# Edward Nelson

For other people named Edward Nelson, see Edward Nelson (disambiguation).

Edward Nelson (born May 4, 1932, in Decatur, Georgia) 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. He has also written on the relationship between religion and mathematics.

## Career

Nelson 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 1956–1959. He has held a position at Princeton University from 1959 to the present, attaining the rank of professor there in 1964.

In 2012 he became a fellow of the American Mathematical Society.[1]

## Early work

Nelson has made contributions to the theory of infinite dimensional group representations, the mathematical treatment of quantum field theory, the use of stochastic processes 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 $\chi$, 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?[2] 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.

## 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 Theory—a version of a portion of Abraham Robinson's non-standard 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.

In September 2011, Nelson announced that he had proved that Peano arithmetic was logically inconsistent.[3] An error was found in the proof, and he retracted the claim.[4]

## Publications

• Dynamical theory of Brownian motion. Princeton University Press 1967
• Topics in Dynamics 1: Flows. Princeton University Press 1969
• Tensor analysis. Princeton University Press 1967
• Quantum Fluctuations. Princeton University Press, 1985
• Predicative Arithmetic. Princeton University Press, 1986[5]
• Radically elementary probability theory. Annals of Mathematical Studies, Princeton 1987

## Notes

1. ^ List of Fellows of the American Mathematical Society, retrieved 2013-02-24.
2. ^ p.23, Soifer, Alexander (2008); The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators; New York: Springer. ISBN 978-0-387-74640-1
3. ^ http://www.cs.nyu.edu/pipermail/fom/2011-September/015816.html
4. ^ http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html
5. ^ Wilkie, A. J. (1990). "Review: Predicative arithmetic, by Edward Nelson". Bull. Amer. Math. Soc. (N.S.) 22 (2): 326–331. doi:10.1090/s0273-0979-1990-15900-2.