph.D.: Yale University, 1951



Academic Employment:


Instructor:                             Yale University, 1951-1953


Assistant Professor:             Yale University, 1953 -1956


Associate Professor:             New York University, 1957 -1958


Professor:                              New York University, 1958 - present



Currently: Professor, Mathematics and Computer Science, Courant Institute,

New York University


Administrative Service:


Former Director, NYU Multimedia Center, 1995-96


Chairman, Computer Science Department, Courant Institute of Mathematical Sciences, New York University, 1969-1977


Former Director, DARPA/ISTO, Arlington VA, 1987-89.


Former Chairman, Computer Science Board, National Research Council


Technical Program Chairman, AFIPS Spring Joint Computer Conference, 1972




Honors:                                 Sloane Fellow, 1961-1962

Wilbur Cross Medal, Yale University

Townsend Harris Medal, City University of New York

Steele Prize, American Mathematical Society, 1981

(shared with N. Dunford)

Mayor's Medal for Contributions to Science and Technology,                         New York City,1986

                                                Member, National Academy of Sciences

                                                Member, American Academy of Arts and Sciences

                                                Corporate Fellow, Thinking Machines Corp.



Recent Grants:  Graduate Traineeeships Grant, National Science Foundation, 1994-present. $750,000 over 5 years.


Center for Advanced Technology Grant, New York State Science and Technology Foundation, , 1994-present. $5,000,000 over 5 years.  I was the lead developer of this major grant, and, as initial Director of the Center for Advanced Technology which it created, responsible for raising $1,000,000 in matching funds annually for the first two years of the Center's existence. This involved a wide variety of activities, including creation of a program of industrial training Seminars, which metamorphosed into the substantial Multimedia program now offered by NYU's School of Continuing Education.


Current Interests and Activities.  I hope at this point to concentrate my attention on a limited number of larger projects: (i) Development of multimedia educational materials. Following on a preliminary design study of a complete set of interactive multimedia materials for a much expanded version of the current high-school mathematics curriculum (to include matrices, splines, algebra thru Sturm's theorem and some calculus), I have developed a plan for realizing the whole in about 5,000 interactive screens. Since current authoring tools are not adequate for a project of this size, this led to a second project, (ii) development of a new authoring tool suite, incorporating systematic notions of Œkeyboardless programming' and some new database ideas. This is joint work with Salvatore Paxia and Giuseppe DiMauro, actively in progress for the last 2 years, and now well advanced.  An expanded edition of the book Higher Level Programming: An Introduction to the Set-Theoretic Language SETL, describing this work (probably in 2 volumes,) is under development. (iii) Studies of visual perception. My increasingly active involvement in this area reflects the fact that such studies can be conducted very effectively using current multimedia tools. (iv) ŒLegion of Science': I have developed some innovative ideas concerning the potential use of desktop computers for mass (national scale) scientific research activities, and have begun working with a few New York area high schools to get this started. (v) Computational logic. Using the new set-based multimedia tools resulting from project (ii), I hope to complete previously initiated work on a major new proof-verifier system growing out of the series of papers with Ferro, Cantone, Omodeo, et al listed below.


Editorial Board Memberships:


Advances in Applied Mathematics;

Journal of Programming Languages;

Discrete and Computational Geometry;

Computers and Mathematics with Applications;

Psychoanalysis and Contemporary Thought




(Chronological order)



Books (Mathematics and Computer Science)


Linear Operators (three volumes), Interscience Publishers.

            vol. I. General Theory, 1958.

            vol. II. Spectral Theory in Hilbert Space, 1963.

            vol. III. Spectral Theory in Banach Spaces, 1970.

Republished by John Wiley & Sons in three volumes as part of the Wiley Classics

Library, New York, 1988.


Matrices and Vectors for High-Schools and Colleges, McGraw-Hill Publishing Co., New York, 1961.


Relativity in Illustrations, NYU Press, 1962.


Lectures on the Mathematical Method in Analytical Economics, Gordon and Breach Publishers, New York, 1962.


Theory of Money, Gordon and Breach Publishers, New York, 1965.


W-* Algebras, Gordon and Breach Publishers, New York, 1967.


Nonlinear Functional Analysis, Gordon and Breach Publishers, New York, 1968.


Differential Geometry and Topology, Gordon and Breach Publishers, New York, 1969.


