Howard Levi (November 9, 1916 in New York City – September 11, 2002 in New York City) was an American mathematician who worked mainly in algebra and mathematical education.^[1] Levi was very active during the educational reforms in the United States, having proposed several new courses to replace the traditional ones.

Biography

Levi earned a Ph.D. in mathematics from Columbia University in 1942 as a student of Joseph Fels Ritt.^[2] Soon after obtaining his degree, he became a researcher on the Manhattan Project.^[3][4]

At Wesleyan University he led a group that developed a course of geometry for high school students that treated Euclidean geometry as a special case of affine geometry.^[5][6] Much of the Wesleyan material was based on his book Foundations of Geometry and Trigonometry.^[7]

His book Polynomials, Power Series, and Calculus, written to be a textbook for a first course in calculus,^[8] presented an innovative approach, and received favorable reviews by Leonard Gillman, who wrote "[...] this book, with its wealth of imaginative ideas, deserves to be better known."^[9][10]

Levi's reduction process is named after him.^[11]

In his last years, he tried to find a proof of the four color theorem that did not rely on computers.^[3]

Selected publications

Books

• Elements of Algebra (Chelsea Publishing Company, 1953, 1956, 1960, 1961)^{[12][13][14][15]}
• Elements of Geometry (Columbia University Press, 1956)
• Foundations of Geometry and Trigonometry (Prentice-Hall, 1956 and 1960)^[16][17]
• Fundamental Concepts of Mathematics (1957)
• Modern Coordinate Geometry: A Wesleyan Experimental Curricular Study (co-authored with C. Robert Clements, Harry Sitomer, et al., for the School Mathematics Study Group, 1961)
• Polynomials, Power Series, and Calculus (Van Nostrand, 1967, 1968)
• Topics in Geometry (1968, 1975)^[18]

Articles

• "On the values assumed by polynomials". Bull. Amer. Math. Soc. 45 (1939), no. 8, pp. 570–575. (LINK)
• "Composite polynomials with coefficients in an arbitrary field of characteristic zero". Amer. J. Math. 64 (1942), no. 1, pp. 389–400. (LINK)
• "On the structure of differential polynomials and on their theory of ideals". T. Am. Math. Soc. 51 (1942), pp. 532–568. (LINK)
• "A characterization of polynomial rings by means of order relations". Amer. J. Math. 65 (1943), no. 2, pp. 221–234. (LINK)
• "Exact nth derivatives". Bull. Amer. Math. Soc. 49 (1943), no. 8, pp. 631–636. (LINK)
• "The low power theorem for partial differential polynomials". Annals of Mathematics, Second Series, Vol. 46, no. 1 (1945), pp. 113–119. (LINK)
• "A geometric construction of the Dirichlet kernel". Trans. N. Y. Acad. Sci., Volume 36, Issue 7 (1974), Series II, pp. 640–643. Levi Howard (1974). "A Geometric Construction of the Dirichlet Kernel". Transactions of the New York Academy of Sciences. 36 (7 Series II): 640–643. doi:10.1111/j.2164-0947.1974.tb03023.x.
• "An algebraic reformulation of the four color theorem." (published posthumously by Don Coppersmith, Melvin Fitting, and Paul Meyer) (LINK)

Expository writing

• "Why Arithmetic Works.", The Mathematics Teacher, Vol. 56, No. 1 (January 1963), pp. 2–7. (LINK)
• "Plane Geometries in Terms of Projections.", Proc. Am. Math. Soc, 1965, Vol. 16, No. 3, pp. 503–511. (LINK)
• "An Algebraic Approach to Calculus.", Trans. N. Y. Acad. Sci., Volume 28, Issue 3 Series II, pp. 375–377, January 1966 Levi Howard (1966). "An Algebraic Approach to Calculus". Transactions of the New York Academy of Sciences. 28 (3 Series II): 375–377. doi:10.1111/j.2164-0947.1966.tb02349.x.
• "Classroom Notes: Integration, Anti-Differentiation and a Converse to the Mean Value Theorem", Amer. Math. Monthly 74 (1967), no. 5, 585–586. (LINK)
• "Foundations of Geometric Algebra", Rendiconti di Matematica, 1969, Vol. 2, Serie VI, pp. 1–32.
• "Geometric Algebra for the High School Program.", Educational Studies in Mathematics, June 1971, Volume 3, Issue 3–4, pp 490–500. (LINK)
• "Geometric Versions of Some Algebraic Identities.", Ann. N. Y. Acad. Sci., Vol. 607, pp. 54–60, November 1990.