Howard Levi (November 9, 1916 in

New York City New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the L ...

– September 11, 2002 in New York City) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who worked mainly in

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

and

mathematical education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although rese ...

. 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 Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

in 1942 as a student of

Joseph Fels Ritt Joseph Fels Ritt (August 23, 1893 – January 5, 1951) was an American mathematician at Columbia University in the early 20th century. He was born and died in New York. After beginning his undergraduate studies at City College of New York, Rit ...

. Soon after obtaining his degree, he became a researcher on the

Manhattan Project The Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons. It was led by the United States with the support of the United Kingdom and Canada. From 1942 to 1946, the project w ...

Melvin Fitting Melvin Fitting (born January 24, 1942) is a logician with special interests in philosophical logic and tableau proof systems. He was a professor at City University of New York, Lehman College and the Graduate Center. from 1968 to 2013. At the ...

�
The Four Color Theorem
/ref> At

Wesleyan University Wesleyan University ( ) is a Private university, private liberal arts college, liberal arts university in Middletown, Connecticut. Founded in 1831 as a Men's colleges in the United States, men's college under the auspices of the Methodist Epis ...

he led a group that developed a course of geometry for high school students that treated

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

as a special case of

affine geometry In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of ''parallel lines'' is one of the main properties that is inde ...

. Much of the Wesleyan material was based on his book ''Foundations of Geometry and Trigonometry''. His book ''Polynomials, Power Series, and Calculus'', written to be a textbook for a first course in

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

, presented an innovative approach, and received favorable reviews by

Leonard Gillman Leonard E. Gillman (January 8, 1917 – April 7, 2009) was an American mathematician, emeritus professor at the University of Texas at Austin. He was also an accomplished classical pianist. Biography Early life and education Gillman was born i ...

, who wrote " ..this book, with its wealth of imaginative ideas, deserves to be better known."

Levi's reduction process Levi Strauss & Co. () is an American clothing company known worldwide for its Levi's () brand of denim jeans. It was founded in May 1853 when German-Jewish immigrant Levi Strauss moved from Buttenheim, Bavaria, to San Francisco, California, to ...

is named after him. In his last years, he tried to find a proof of the

four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sh ...

that did not rely on computers.

Selected publications

Books

* ''Elements of Algebra'' (

Chelsea Publishing Company The Chelsea Publishing Company was a publisher of mathematical books, based in New York City New York, often called New York City or NYC, is the most populous city in the United States. With a 2020 population of 8,804, ...

, 1953, 1956, 1960, 1961) * ''Elements of Geometry'' (

Columbia University Press Columbia University Press is a university press based in New York City, and affiliated with Columbia University. It is currently directed by Jennifer Crewe (2014–present) and publishes titles in the humanities and sciences, including the fiel ...

, 1956) * ''Foundations of Geometry and Trigonometry'' (

Prentice-Hall Prentice Hall was an American major educational publisher owned by Savvas Learning Company. Prentice Hall publishes print and digital content for the 6–12 and higher-education market, and distributes its technical titles through the Safari B ...

, 1956 and 1960) * ''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 The School Mathematics Study Group (SMSG) was an American academic think tank focused on the subject of reform in mathematics education. Directed by Edward G. Begle and financed by the National Science Foundation, the group was created in the wak ...

, 1961) * ''Polynomials, Power Series, and Calculus'' (Van Nostrand, 1967, 1968) * ''Topics in Geometry'' (1968, 1975)

Articles

* "On the values assumed by polynomials". ''

Bull. Amer. Math. Soc. The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...

'' 45 (1939), no. 8, pp. 570–575. (LINK)
/small> * "Composite polynomials with coefficients in an arbitrary field of characteristic zero". '' Amer. J. Math.'' 64 (1942), no. 1, pp. 389–400. (LINK)
/small> * "On the structure of differential polynomials and on their theory of ideals". '' T. Am. Math. Soc.'' 51 (1942), pp. 532–568. (LINK)
/small> * "A characterization of polynomial rings by means of order relations". ''Amer. J. Math.'' 65 (1943), no. 2, pp. 221–234. (LINK)
/small> * "Exact nth derivatives". ''Bull. Amer. Math. Soc.'' 49 (1943), no. 8, pp. 631–636. (LINK)
/small> * "The low power theorem for partial differential polynomials". ''

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...

'', Second Series, Vol. 46, no. 1 (1945), pp. 113–119. (LINK)
/small> * "A geometric construction of the Dirichlet kernel". ''Trans. N. Y. Acad. Sci.'', Volume 36, Issue 7 (1974), Series II, pp. 640–643. * "An algebraic reformulation of the four color theorem." (published posthumously by

Don Coppersmith Don Coppersmith (born 1950) is a cryptographer and mathematician. He was involved in the design of the Data Encryption Standard block cipher at IBM, particularly the design of the S-boxes, strengthening them against differential cryptanalysis ...

, and Paul Meyer) (LINK)
/small>

Expository writing

* "Why Arithmetic Works.", ''The Mathematics Teacher'', Vol. 56, No. 1 (January 1963), pp. 2–7. (LINK)
/small> * "Plane Geometries in Terms of Projections.", '' Proc. Am. Math. Soc'', 1965, Vol. 16, No. 3, pp. 503–511. (LINK)
/small> * "An Algebraic Approach to Calculus.", ''Trans. N. Y. Acad. Sci.'', Volume 28, Issue 3 Series II, pp. 375–377, January 1966 * "Classroom Notes: Integration, Anti-Differentiation and a Converse to the Mean Value Theorem", '' Amer. Math. Monthly'' 74 (1967), no. 5, 585–586. (LINK)
/small> * "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 ''Educational Studies in Mathematics'' is a peer-reviewed academic journal covering mathematics education. It was established by Hans Freudenthal in 1968. The journal is published by Springer Science+Business Media and the editors-in-chief are Su ...

'', June 1971, Volume 3, Issue 3–4, pp 490–500. (LINK)
/small> * "Geometric Versions of Some Algebraic Identities.", ''Ann. N. Y. Acad. Sci.'', Vol. 607, pp. 54–60, November 1990.

References

{{DEFAULTSORT:Levi, Howard 1916 births 2002 deaths 20th-century American mathematicians 21st-century American mathematicians Columbia University alumni American textbook writers Algebraists Columbia University faculty Hunter College faculty Lehman College faculty Manhattan Project people Mathematics educators