Om Qualitative Computing
High technology industries are in desperate need for adequate tools to assess the validity of simulations produced by ever faster computers for perennial unstable problems. In order to meet these industrial expectations, applied mathematicians are facing a formidable challenge summarized by these words — nonlinearity and coupling. This book is unique as it proposes truly original solutions: (1) Using hypercomputation in quadratic algebras, as opposed to the traditional use of linear vector spaces in the 20th century; (2) complementing the classical linear logic by the complex logic which expresses the creative potential of the complex plane.The book illustrates how qualitative computing has been the driving force behind the evolution of mathematics since Pythagoras presented the first incompleteness result about the irrationality of √2. The celebrated results of Gödel and Turing are but modern versions of the same idea: the classical logic of Aristotle is too limited to capture the dynamics of nonlinear computation. Mathematics provides us with the missing tool, the organic logic, which is aptly tailored to model the dynamics of nonlinearity. This logic will be the core of the “Mathematics for Life” to be developed during this century.Contents:Introduction to Qualitative ComputingHypercomputation in Dickson AlgebrasVariable Complexity within Noncommutative Dickson AlgebrasSingular Values for the Multiplication MapsComputation Beyond Classical LogicComplexification of the ArithmeticHomotopic Deviation in Linear AlgebraThe Discrete and the ContinuousArithmetic in the Four Dickson Division AlgebrasThe Real and the ComplexThe Organic Logic of HypercomputationThe Organic Intelligence in NumbersReadership: Graduate students and researchers in applied and pure mathematics.