Howard Jerome Keisler (born 3 December 1936) is 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 ...

, currently professor emeritus at

University of Wisconsin–Madison A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United Stat ...

. His research has included

model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...

and

non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...

. His Ph.D. advisor was

Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

at Berkeley; his dissertation is ''Ultraproducts and Elementary Classes'' (1961). Following

Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...

's work resolving what had long been thought to be inherent logical contradictions in the literal interpretation of

Leibniz's notation In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just a ...

that Leibniz himself had proposed, that is, interpreting "dx" as literally representing an

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...

ly small quantity, Keisler published '' Elementary Calculus: An Infinitesimal Approach'', a first-year calculus textbook conceptually centered on 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 Albert Henkin (April 19, 1921, Brooklyn, New York - November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the theory of types. He was an active schola ...

) to what are now called Henkin–Keisler models. He is also known for the Rudin–Keisler ordering along with

Mary Ellen Rudin Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award, which is awarded annually to a young res ...

. He held the named chair of

Vilas Professor of Mathematics Vilas may refer to: People ;Last name * Vilas Nande (fl.2000), musician * Charles Nathaniel Vilas (died 1931), American philanthropist in New Hampshire for whom the Vilas Bridge was named *Dane Vilas (born 1985), South African cricketer * Faith V ...

at Wisconsin. Among Keisler's graduate students, several have made notable mathematical contributions, including

Frederick Rowbottom Frederick Rowbottom (16 January 1938 – 12 October 2009) was a British logician and mathematician. The large cardinal notion of Rowbottom cardinals is named after him. Biography After graduating from Cambridge University, Rowbottom studied unde ...

who discovered

Rowbottom cardinal In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number. An uncountable cardinal number \kappa is said to be ''\lambda- Rowbottom'' if for every function ''f'': kappa;sup><ω → λ (whe ...

s. Several others have gone on to careers in computer science research and product development, including: Michael Benedikt, a professor of computer science at the

University of Oxford , mottoeng = The Lord is my light , established = , endowment = £6.1 billion (including colleges) (2019) , budget = £2.145 billion (2019–20) , chancellor ...

, Kevin J. Compton, a professor of computer science at the

University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

, Curtis Tuckey, a developer of software-based collaboration environments;

Joseph Sgro Joseph A. Sgro (born in San Diego, California) is an American mathematician, neurologist / neurophysiologist, and an engineering technologist / entrepreneur in the field of frame grabbers, high-speed cameras, smart cameras, image processors, co ...

, a neurologist and developer of

vision processor Vision, Visions, or The Vision may refer to: Perception Optical perception * Visual perception, the sense of sight * Visual system, the physical mechanism of eyesight * Computer vision, a field dealing with how computers can be made to gain und ...

hardware and software, and Edward L. Wimmers, a database researcher at

IBM Almaden Research Center IBM Research is the research and development division for IBM, an American multinational information technology company headquartered in Armonk, New York, with operations in over 170 countries. IBM Research is the largest industrial research o ...

. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...

. His son Jeffrey Keisler is a

Fulbright The Fulbright Program, including the Fulbright–Hays Program, is one of several United States Cultural Exchange Programs with the goal of improving intercultural relations, cultural diplomacy, and intercultural competence between the people ...

Distinguished Chair at the University of Massachusetts, Boston, College of Management.

Publications

* Chang, C. C.; Keisler, H. J
''Continuous Model Theory''
Annals of Mathematical Studies, 58, Princeton University Press, 1966. xii+165 pp. *''Model Theory for Infinitary Logic'', North-Holland, 1971 *Chang, C. C.; Keisler, H. J

Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. xvi+650 pp. ; 1st edition 1973; 2nd edition 1977 *''Elementary Calculus: An Infinitesimal Approach.'' Prindle, Weber & Schmidt, 1976/1986. Available online a

*''An Infinitesimal Approach to Stochastic Analysis'', American Mathematical Society Memoirs, 1984 *Keisler, H. J.; Robbin, Joel. ''Mathematical Logic and Computability'', McGraw-Hill, 1996 * Fajardo, Sergio; Keisler, H. J. ''Model Theory of Stochastic Processes'', Lecture Notes in Logic, Association for Symbolic Logic. 2002

References

External links

*
Keisler's home page
20th-century American mathematicians 21st-century American mathematicians Living people Model theorists University of Wisconsin–Madison faculty Fellows of the American Mathematical Society 1936 births Tarski lecturers {{US-mathematician-stub

Publications

See also

References

External links