Keynote Title: The Infinity Computer and Numerical Computations with Infinite and Infinitesimal Numbers
Keynote Lecturer: Dr. Yaroslav D. Sergeyev
Keynote Chair: Dr. Rolf Pfeifer
Presented on: 05-10-2012
Abstract: A new methodology allowing one to execute numerical computations with finite, infinite, and infinitesimal numbers on a new type of a computer – the Infinity Computer – is introduced. A calculator using the Infinity Computer technology is presented during the talk. The new approach is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks that is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework (different from that of the non-standard Analysis). The new methodology evolves ideas of Cantor and Levi-Civita in a more applied way and (among other things) introduces infinite integers that possess both cardinal and ordinal properties as usual finite numbers. Note that foundations of the Set Theory dealing with infinity have been developed starting from the end of the XIX-th century until more or less the first decades of the XX-th century. Foundations of the classical Analysis dealing both with infinity and infinitesimal quantities have been developed even earlier, more than 200 years ago, with the goal to develop mathematical tools allowing one to solve problems arising in the real world in that remote time. As a result, these parts of Mathematics reflect ideas that people had about Physics (including definitions of the notions continuous and discrete) more than 200 years ago. Thus, these mathematical tools do not include numerous achievements of Physics of the XX-th century. Even the brilliant non-standard Analysis of Robinson made in the middle of the XX-th century has been also directed to a reformulation of the classical Analysis (i.e., Analysis created two hundred years before Robinson) in terms of infinitesimals and not to the creation of a new kind of Analysis that would incorporate new achievements of Physics. The point of view on infinite and infinitesimal quantities presented in this talk uses strongly two methodological ideas borrowed from the modern Physics: relativity and interrelations holding between the object of an observation and the tool used for this observation. The latter is directly related to connections between different numeral systems used to describe mathematical objects and the objects themselves. The new computational methodology gives the possibility both to execute numerical (not symbolic) computations of a new type and simplifies fields of Mathematics where the usage of the infinity and/or infinitesimals is necessary (e.g., divergent series, limits, derivatives, integrals, measure theory, probability theory, fractals, etc.). In particular, a number of results related to the First Hilbert Problem and Turing machines are established and a new concept of continuity better reflecting the modern view of physicists on the world around us is introduced. Numerous examples and applications are given: approximation, automatic differentiation, cellular automata, differential equations, linear and non-linear optimization, fractals, percolation, processes of growth in biological systems, etc.
Presented at the following Conference: IJCCI, International Joint Conference on Computational Intelligence
Conference Website: ijcci.org