Howard Jerome Keisler
Encyclopedia
H. Jerome Keisler is an American mathematician
, currently professor emeritus at University of Wisconsin–Madison
. His research has included model theory
and non-standard analysis
.
His Ph.D. advisor was Alfred Tarski
at Berkeley
; his dissertation is Ultraproducts and Elementary Classes (1961).
Following Abraham Robinson
's work resolving what had long been thought to be inherent logical contradictions in the literal interpretation of Leibniz's notation that Leibniz himself had proposed, that is, interpreting "dx" as literally representing an infinitesimal
ly small quantity, Keisler published Elementary Calculus: An Infinitesimal Approach
, a first-year calculus textbook conceptually centered around the use of infinitesimals, rather than the epsilon, delta approach, for developing the calculus.
He is also known for extending the Henkin construction (of Leon Henkin
) to what are now called Henkin-Keisler models.
He held the named chair of Vilas Professor of Mathematics at Wisconsin.
Among Keisler's graduate students, several have made notable mathematical contributions, including Frederick Rowbottom
discovered Rowbottom cardinal
s. Several others have gone on to careers in computer science research and product development, including: Kevin J. Compton, a professor of computer science at the University of Michigan
, Curtis Tuckey, a developer of software-based collaboration environments; Joseph Sgro
, a neurologist and developer of vision processor hardware and software, and Edward L. Wimmers, a database researcher at IBM Almaden Research Center.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, currently professor emeritus at University of Wisconsin–Madison
University of Wisconsin–Madison
The University of Wisconsin–Madison is a public research university located in Madison, Wisconsin, United States. Founded in 1848, UW–Madison is the flagship campus of the University of Wisconsin System. It became a land-grant institution in 1866...
. His research has included model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
.
His Ph.D. advisor was Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
at Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
; his dissertation is Ultraproducts and Elementary Classes (1961).
Following Abraham Robinson
Abraham Robinson
Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
's work resolving what had long been thought to be inherent logical contradictions in the literal interpretation of Leibniz's notation that Leibniz himself had proposed, that is, interpreting "dx" as literally representing an infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
ly small quantity, Keisler published Elementary Calculus: An Infinitesimal Approach
Elementary Calculus: An Infinitesimal Approach
Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of Abraham Robinson's non-standard analysis...
, a first-year calculus textbook conceptually centered around the use of infinitesimals, rather than the epsilon, delta approach, for developing the calculus.
He is also known for extending the Henkin construction (of Leon Henkin
Leon Henkin
Leon Albert Henkin was a logician at the University of California, Berkeley. He was principally known for the "Henkin's completeness proof": his version of the proof of the semantic completeness of standard systems of first-order logic.-The completeness proof:Henkin's result was not novel; it had...
) to what are now called Henkin-Keisler models.
He held the named chair of Vilas Professor of Mathematics at Wisconsin.
Among Keisler's graduate students, several have made notable mathematical contributions, including Frederick Rowbottom
Frederick Rowbottom
Frederick Rowbottom was a British logician and mathematician. The large cardinal notion of Rowbottom cardinals is named after him.-Biography:...
discovered Rowbottom cardinal
Rowbottom cardinal
In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number.An uncountable cardinal number κ is said to be Rowbottom if for every function f: [κ]...
s. Several others have gone on to careers in computer science research and product development, including: Kevin J. Compton, a professor of computer science at the University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
, Curtis Tuckey, a developer of software-based collaboration environments; Joseph Sgro
Joseph Sgro
Joseph A. Sgro is a mathematician, neurologist and a technologist/entrepreneur in the field of frame grabbers, high speed smart cameras, vision processors, and related computer vision and machine vision technologies.Sgro began his career as an academic researcher in advanced mathematics and logic...
, a neurologist and developer of vision processor hardware and software, and Edward L. Wimmers, a database researcher at IBM Almaden Research Center.
Publications
- Chang, C. C.Chen Chung ChangChen Chung Chang is a mathematician who works in model theory. He obtained his PhD from Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text on model theory. Chang's conjecture is named after him...
; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. xvi+650 pp. ISBN: 0-444-88054-2 - Elementary Calculus: An Infinitesimal Approach. Prindle, Weber & Schmidt, 1976/1986. Available online at http://www.math.wisc.edu/~keisler/calc.html.
See also
- Criticism of non-standard analysisCriticism of non-standard analysisNon-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors. The evaluation of non-standard analysis in the literature has varied greatly...
- Non-standard calculusNon-standard calculusIn mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus...
- Elementary Calculus: An Infinitesimal ApproachElementary Calculus: An Infinitesimal ApproachElementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of Abraham Robinson's non-standard analysis...