Programming Languages and Their Compilers (with John Cocke),

Courant Institute Lecture Notes, 1968-69.


On Programming: An Interim Report on the SETL Project.

            Installment I: Generalities. New York University, 1973.           

            Installment II: The SETL Language and Examples of its Use.

            Courant Institute Lecture Notes, November 1973.



Higher Level Programming: An Introduction to the Set-Theoretic Language SETL (with E. Schonberg, E. Dubinsky, and R. Dewar), Springer-Verlag Publishers, 1986.

Discrete Thoughts: Essays on Mathematics, Science, and Philosophy (anthology of previous papers; with M. Kac and G. C. Rota), Birkhauser Publishers, Boston MA, 1986.


Planning, Geometry, and Complexity of Robot Motion, (anthology of previous papers; with

 M. Sharir and J. Hopcroft), Ablex Publishing Corp., Norwood NJ, 1987.


Advances in Robotics, vol. 1, J. T. Schwartz, C. K. Yap, editors. Lawrence Erlbaum Associates, Hillsdale NJ, 1987.



Multimedia Electronic Publications


Animated Relativity, (with Diana Schwartz; Multimedia version of Relativity in Illustrations). The Multimedia Library, New York 1992.


The Electronic Dictionary of Musical Themes, (with Diana Schwartz; Multimedia version of The Dictionary of Musical Themes by H. Barlow and S. Morgenstern, Crown Publishers, New York). The Multimedia Library, New York 1992.


 Papers, Computer Science


Large Parallel Computers, J.A.C.M., vol. 13, pp. 25-32, 1966.


Sharer: A Time Sharing for the CDC 6600 (with M. Harrison), C.A.C.M., vol. 10, pp. 659-665, 1967.


Programming Considerations for Parallel Computers (with E. Draughon, R. Grishman, and A. Stein), NYU Technical Report, 1967 (Became basis for NYU ŒUltracomputer' project.)


An Overview of Bugs, Courant Computer Science Symposium 1, pp. 1-16, Prentice-Hall Publishers, 1970.


Semantic Definition Methods and The Evolution of Programming Languages, Courant Computer Symposium 2, pp. 1- 23, Prentice-Hall Publishers, 1971.


Principles of Specification Language Design, with some Observations Concerning the Utility of Specification Languages, Courant Computer Symposium 4, pp. 1-37, Prentice-Hall Publishers, 1971.


Abstract and Concrete Problems in the Theory of Files, Courant Computer Symposium 6, pp. 1- 22,  Prentice-Hall Publishers, 1971.


Semantic and Syntactic Issues in Programming, American Mathematical Society Bulletin, vol. 88, pp. 195-206, 1974.


Automatic and Semiautomatic Optimization of SETL, Proc. of 1974 Symposium on Very High Level Languages, SIGPLAN Notices, vol. 9, no. 4, pp. 43-49, 1974.


An Introduction to the SETL Programming Language (with K. Kennedy), Computing and Mathematics with Applications,  vol. 1, pp. 97-119, 1975.


Optimization of Very High Languages: Part I. Value Transmission and its Corollaries, Journal of Computer Languages, vol. 1, pp. 161-194, 1975.


Optimization of Very High Level Languages: Part II. Deducing Relationships of Inclusion and Membership, Journal of Computer Languages, vol. 1, pp. 197-218, 1976.


Automatic Data Structure Choice in a Language of Very High Level, C.A.C.M., pp.722-728, 1975.


Program Genesis and the Design of Programming Languages. In:  Current Trends in Programming Methodology, Raymond T. Yeh, ed., vol. IV, pp. 185-215, 1976.


What Programmers Should Know, Journal of Programming Languages, vol. 2, pp. 21-25, 1976.


A Fast Switch (with S. Coppage), American Math. Monthly, vol. 83, pp. 711-718, 1976.


A History of Computers (review article), Enciclopedia Italiana, 1976.


Expression Continuity and the Formal Differentiation of Algorithms (with Robert Paige), Proceedings of the SIGPLAN/SIGACT Symposium on Principles of Programming Languages, pp. 58-71, 1977.


Correct-Program Technology Extensibility of Verifiers, (with M. Davis: Appendix by E. Deak), NYU Courant Computer Science Report No. 12, 1977.


