Edward Nelson
Encyclopedia
Edward Nelson is a professor in the Mathematics Department at Princeton University
. He is known for his work on mathematical physics
and mathematical logic
. In mathematical logic, he is noted especially for his internal set theory
, and his controversial views on ultrafinitism
and the consistency
of arithmetic.
, where he worked with Irving Segal
.
He was a member of the Institute for Advanced Study
from 1956–1959. He has held a position at Princeton University
from 1959 to the present, attaining the rank of professor there in 1964.
s, the mathematical treatment of quantum field theory
, the use of stochastic process
es in quantum mechanics
, and the reformulation of probability theory
in terms of non-standard analysis
.
For many years he worked on mathematical physics
and probability theory, and still has a residual interest in these fields, particularly in possible extensions of stochastic mechanics to field theory.
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 in such a way 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 also a favorite of Paul Erdős
, who mentioned it frequently in his problems lectures.
—a version of a portion of Abraham Robinson
's nonstandard analysis) in a natural way to include 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 centers on fragments of arithmetic, studying the divide between those theories interpretable in Raphael Robinson's Arithmetic
and those that are not; computational complexity
, including the problem of whether P is equal to NP or not; and automated proof checking.
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. He is known for his work on mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
and mathematical logic
Mathematical 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...
. In mathematical logic, he is noted especially for his internal set theory
Internal set theory
Internal set theory is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can be...
, and his controversial views on ultrafinitism
Ultrafinitism
In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism is a form of finitism. There are various philosophies of mathematics which are called ultrafinitism...
and the consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
of arithmetic.
Career
Nelson received his Ph.D. in 1955 from the University of ChicagoUniversity of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
, where he worked with Irving Segal
Irving Segal
Irving Ezra Segal was a mathematician known for work on theoretical quantum mechanics.He was at the Massachusetts Institute of Technology...
.
He was a member of the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
from 1956–1959. He has held a position at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
from 1959 to the present, attaining the rank of professor there in 1964.
Early work
Nelson has made contributions to the theory of infinite dimensional group representationGroup representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
s, the mathematical treatment of quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the use of stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, and the reformulation of probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
in terms 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:...
.
For many years he worked on mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
and probability theory, and still has a residual interest in these fields, particularly in possible extensions of stochastic mechanics to field theory.
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 in such a way 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 GardnerMartin Gardner
Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature , philosophy, scientific skepticism, and religion...
in his October 1960 Mathematical Games column. The chromatic number problem, also now known as the Hadwiger–Nelson problem
Hadwiger–Nelson problem
In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance one from each other have the same color. The answer is unknown, but has been narrowed down to...
, was also a favorite of Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
, who mentioned it frequently in his problems lectures.
Work on foundations
In recent years he has been working on mathematical logic and the foundations of mathematics. One of his goals is to extend IST (Internal Set TheoryInternal set theory
Internal set theory is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can be...
—a version of a portion of 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 nonstandard analysis) in a natural way to include 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 centers on fragments of arithmetic, studying the divide between those theories interpretable in Raphael Robinson's Arithmetic
Robinson arithmetic
In mathematics, Robinson arithmetic, or Q, is a finitely axiomatized fragment of Peano arithmetic , first set out in R. M. Robinson . Q is essentially PA without the axiom schema of induction. Since Q is weaker than PA, it is incomplete...
and those that are not; computational complexity
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
, including the problem of whether P is equal to NP or not; and automated proof checking.