Abraham Robinson
Encyclopedia
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.
, Germany
, which is now Wałbrzych, in Poland
. In 1933, he immigrated to British Mandate of Palestine, where he earned a first degree from the Hebrew University. Robinson was in France
when the Nazis
invaded during World War II
, and escaped by train and on foot, being alternately questioned by French soldiers suspicious of his German passport and asked by them to share his map, which was more detailed than theirs. While in London
, he joined the Free French Air Force
and contributed to the war effort by teaching himself aerodynamics
and becoming an expert on the airfoil
s used in the wings of fighter planes.
After the war, Robinson worked in London
, Toronto
, and Jerusalem, but ended up at University of California, Los Angeles
in 1962.
to attack problems in analysis
and abstract algebra
. He "introduced many of the fundamental notions of model theory
". Using these methods, he found a way of using formal logic
to show that there are self-consistent nonstandard models of the real number system which include infinite and infinitesimal numbers. Others, such as Wilhelmus Luxemburg
, showed that the same results could be achieved using ultrafilter
s, which made Robinson's work more accessible to mathematicians who lacked training in formal logic. Robinson's book Non-standard Analysis was published in 1966. Robinson was strongly interested in the history and philosophy of mathematics, and often remarked that he wanted to get inside the head of Leibniz
, the first mathematician to attempt to articulate clearly the concept of infinitesimal numbers
.
While at UCLA his colleagues remember him as working hard to accommodate PhD
students of all levels of ability by finding them projects of the appropriate difficulty. He was courted by Yale
, and after some initial reluctance, he moved there in 1967. He died of pancreatic cancer
in 1974.
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....
who is most widely known for development of 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:...
, a mathematically rigorous system whereby 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...
and infinite
Transfinite number
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these...
numbers were incorporated into mathematics.
Biography
He was born to a Jewish family with strong Zionist beliefs, in WaldenburgWaldenburg
Waldenburg is the name of several places:* Waldenburg, Saxony, Germany* Waldenburg, Baden-Württemberg* Waldenburg, Switzerland*the former German name for Wałbrzych, Poland* Waldenburg, Arkansas, USAWaldenburg is also a surname:* Louis Waldenburg...
, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
, which is now Wałbrzych, in Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
. In 1933, he immigrated to British Mandate of Palestine, where he earned a first degree from the Hebrew University. Robinson was in France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
when the Nazis
Nazi Germany
Nazi Germany , also known as the Third Reich , but officially called German Reich from 1933 to 1943 and Greater German Reich from 26 June 1943 onward, is the name commonly used to refer to the state of Germany from 1933 to 1945, when it was a totalitarian dictatorship ruled by...
invaded during World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, and escaped by train and on foot, being alternately questioned by French soldiers suspicious of his German passport and asked by them to share his map, which was more detailed than theirs. While in London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
, he joined the Free French Air Force
Free French Air Force
The Free French Air Force was the air arm of the Free French Forces during the Second World War.-Fighting for Free France — the FAFL in French North Africa :...
and contributed to the war effort by teaching himself aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...
and becoming an expert on the airfoil
Airfoil
An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....
s used in the wings of fighter planes.
After the war, Robinson worked in London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
, Toronto
Toronto
Toronto is the provincial capital of Ontario and the largest city in Canada. It is located in Southern Ontario on the northwestern shore of Lake Ontario. A relatively modern city, Toronto's history dates back to the late-18th century, when its land was first purchased by the British monarchy from...
, and Jerusalem, but ended up at University of California, Los Angeles
University of California, Los Angeles
The University of California, Los Angeles is a public research university located in the Westwood neighborhood of Los Angeles, California, USA. It was founded in 1919 as the "Southern Branch" of the University of California and is the second oldest of the ten campuses...
in 1962.
Work in model theory
He become known for his approach of using the methods of mathematical logicMathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
to attack problems in analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
and abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. He "introduced many of the fundamental notions of model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
". Using these methods, he found a way of using formal logic
Formal logic
Classical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...
to show that there are self-consistent nonstandard models of the real number system which include infinite and infinitesimal numbers. Others, such as Wilhelmus Luxemburg
Wilhelmus Luxemburg
Wilhelmus Luxemburg is Professor of Mathematics, Emeritus at the California Institute of Technology. He received his B.A. from the University of Leiden in 1950; his M.A., in 1953; his Ph.D., from the Delft Institute of Technology, in 1955. He was Assistant Professor at Caltech during 1958-60;...
, showed that the same results could be achieved using ultrafilter
Ultrafilter
In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
s, which made Robinson's work more accessible to mathematicians who lacked training in formal logic. Robinson's book Non-standard Analysis was published in 1966. Robinson was strongly interested in the history and philosophy of mathematics, and often remarked that he wanted to get inside the head of Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
, the first mathematician to attempt to articulate clearly the concept of infinitesimal numbers
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...
.
While at UCLA his colleagues remember him as working hard to accommodate PhD
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
students of all levels of ability by finding them projects of the appropriate difficulty. He was courted by Yale
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
, and after some initial reluctance, he moved there in 1967. He died of pancreatic cancer
Pancreatic cancer
Pancreatic cancer refers to a malignant neoplasm of the pancreas. The most common type of pancreatic cancer, accounting for 95% of these tumors is adenocarcinoma, which arises within the exocrine component of the pancreas. A minority arises from the islet cells and is classified as a...
in 1974.