Howard Levi | |
---|---|

Born | November 9, 1916 |

Died | September 11, 2002 New York City | (aged 85)

Nationality | American |

Alma mater | Columbia University |

Known for | Levi's reduction process |

Scientific career | |

Fields | Mathematics: differential algebra |

Institutions | Columbia University City University of New York |

Doctoral advisor | Joseph Fels Ritt |

**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.

## References

**^***Notices of the AMS*, June/July 2003, Volume 50, Number 6, p. 705.**^**Howard Levi at the Mathematics Genealogy Project- ^
^{a}^{b}Melvin Fitting – The Four Color Theorem **^**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).**^**Sinclair, Nathalie (2008).*The History of the Geometry Curriculum in the United States*. IAP. p. 64. ISBN 978-1-59311-697-2.**^**Sitomer, H. – Coordinate geometry with an affine approach, Mathematics Teacher 57 (1964), 404–405.**^**C. Ray Wylie, An Affine Approach to Euclidean Geometry (p. 237 from the PDF document, p. 231 from the document itself)**^**Levi, Howard — An Experimental Course in Analysis for College Freshmen**^**Gillman, Leonard (1993). "An Axiomatic Approach to the Integral" (PDF).*The American Mathematical Monthly*.**100**(1): 16–25. doi:10.2307/2324809. JSTOR 2324809.**^**Gillman, Leonard (1974). "Review:*Polynomials, Power Series, and Calculus*by Howard Levi".*The American Mathematical Monthly*.**81**(5): 532–533. doi:10.2307/2318616. JSTOR 2318616.**^**Mead, D. G. (December 1973). "The Equation of Ramanujan-Nagell and [y^{2}]" (PDF).*Proceedings of the American Mathematical Society*.**41**(2): 333–341. doi:10.2307/2039090. JSTOR 2039090.**^**Halmos, Paul R. (1955). "Review:*Elements of algebra*by Howard Levi".*Bull. Amer. Math. Soc*.**61**(3): 245–247. doi:10.1090/S0002-9904-1955-09905-1.**^**Lott, Fred W. (1955). "Review:*Elements of algebra*by Howard Levi".*The Mathematics Teacher*.**48**(5): 353–354. JSTOR 27954922.**^**Lee, Herbert L. (1955). "Review:*Elements of algebra*by Howard Levi".*The Scientific Monthly*.**80**(6): 387. JSTOR 21575.**^**Rajaratnam, Nageswari (1960). "Review:*Elements of algebra*by Howard Levi".*The Mathematics Teacher*.**53**(7): 585–586. JSTOR 27956256.**^**Dickson, Douglas G. (1962). "Review:*Foundations of Geometry and Trigonometry*by Howard Levi".*Science Magazine*.**137**(3533): 846–847. doi:10.1126/science.137.3533.846-d. PMID 17787326.**^**Bezuszka, S. J. (1965). "Review:*Foundations of Geometry and Trigonometry*by Howard Levi".*The American Mathematical Monthly*.**72**(5): 565. doi:10.2307/2314158. JSTOR 2314158.**^**Chakerian, G. D. (1969). "Review:*Topics in Geometry*by Howard Levi".*The American Mathematical Monthly*.**76**(8): 962. doi:10.2307/2317992. JSTOR 2317992.