Metamathematical Extensibility for Theorem Verifiers (with M. Davis), Courant Computer Science Report 12B, 1977. (Republished in Journal of Computer Mathematics, vol. 5, pp. 217-230, 1979.)


SETL as a Tool for Generation of Quality Software (with R. Dewar, A. Grand, S.C. Liu, and E. Schonberg), Proc. of IFIP Working Conf. on Constructing Quality Software, Novosibirsk, U.S.S.R., May 1977. (Proceedings republished by North-Holland Publishing Co., December 1978).


A Note on Monte Carlo Primality Tests and Algorithmic Information Theory (with G. Chaitin), Comm. Pure and Applied Mathematics, vol. 31, pp. 521-527, 1978.


Realizing the Potential of Microstructures: Connection Patterns, Algorithms, Languages, and Limitations, Proceedings of NSF Workshop on Opportunities in Microelectronics, November, 1978 (Revived the NYU ŒUltracomputer' project).


A Design for Optimizations of the Bitvectoring Class (with M. Sharir), NYU Courant Computer Science Report No. 17, 1979. (Subsequently incorporated in the SETL optimizer implementation).


Programming by Refinement, as Exemplified by the SETL Representation Sublanguage (with R. Dewar, A. Grand, S.D. Liu, and E. Schonberg), ACM TOPLAS, vol. 1, pp. 27-49, 1979.


Automatic Data Structure Selection in SETL (with M. Sharir), Proc. ACM Conf. on Principles of Programming Languages, 1979.


Fast Probabilistic Algorithms for Verification of Polynomial Identities, J.A.C.M., vol. 27, pp. 701-717, 1980.


Decision Procedures for Elementary Sublanguages of Set Theory


Part I: Multi-Level Syllogistic and its Extensions (with A. Ferro and E. Omodeo), Comm. Pure and Appl. Math., vol. 33, pp. 599-608, 1980.


Part II: A Decidable Class of Quantified Formulae (with A. Ferro, M. Breban and E. Omodeo), Comm. Pure and Appl. Math., vol. 34, pp. 177-195,1981.


Part V: Multilevel Syllogistic Extended by the General Union Operator, (with D. Cantone and A. Ferro), J. Comp. System Sciences, vol. 34, pp. 1-18, 1987.


Part VI: Multilevel Syllogistic Extended by the Powerset Operator, (with D. Cantone and A. Ferro), Comm. Pure Appl. Math., vol. 38, pp. 549-571, 1985.


Ultracomputers. ACM TOPLAS, vol. 2, pp.484-521, 1980.


An  Automatic Technique for Selection of Data Representations in SETL Programs (with E. Schonberg and M. Sharir), ACM TOPLAS, vol. 3, pp. 126-143,1981.


Finding the Minimum Distance Between Two Convex Polygons, Info. Processing Letters, vol. 15, pp. 168-170, 1981.


On the Piano Movers' Problem: I. The special Case of a Rigid Polygonal Body Moving Amidst Polygonal Barriers (with M. Sharir), NYU Technical Report No. 39, September, 1981.  Also in Comm. Pure Appl. Math., vol. 36, pp. 345-398, 1983.  Also in Office of Naval Research:  Forty Years of Excellence in Support of Naval Science; 40th anniversary, 1946-1986, ONR, Dept. of the Navy,  pp. 404-457.


Networks and Algorithms for Large-Scale Parallel Computation (with A. Gottlieb),  IEEE Computer Magazine, January, 1982,  pp. 27-36.


Artificial Intelligence, appears as Chapter 17,  pp. 747-753, ³Prospects for New Technologies², in Outlook for Science and Technology: The Next Five Years, a report of

the National Research Council, National Academy of Sciences, W.H. Freeman and Co., Publishers, 1982.


On the Piano Movers' Problem: II. General Techniques for Computing Topological Properties of Real Algebraic Manifolds (with M. Sharir), Adv. Appl. Math., vol. 4,

pp. 298-351, 1983.


Experience with the SETL Optimizer (with S. Freudenberger and M. Sharir), ACM TOPLAS, vol. 5, pp. 26-45, 1983.


Languages and System for Computer-Aided Instruction, Machine-Mediated Learning, vol. 1, No. 1, 1983.


