E. H. Moore | |
---|---|

Born | Marietta, Ohio, U.S. | January 26, 1862

Died | December 30, 1932 70) | (aged

Alma mater | Yale University (BA, PhD) |

Known for | "General analysis", Moore–Smith convergence of nets in topology, Moore family and hull operator, Moore–Penrose inverse, Galois representation of finite fields, Axiomatic systems |

Awards | AMS Colloquium Lecturer, 1906 |

Scientific career | |

Fields | Mathematics |

Institutions | University of Chicago 1892–31 Yale University 1887–89 Northwestern University 1886–87, 1889–92 |

Thesis | Extensions of Certain Theorems of Clifford and Cayley in the Geometry of n Dimensions (1885) |

Doctoral advisor | Hubert Anson Newton |

Doctoral students | George Birkhoff Leonard Dickson T. H. Hildebrandt D. N. Lehmer Robert Lee Moore Oswald Veblen Anna Wheeler |

Other notable students | Anne Bosworth |

**Eliakim Hastings Moore** ( /ɪˈlaɪəkɪm/ ; January 26, 1862 – December 30, 1932), usually cited as **E. H. Moore** or **E. Hastings Moore**, was an American mathematician.

Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, discovered mathematics through a summer job at the Cincinnati Observatory while in high school. He subsequently studied mathematics at Yale University, where he was a member of Skull and Bones ^{ [1] }^{: 47–8 } and obtained a BA in 1883 and the PhD in 1885 with a thesis supervised by Hubert Anson Newton, on some work of William Kingdon Clifford and Arthur Cayley. Newton encouraged Moore to study in Germany, and thus he spent an academic year at the University of Berlin, attending lectures by Leopold Kronecker and Karl Weierstrass.

On his return to the United States, Moore taught at Yale and at Northwestern University. When the University of Chicago opened its doors in 1892, Moore was the first head of its mathematics department, a position he retained until his death in 1932. His first two colleagues were Oskar Bolza and Heinrich Maschke. The resulting department was the second research-oriented mathematics department in American history, after Johns Hopkins University.

Moore first worked in abstract algebra, proving in 1893 the classification of the structure of finite fields (also called Galois fields). Around 1900, he began working on the foundations of geometry. He reformulated Hilbert's axioms for geometry so that points were the only primitive notion, thus turning David Hilbert's primitive lines and planes into defined notions. In 1902, he further showed that one of Hilbert's axioms for geometry was redundant. His work on axiom systems is considered one of the starting points for metamathematics and model theory. After 1906, he turned to the foundations of analysis. The concept of a closure operator first appeared in his 1910 *Introduction to a form of general analysis*.^{ [2] } He also wrote on algebraic geometry, number theory, and integral equations.^{ [3] }

At Chicago, Moore supervised 31 doctoral dissertations, including those of George Birkhoff, Leonard Dickson, Robert Lee Moore (no relation), and Oswald Veblen. Birkhoff and Veblen went on to lead departments at Harvard and Princeton, respectively. Dickson became the first great American algebraist and number theorist. Robert Moore founded American topology. According to the Mathematics Genealogy Project, as of December 2012, E. H. Moore had over 18,900 known "descendants."

Moore convinced the New York Mathematical Society to change its name to the American Mathematical Society, whose Chicago branch he led. He presided over the AMS, 1901–02, and edited the *Transactions of the American Mathematical Society*, 1899–1907. He was elected to the National Academy of Sciences, the American Academy of Arts and Sciences, and the American Philosophical Society. He was an Invited Speaker at the International Congress of Mathematicians in 1908 in Rome and in 1912 in Cambridge, England.

The American Mathematical Society established a prize in his honor in 2002.

- Moore–Penrose inverse
- Moore–Smith sequence
- Moore matrix over a finite field
- Moore determinant of a Hermitian matrix over a quaternion algebra

- ↑ "Obituary Record of Graduates of Yale University Deceased during the Year 1932–1933" (PDF). Yale University. October 15, 1933. Retrieved April 18, 2011.
- ↑ T.S. Blyth,
*Lattices and Ordered Algebraic Structures*, Springer, 2005, ISBN 1-85233-905-5, p. 11 - ↑ Bliss, G. A. (1934). "The scientific work of Eliakim Hastings Moore".
*Bulletin of the American Mathematical Society*.**40**(7): 501–514. doi: 10.1090/s0002-9904-1934-05872-5 . MR 1562892.

**David Hilbert** was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics.

**Euclidean geometry** is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the *Elements*. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is *proved* from axioms and previously proved theorems.

**Linear algebra** is the branch of mathematics concerning linear equations such as:

**Mathematical logic** is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

In mathematics, **affine geometry** is what remains of Euclidean geometry when ignoring the metric notions of distance and angle.

A **finite geometry** is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.

**Saunders Mac Lane** was an American mathematician who co-founded category theory with Samuel Eilenberg.

**Oswald Veblen** was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof of the theorem, many now also consider Camille Jordan's original proof rigorous.

**Marshall Harvey Stone** was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras.

**Hilbert's axioms** are a set of 20 assumptions proposed by David Hilbert in 1899 in his book *Grundlagen der Geometrie* as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.

**Robert Lee Moore** was an American mathematician who taught for many years at the University of Texas. He is known for his work in general topology, for the Moore method of teaching university mathematics, and for his racist treatment of African-American mathematics students.

**Leonard Eugene Dickson** was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory, *History of the Theory of Numbers*.

**Joseph Henry Maclagan Wedderburn** FRSE FRS was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras. He also worked on group theory and matrix algebra.

**Gilbert Ames Bliss**,, was an American mathematician, known for his work on the calculus of variations.

**Michael Jerome Hopkins** is an American mathematician known for work in algebraic topology.

**Foundations of geometry** is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term **axiomatic geometry** can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

**George David Birkhoff** was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during his time he was considered by many to be the preeminent American mathematician.

**John Wesley Young** was an American mathematician who, with Oswald Veblen, introduced the axioms of projective geometry, coauthored a 2-volume work on them, and proved the Veblen–Young theorem. He was a proponent of Euclidean geometry and held it to be substantially "more convenient to employ" than non-Euclidean geometry. His lectures on algebra and geometry were compiled in 1911 and released as *Lectures on Fundamental Concepts of Algebra and Geometry.*

In mathematics, **continuous geometry** is an analogue of complex projective geometry introduced by von Neumann, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., *n*, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.

In geometry, the **point–line–plane postulate** is a collection of assumptions (axioms) that can be used in a set of postulates for Euclidean geometry in two, three or more dimensions.

- Ivor Grattan-Guinness (2000)
*The Search for Mathematical Roots 1870–1940*. Princeton University Press. - Karen Parshall and David E. Rowe (1994)
*The emergence of the American mathematical research community, 1876–1900 : J. J. Sylvester, Felix Klein, and E. H. Moore*, American Mathematical Society.

- O'Connor, John J.; Robertson, Edmund F., "E. H. Moore",
*MacTutor History of Mathematics archive*, University of St Andrews - E. H. Moore at the Mathematics Genealogy Project
- E. H. Moore —
*Biographical Memoirs*of the National Academy of Sciences - David Lindsay Roberts,
*Moore´s early twentieth century program for reform in mathematics education*, American Mathematical Monthly, vol. 108, 2001, pp. 689–696 - Guide to the Eliakim Hastings Moore Papers 1899-1931 at the University of Chicago Special Collections Research Center

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