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Numerical infinities and infinitesimals open new horizons in computations and give unexpected answers to 2 Hilbert problems

November 22, 2017

In the last issue of the prestigious journal EMS Surveys in Mathematical Sciences published by the European Mathematical Society there appeared a 102 pages long paper entitled "Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems" written by Yaroslav D. Sergeyev, Professor at Lobachevsky State University in Nizhni Novgorod, Russia and Distinguished Professor at the University of Calabria, Italy (see his Brief Bio below). The paper describes a recent computational methodology introduced by the author paying a special attention to the separation of mathematical objects from numeral systems involved in their representation. It has been introduced with the intention to allow people to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor's and non-standard analysis views and is based on the Euclid's Common Notion no. 5 "The whole is greater than the part" applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The non-contradictory of the approach has been proved by the famous Italian logician Prof. Gabriele Lolli.

This computational methodology uses a new kind of supercomputer called the Infinity Computer (patented in USA and EU) working numerically (traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. There exists its working software prototype. The appearance of the Infinity Computer changes drastically the entire panorama of numerical computations enlarging horizons of what can be computed to different numerical infinities and infinitesimals. It is argued in the paper that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. Numerous numerical examples and theoretical illustrations are given.

In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems giving unexpected answers to both problems. The effect of employing the new methodology in the study of the above Hypotheses is comparable to the dissolution of computational problems posed in Roman numerals (e.g. X - X cannot be computed in Roman numerals since zero is absent in their numeral system) once a positional system capable of expressing zero is adopted. More papers on a variety of topics using the new computational methodology can be found at the Infinity computer web page:

Yaroslav D. Sergeyev, Ph.D., D.Sc., D.H.C. is President of the International Society of Global Optimization. His research interests include numerical analysis, global optimization, infinity computing and calculus, philosophy of computations, set theory, number theory, fractals, parallel computing, and interval analysis. Prof. Sergeyev was awarded several research prizes (Khwarizmi International Award, 2017; Pythagoras International Prize in Mathematics, Italy, 2010; EUROPT Fellow, 2016; Outstanding Achievement Award from the 2015 World Congress in Computer Science, Computer Engineering, and Applied Computing, USA; Honorary Fellowship, the highest distinction of the European Society of Computational Methods in Sciences, Engineering and Technology, 2015; The 2015 Journal of Global Optimization (Springer) Best Paper Award; Lagrange Lecture, Turin University, Italy, 2010; MAIK Prize for the best scientific monograph published in Russian, Moscow, 2008, etc.).

His list of publications contains more than 250 items (among them 6 books). He is a member of editorial boards of 6 international journals and co-editor of 8 special issues. He delivered more than 60 plenary and keynote lectures at prestigious international congresses. He was Chairman of 7 international conferences and a member of Scientific Committees of more than 60 international congresses.

Lobachevsky University

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