On the Piano Movers' Problem: III. Coordinating the Motion of Several Independent Bodies. The Special Case of Circular Bodies Moving Amidst Polygonal Barriers (with M. Sharir), Robotics Research vol. 2, pp. 46-75, 1983.


The Next Generation of Robots (with D. Grossman), in Frontiers in Science and Technology: A Report by the Committee on Science, Engineering, and Public Policy of the National Academy of Sciences, pp. 185-209, W.H. Freeman and Co., 1983.


Mathematical Problems and Training in Robotics, (with M. Sharir), Notices Amer. Math. Soc., vol. 30, pp. 478-481, 1983.


Software Prototyping Using the SETL Programming Language (with P. Kruchten and E.Schonberg), IEEE Software, vol. 1, no. 4, pp. 66-75, 1984.


On the Piano Movers' Problem: IV. The Case of a Rod Moving in Three-Dimensional Space Amidst Polyhedral Obstacles (with M. Sharir), Comm. Pure Appl. Math., vol. 37,

pp. 815-848, 1984.


On the Complexity of Motion Planning for Multiple Independent Objects; PSPACE hardness of the ³Warehouseman's Problem² (with J.E. Hopcroft, and M. Sharir), Intl. J. Robotics Research, vol. 3, pp. 76-88, 1984.


Large-scale Parallel Computers, in: Science, Computers, and the Information Onslaught, pp. 129-139, Academic press, New York City, 1984.


Efficient Detection of Intersections among Spheres (with J. Hopcroft and M. Sharir), NYU Technical Report S9, Robotics Report 7.  Also in Intl. J. of Robotics Research, vol. 2, No. 4, pp. 77-80, 1985.


Preliminary Implementation of a Ratio Image Depth Sensor (with C. M. Bastuscheck), NYU Technical Report No. 124, June, 1984.  Also in Techniques for 3-D Machine Perception, ed. by A. Rosenfeld, Elsevier Science Publishers, 1986.


Decision Procedures for some Fragments of Set Theory (with A. Ferro and E. Omodeo), Proceedings of the 5th Conference on Automated Deduction, Lecture Notes on Computer Science, vol. 87, pp. 88-96, 1986, Springer Verlag, New York.


Two Dimensional Model-Based Boundary Matching Using Footprints (with A. Kalvin, E. Schonberg, and M. Sharir), NYU Technical Report No. 162, May, 1985. Also in Intl. J. Robotics Research, vol. 5, pp. 38-55, 1986.


Identification of Partially Obscured Objects in Two Dimensions by Matching of Noisy ŒCharacteristic Curves', (with M. Sharir), NYU Technical Report No. 165, June, 1985.

Also in Intl. J. Robotics Research, vol. 6, pp. 29-44, 1987.


Object Recognition by 3-Dimensional Curve Matching, (with M. Sharir, E. Schonberg, and C. M. Bastuscheck), NYU Technical Report No. 203, January, 1986), also in Intl. J. of Intell. Systems, vol. 1, pp. 105-132, 1986.


Mathematical Problems in Robotics, in Frontiers of Supercomputing, Edited by N. Metropolis, D.H. Sharp, W.J. Worlton, and K.R. Ames, pp. 263-282, University of California Press, 1986.


On the Existence and Synthesis of Multifinger Positive Grips (with B. Mishra and M. Sharir). NYU Technical Report No. 259, 1986. Also in Algorithmica, vol. 2, pp. 541-558, 1987.


Problems and Perspectives in Robotics, in Mathematics and Computer Science, Proceedings of the CWI Symposium of November 1983, (ed. J. W. De Bakker, M. Hazewinkel, and J. K. Lenstra), pp. 66-87, Amsterdam, Elsevier Science Publishing Company, 1986.


Improved Shape-Signature and Matching Methods for Model-Based Robotic Vision (with H. Wolfson), in Proc. of the Workshop on Space Telerobotics, (ed. by G. Rodriguez), vol. II, pp. 103-109, NASA, Jet Propulsion Lab, Pasadena CA, 1987.


T-Theorem Proving (with Alberto Policriti), Journal of Symbolic Computation, 1996.


Decision Algorithms for Some Fragments of Analysis and Related Areas (with D. Cantone, A. Ferro, and E. Omodeo). Comm. Pure Appl. Math., vol. 40, pp. 281-300, 1987.


