An Invitation to Qualitative Computing

This report is Chapter 1 in the author’s book Qualitative Computing: A computational journey into non linearity published by World Scientific, Singapore (2011). It presents an historical survey of the evolution of mathematical computation since Sumer four millennia ago. It shows how the laws of logic have evolved in the mind of mathematicians. This evolution has been driven by three major ideas. First idea: multiplication, that is nonlinearity, leading to √ 2 (irrationality in Ancient Greece) and √ −1 (complexity in Renaissance Italy). Second idea: zero, a difficult concept (there exists the non existence is paradoxical) turned into a number in India (7th century). Third idea: actual infinity (the “inverse” of zero) embraced by Middle Ages Europe. This was the milestone that unleashed the power of modern analysis and made possible the rapid development of western technology. This evolution, which appears necessary only in retrospect, was not at all obvious and demanded an immense courage and a lot of soul-searching on the part of mathematicians in the past. The evolution is fragile and may stop at any time. For example, it was dramatically slowed down at the beginning of the 20th century when logic took precedence over computation, leading to a purely axiomatic view of mathematics with many adverse consequences for a sound understanding of life on earth. Thanks to countless unfashionable numerical analysts working at the margins of main-stream mathematics, the spirit of mathematical computation was kept alive during the reign of the all-axiomatic ideology. New fashion and the pressing need to overcome the limitations of our planet may soon call it back in the open.