Archimedean and Non-Archimedean Approaches to Mathematical Modeling

Abstract

This chapter presents the two most relevant approaches to mathematical modeling: the long standing and by now classic Archimedean model, based on the Archimedean Principle, named by Otto Stolz after the ancient Greek mathematician Archimedes de Syracuse: an Archimedean space has no infinitely large or infinitely small elements. We illustrate this model with an example from the signed qualitative mathematical modeling, that is, the so-called Jacobian Feedback Loop Methodology which stresses qualitative understanding of systems as the primary goal rather the numerical prediction and the more familiar large scale quantitative methods. The other approach we present is the Non-Archimedean/Ultrametric/p-adic mathematical model: it is based on the violation of the Archimedean principle, and it is proving to be more adequate and appropriate for systems with intrinsic hierarchical structure; the associated geometry is non-intuitive, where the topological concept of openess/closeness is rendered meaningless (clopen balls topology), all triangles are isosceles, and the space is totally disconnected; we present the most recent areas of applications from mathematical physics (spin glasses, p-adic string theory,…) to p-adic arithmetic dynamics to the analysis of Mental spaces (p-adic cognitive sciences).

We introduce the chapter with a note on the influence of cultures (e.g., French and Russian) on mathematical concepts and models, as well as a note of the upcoming post-human mathematics to be featured by computer/artificial mathematical creativity.