On the Lower Envelope of Bivariate Functions and Its Applications (with M. Sharir), Proc. of 28th IEEE Symp. of Foundations of Computer Science, 1987, pp. 27-37.


Precise Implementation of CAD Primitives Using Rational Parametrizations of Standard Surfaces (with S. Ocken and M. Sharir), in Planning, Geometry, and Complexity of Robot Motion, (J. Hopcroft, J. Schwartz and M. Sharir, eds.), 1987.


Identification of Partially Obscured Objects in Two Dimensions by Matching of Noisy ŒCharacteristic Curves', Intl. J. Robotics Research, vol. 6, pp. 29-44, 1987.


Dreamworld. Daedalus Magazine, vol. 116, No. 3, pp. 165-185, 1987.


The Limits of Artificial Intelligence, in the Encyclopedia of Artificial Intelligence, pp. 488-503, John Wiley & Sons, 1987.


On the Bivariate Function Minimization Problem and Its Applications to Motion Planning, (with M. Sharir) in Lecture Notes in Computer Science, 267.  Automata, languages, and programming. 14th Intl. Colloquium, Karlsruhe, 1987, (ed. by T. Ottmann), pp. 357-363, Springer Verlag Publishers, 1987.


Issues for the ŒNew Connectionism': Developing Relationships Between Neuroscience and Artificial Intelligence. Daedalus Magazine, vol. 117, pp. 123-142, 1988.


Decision Procedures for Elementary Languages of Set Theory, XI. Multilevel Syllogistic Extended by Some Elementary Map Constructs (with D. Cantone), Technical report 374, New York University, 1988.


On Recognition of 3-D Objects from 2-D Images, (with H. Wolfson and Y. Lamdan), NYU Robotics Report 122, Technical Report 322. Also in Proc. of 1988 IEEE Intl. Conf. on Robotics and Automation, pp. 1407-1413, IEEE Computer Society Press, Washington, D.C.


Control of Multifinger Manipulation: Theoretical and Experimental Studies Using a Planar Manipulator (with J. Demmel, G. Lafferriere, and M. Sharir), Robotics Report 126, Technical Report 329. Also in Proc. of 1988 IEEE Intl. Conf. on Robotics and Automation, IEEE Computer Society Press, Washington DC, pp. 390-395.


Finding Effective ŒForce-Targets' for Two-Dimensional Multifinger Frictional Grips, (with M. Sharir). NYU Computer Science Technical Report 379, Robotics Report 156, June, 1988.

 Also in Proc. 25th Allerton Conf. on Communication, Control, and Computing, Sept. 1987.


A Survey of Motion Planning and Related Geometric Algorithms (with M. Sharir), Artificial Intelligence, vol. 37, pp. 157-169, December, 1988; see also Motion Planning and Related Geometric Algorithms in Robotics (with M. Sharir), Proceedings of the International Congress of Mathematicians, Berkeley 1986, pp. 1594-1161.


The Practical and Not-Yet-Practical in Software Engineering, in: Computing Tools for Scientific problem Solving, pp. 23-38, Academic Press, New York, 1990.


Affine Invariant Model-Based Object Recognition, (with Y. Lamdan and H. J. Wolfson), IEEE Transactions on Robotics and Automation, vol. 6, pp. 578-589, 1990.


Decidable Languages of Set Theory: MLS with Monotone Functions. (with Domenico Cantone and Calogero Zarba) PPDP'99 Workshop on Declarative Programming with Sets, Paris, France. September 28, 1999


Papers, Mathematics


A Note on the Space Lp*, Proc. Amer. Math. Soc., vol. 2, No. 2,1951.


Perturbations of Spectral Operations and Applications, I. Bounded Perturbations.

Pacific J. Math., vol. 4, pp. 270-275, 1954.


The Formula for Change of Variables in a Multiple lntegral, Amer. Math. Monthly, vol. 61, pp. 415-488, 1954.


De Rham's Theorem for Arbitrary Spaces, Amer. J. of Math., vol. 77, pp. 29-44, 1955.


Two Perturbation Formulas, Comm. on Pure and Applied Math., vol. 8, pp. 371-376, 1955.


Weak Compactness and Vector Measures, (with N. Dunford and R. Bartle), Canadian J. Math., vol. 7, pp.  289-305, 1955.


