Mathematics and Complex Systems
Richard Foote
Contemporary researchers strive to understand complex physical
phenomena that involve many constituents, may be influenced
by numerous forces, and may exhibit unexpected or emergent behavior.
Often such "complex systems" are macroscopic manifestations
of other systems that exhibit their own complex behavior and
obey more elemental laws. This article 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. In a larger sense, the study
of mathematics itself, which is increasingly surpassing the
capacity of researchers to verify "by hand," may be the ultimate
complex system.
