Exploring the Process of Discovery
(Information at this site represents work done jointly by Wayne Patterson at Howard, Dinesh Sarvate of the College of Charleston, and Susanne Wetzel of the Stevens Institute of Technology.)
Since early 2009, there has been in progress a unique development in how information is communicated, and how discoveries can be made when the participants in a discussion may number in the tens or hundreds.
It is currently an issue of considerable discussion in the academic community regarding how discoveries are made, but it will undoubtedly lead to considerations of intellectual property, attribution, credit, secrecy, and/or plagiarism. Thus it will soon be of interest in the cybersecurity community.
Last year, Timothy Gowers, Cambridge mathematician and 1998 Fields Medal winner, put forth the challenge to the mathematics community to examine the manner by which research is conducted in the field:
http://gowers.wordpress.com/2009/01/27/ismassivelycollaborativemathematicspossible/
He asked an interesting question: Is it possible to discover new theorems in mathematics, or improve on the proofs of existing theorems, by using a “polymath” process, in effect by creating a virtual organization to work towards a specific result?
Our interest has been in studying the evolution of these polymath projects, in order to gain an understanding of how discovery occurs.
In order to develop some basis for comparison of different polymath projects, we developed a graphtheoretic model to describe an individual project. Since it is important to observe the polymath project as it evolves over time, the model is actually a sequence of graphs parametrized by time.
We have built models for two specific polymath projects that have been among those most widely discussed. They are:
IMOq6: sixth problem from the 2009 International Mathematics Olympiad, which states: Let
be distinct positive integers and let be a set of
positive integers not containing A grasshopper is to jump along the real axis, starting at the point and making jumps to the right with lengths
in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in .
During the Olympiad competition, only three of 565 participants completely solved the problem. Subsequently, Terence Tao of UCLA (and another Fields Medalist), launched a polymath project to address this problem and to see respondents would be interested in participating when the mathematical challenge was not likely to lead to a research publication. 70 persons participated in the solution of the problem, which involved 278 posts in 35 hours to reach the result.
DHJ: An earlier, and more noted, polymath project launched by Gowers sought a solution to an important problem in discrete mathematics, the density HalesJewitt theorem, which states in one version: for any positive integers n and c there is a number H such that if the cells of a Hdimensional n×n×n×...×n cube are colored with c colors, there must be one row, column, or diagonal of length n all of whose cells are the same color. In a more familiar context, this is a statement that the higherdimensional, multiplayer, n {row, column, diagonal} generalization of the game of tictactoe cannot end in a draw, independently of n and c and who plays first, provided only that it is played in sufficiently high dimension H.
A proof of DHJ has been known since 1991, but researchers in this field have felt that the existing proof was not satisfactory to many
We have built models for these two polymath projects using Mathematica 7 (Wolfram Corporation). The animations generated by these models are available here. If you do not have a copy of Mathematica to run these animations, you can download the Mathematica Player 7 at:
http://www.wolfram.com/products/player/
DHJ Polymath Project Mathematica Animation
IMOQ6 Polymath Project Mathematica Animation
References:
Timothy Gowers, Gowers’ Weblog, Is massively collaborative mathematics possible?,
http://gowers.wordpress.com/2009/01/27/ismassivelycollaborativemathematicspossible/ Timothy Gowers and Michael Nielsen, Massively collaborative mathematics, Nature461, 879881 (15 October 2009)  doi:10.1038/461879a; Published online 14 October 2009.
Terence Tao, IMO 2009 Q6 as a Minipolymath project, http://terrytao.wordpress.com/2009/07/20/imo2009q6asaminipolymathproject/