Convergence Almost Everywhere of Operator Averages, (with N. Dunford),  J. of Rat. Mech., vol. 5, pp. 129-178, 1956.


On Mautner's Eigenfunction Expansion, (with W.G. Bade), Proc. Natl. Acad. Sci., vol. 42, No. 8, pp.519-525, 1956.


Riemann's Method in the Theory of Special Functions, Bull. Amer. Math. Soc., vol. 62,

pp. 531-540, 1956.


A Vector-Valued Random Ergodic Theorem (with Anatole Beck), Proc. Amer. Math. Soc., vol 8, pp. 1049-1059, 1958.


Another Proof of E. Hopf's Ergodic Lemma, Comm. Pure and Applied Math., Vol 12,

pp.  399-401, 1959.


On Nash's Implicit Function Theorem, Comm. Pure and Applied Math., Vol. 13, pp. 509-530, 1960.


Some Non-Selfadjoint Operators, I. Comm. Pure and Applied Math.,  Vol. 13,  pp. 609-641, 1960.


Free Quantized Lorentzian Fields, Journal of Mathematical Physics, Vol. 2, pp. 271-290, 1961.


Some Non-Selfadjoint Operators, II. A Class of Operators Amenable to Friedrichs Method, Comm. Pure and Applied Math., Vol. 14, pp. 619-626, 1961.


Compact Positive Mappings in Lebesgue Spaces, Comm. Pure and Applied Math., Vol. 14, pp. 693-705, 1961.


A Remark on Inequalities of  Calderon-Zygmund Type for Vector-Valued Functions, Ibid., Vol. 14, pp. 785-799, 1961.


The Pernicious Influence of Mathematics on Science, in: Proc. of 1960 International Congress on Logic, Philosophy, and the Methodology of Science, Stanford University Press, pp. 356-360, 1961.


On Spectral Operators in Hilbert Space With Compact Imaginary Part, Comm. Pure and Applied Math., Vol. 15, pp. 95-97, 1962.


Some Results on the Spectra and Spectral Resolution of Singular Integral Operators, Comm. Pure and Applied Math., Vol. 15, pp. 159-172, 1962.


Subdiagonalisation of Operators in Hilbert Space with Compact Imaginary Part, Comm. Pure and Applied Math., Vol. 15, pp. 75-90, 1962.


Two Finite, non-Hyperfinite,  Non-isomorphic Factors, Comm. Pure and Applied Math., Vol. 16,  pp. 19-26, 1963.


Non-isomorphic Factors of Type III, Comm. Pure and Applied Math., Vol. 16, pp. 111-120, 1963.


Type II Factors in a Direct Integral Composition, Comm. Pure and Applied Math., Vol. 16, pp. 247-252, 1963.


Compact Nonlinear Mappings of Complex B-spaces and a Lemma of Jane Cronin, Comm. Pure and Applied Math.,  Vol. 16, pp. 253-260, 1963.


Generators of the Convolution Algebra L1, (with H. S. Shapiro and D. J. Newman), Trans. Amer. Math. Soc., vol. 107, pp. 466-484, 1963.


Generalizing the Lusternik-Schnirelman Theory of Critical Points, Comm. Pure and Applied Math., Vol. 17, pp. 307-315, 1964.


On the Norm of a Finite Boolean Algebra of Projections, with Applications to a Theorem of Kreiss and Morton (with C. McCarthy), Comm. Pure and Applied Math.,  Vol. 18,

pp. 191-201, 1965.


A Remark on Morse Theory, Comm. Pure and Applied Math., Vol. 19, pp. 139-143, 1966.


Intersection-Theoretic Principles for the Existence of Fixed Points and Critical Points, Proceedings of the Georgetown Symposium on Differential Equations, 1966.


Representations of the Algebra of All Operators in Hilbert Space, and Related Analytic Function Algebras, Comm. Pure and Applied Math., Vol. 20, pp. 457-492, 1967.


Recent Progress in the Structure Theory of Factors, in: Functional Analysis, Proceedings of a Symposium, pp. 37-52, Academic Press, New York, 1970.


Papers, General Interest


America's Economic-Technical Agenda for the 1990's, Daedalus Magazine, vol. 131, No. 1, pp. 139-165, 1992.


Mathematics as a Tool for Economic Understanding, in Mathematics Today, Lynn Steen (ed), Springer Verlag 1978.

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