Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at
Princeton University. He was known for his work on
mathematical physics and
mathematical logic. In mathematical logic, he was noted especially for his
internal set theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Ne ...
, and views on
ultrafinitism
In the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism,International Workshop on Logic and Computational Complexity, ''Logic and Computational Complexity'', Springer, 1995, p. 31. strict formalism,St. Iwan (2000),On the U ...
and the
consistency of
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
. In
philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people ...
he advocated the view of
formalism rather than
platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at le ...
or
intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
. He also wrote on the relationship between religion and mathematics.
Biography
Edward Nelson was born in
Decatur, Georgia in 1932. He spent his early childhood in
Rome where his father worked for the Italian YMCA. At the advent of
World War II, Nelson moved with his mother to
New York City, where he attended high school at the
Bronx High School of Science
The Bronx High School of Science, commonly called Bronx Science, is a public specialized high school in The Bronx in New York City. It is operated by the New York City Department of Education. Admission to Bronx Science involves passing the Spe ...
. His father, who spoke fluent
Russian
Russian(s) refers to anything related to Russia, including:
*Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries
* Rossiyane (), Russian language term for all citizens and p ...
, stayed in
St. Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
in connection with issues related to
prisoners of war. After the war, his family returned to Italy and he attended the
Liceo Scientifico Giovanni Verga in Rome.
He received his Ph.D. in 1955 from the
University of Chicago, where he worked with
Irving Segal. He was a member of the
Institute for Advanced Study from 1956 to 1959. He held a position at
Princeton University starting in 1959, attaining the rank of professor there in 1964 and retiring in 2013.
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, meetings, ...
. He died in
Princeton, New Jersey on September 10, 2014.
Academic work
Stochastic quantum mechanics
Nelson made contributions to the theory of infinite-dimensional
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s, the mathematical treatment of
quantum field theory, the use of
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es in
quantum mechanics, and the reformulation of
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
in terms of
non-standard analysis. For many years he worked on
mathematical physics and probability theory, and he retained a residual interest in these fields, particularly in connection with possible extensions of stochastic mechanics to
field theory.
Four color problem
In 1950, Nelson formulated a popular variant of the
four color problem: What is the chromatic number, denoted
, of the plane? In more detail, what is the smallest number of colors sufficient for coloring the points of the Euclidean plane such that no two points of the same color are unit distance apart? We know by simple arguments that 4 ≤ ''χ'' ≤ 7. The problem was introduced to a wide mathematical audience by
Martin Gardner in his October 1960
Mathematical Games column. The chromatic number problem, also now known as the
Hadwiger–Nelson problem, was a favorite of
Paul Erdős, who mentioned it frequently in his problems lectures. In 2018,
Aubrey de Grey showed that ''χ'' ≥ 5.
[.]
Foundations of mathematics
In the later part of his career, he worked on mathematical logic and the foundations of mathematics. One of his goals was to extend IST (
Internal Set Theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Ne ...
—a version of a portion of
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 reincorpo ...
's
non-standard analysis) in a natural manner that includes external functions and sets, in a way that provides an external function with specified properties unless there is a finitary obstacle to its existence. Other work centered on fragments of arithmetic, studying the divide between those theories interpretable in
Raphael Robinson's arithmetic and those that are not;
computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...
, including the problem of whether
P is equal to NP or not; and
automated proof checking
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof edito ...
.
In September 2011, Nelson announced that he had proved that
Peano arithmetic was logically inconsistent. An error was found in the proof by
Terry Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
, and Nelson retracted the claim.
Publications
Selected papers
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Books
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See also
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Hidden variable theory
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Influence of non-standard analysis
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Stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
*
Stochastic quantum mechanics
*
Stochastic electrodynamics
Stochastic electrodynamics (SED) is a variant of classical electrodynamics (CED) of theoretical physics. SED consists of a set of controversial theories that posit the existence of a classical Lorentz invariant radiation field having statistica ...
References
Notes
Sources
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External links
Curriculum Vitae – Edward NelsonMathematics and Faith – Edward NelsonEdward Nelson's Homepage*
Mathematics and Religion DiscussionThe Limitation of Mental and Physical Reality Discussion
{{DEFAULTSORT:Nelson, Edward
1932 births
Members of the United States National Academy of Sciences
Fellows of the American Mathematical Society
20th-century American mathematicians
21st-century American mathematicians
American logicians
Set theorists
University of Chicago alumni
Princeton University faculty
2014 deaths
Mathematical physicists