Science features "Mathematics and Complex Systems" article by Richard Foote
Release Date: 10-22-2007
Dr. Richard Foote, mathematics professor in UVM's College of Engineering and Mathematical Sciences (CEMS), has published an article in the October 2007 issue of Science.
The article, "Mathematics and Complex Systems," proposes that areas of mathematics even ones based on simple axiomatic foundations have discernible layers, entirely unexpected "macroscopic" outcomes, and both mathematical and physical ramifications profoundly beyond their historical beginnings. Foote proposes that as many areas and results surpass individual researchers' capacity to make verifications "by hand," we may come to realize that mathematics itself is the ultimate complex system.
Foote's paper relates to highly complex physical systems, such as the atmosphere, transportation networks and the Internet, for which researchers have created mathematical models. He reviews the case of finite group theory which features examples such as the Enormous Theorem, requiring 15,000 pages of proofs, and the Monster simple group, containing about 10^54 elements. He proposes that the ways in which mathematical concepts result in complex structures may help us understand the complexity of physical systems and vice versa.
According to Foote, "Each scientist, mathematician, or researcher unfurls the mysteries of nature and humankind in small, deliberate steps. Science embodies the ability to verify, reproduce, and convince others of the veracity of one's discoveries, so the work of scientists is inherently incremental and precise." With this in mind, Foote believes scientists must work toward understanding the "big picture" issues, both in their own disciplines and beyond. "Each of our disciplines," he avers, "must itself exhibit the inherent facets of a complex system, or our research is surely nugatory."