Howard Levi Explained

Howard Levi
Birth Date:	November 9, 1916
Birth Place:	New York City
Death Place:	New York City
Nationality:	American
Fields:	Mathematics differential algebra
Alma Mater:	Columbia University
Work Institutions:	Columbia University City University of New York
Doctoral Advisor:	Joseph Fels Ritt
Known For:	Levi's reduction process

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. Soon after obtaining his degree, he became a researcher on the Manhattan Project.^{[2] [3]}

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.^{[4] [5]} Much of the Wesleyan material was based on his book Foundations of Geometry and Trigonometry.^[6]

His book Polynomials, Power Series, and Calculus, written to be a textbook for a first course in calculus,^[7] 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."^{[8] [9]}

Levi's reduction process is named after him.^[10]

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

Selected publications

Books

• Elements of Algebra (Chelsea Publishing Company, 1953, 1956, 1960, 1961)^{[11] [12] [13] [14]}
• Elements of Geometry (Columbia University Press, 1956)
• Foundations of Geometry and Trigonometry (Prentice-Hall, 1956 and 1960)^{[15] [16]}
• 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)^[17]

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. 10.1111/j.2164-0947.1974.tb03023.x . 36 . 7 Series II . 1974 . Transactions of the New York Academy of Sciences . 640–643 . Levi Howard. A Geometric Construction of the Dirichlet Kernel .
• "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 10.1111/j.2164-0947.1966.tb02349.x . 28 . 3 Series II . 1966 . Transactions of the New York Academy of Sciences . 375–377 . Levi Howard. An Algebraic Approach to Calculus .
• "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.

Notes and References

1. https://www.ams.org/notices/200306/inside.pdf Notices of the AMS, June/July 2003, Volume 50, Number 6, p. 705.
2. [Melvin Fitting]
3. For some details, consult: Mildred Goldberg – Personal recollections of Mildred Goldberg, secretary to the theoretical group, SAM Laboratories, The Manhattan Project; 1943-1946 (Gilder Lehrman Institute of American History).
4. Book: Nathalie Sinclair

   . Nathalie. Sinclair. Nathalie Sinclair. The History of the Geometry Curriculum in the United States. 2008. IAP. 978-1-59311-697-2. 64.

5. Sitomer, H. – Coordinate geometry with an affine approach, Mathematics Teacher 57 (1964), 404–405.
6. C. Ray Wylie, An Affine Approach to Euclidean Geometry (p. 237 from the PDF document, p. 231 from the document itself)
7. https://www.jstor.org/stable/2312681 Levi, Howard — An Experimental Course in Analysis for College Freshmen
8. Gillman, Leonard. Leonard Gillman. An Axiomatic Approach to the Integral. The American Mathematical Monthly. 1993. 100. 1. 16–25. 10.2307/2324809. 2324809.
9. Gillman, Leonard. Leonard Gillman. Review: Polynomials, Power Series, and Calculus by Howard Levi. The American Mathematical Monthly. 1974. 81. 5. 532–533. 2318616. 10.2307/2318616.
10. Mead, D. G.. The Equation of Ramanujan-Nagell and [y²]]. Proceedings of the American Mathematical Society. December 1973. 41. 2. 333–341. 10.2307/2039090. 2039090.
11. Halmos, Paul R.. Paul Halmos. Review: Elements of algebra by Howard Levi. Bull. Amer. Math. Soc.. 1955. 61. 3. 245–247. 10.1090/S0002-9904-1955-09905-1. free.
12. Lott, Fred W.. Review: Elements of algebra by Howard Levi. The Mathematics Teacher. 1955. 48. 5. 353–354. 27954922.
13. Lee, Herbert L.. Review: Elements of algebra by Howard Levi. The Scientific Monthly. 1955. 80. 6. 387. 21575.
14. Rajaratnam, Nageswari. Review: Elements of algebra by Howard Levi. The Mathematics Teacher. 1960. 53. 7. 585–586. 27956256.
15. Dickson, Douglas G.. Review: Foundations of Geometry and Trigonometry by Howard Levi. Science Magazine. 1962. 137. 3533. 846–847. 10.1126/science.137.3533.846-d. 17787326.
16. Bezuszka, S. J.. Review: Foundations of Geometry and Trigonometry by Howard Levi. The American Mathematical Monthly. 1965. 72. 5. 565. 2314158. 10.2307/2314158.
17. Chakerian, G. D.. Review: Topics in Geometry by Howard Levi. The American Mathematical Monthly. 1969. 76. 8. 962. 2317992. 10.2307/2317992.