In this talk, a recent computational methodology (not related to non-standard analysis) is described (see [3, 5-7]). It allows people to work on a computer with infinities and infinitesimals numerically (i.e., not symbolically) in a unique framework and in all the situations requiring these notions. Recall that traditional approaches work with infinities and infinitesimals only symbolically and different notions are used in different situations related to infinity (∞, ordinals, cardinals, etc). The new methodology is based on the Euclid’s Common Notion “The whole is greater than the part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite).
One of the strong advantages of this methodology consists of its usefulness in practical applications (see [1, 2, 7, 9]) that use a new kind of supercomputer called the Infinity Computer patented in several countries. It works numerically with numbers that can have several infinite and infinitesimal parts written in a positional system with an infinite base (called Grossone) using floating-point representation. On a number of examples (paradoxes [5-8], optimization [9], ODEs [1], hybrid systems [2], Turing machines [10], teaching [4,5], etc.), it is shown that the new approach can be useful both in practice in theoretical considerations. In particular, thanks to the new methodology, the accuracy of computations increases drastically, and all kinds of indeterminate forms and divergences are avoided.
It is argued that traditional numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in certain theoretical assertions, as well. The Continuum Hypothesis and some results related to the Riemann zeta function are discussed from the point of view of the grossone methodology. It is also shown that this methodology allows to avoid several classical paradoxes related to infinity and infinitesimals.
The Infinity Calculator working with infinities and infinitesimals numerically is shown during the talk. For more information see https://www.theinfinitycomputer.com and https://www.numericalinfinities.com
Selected references
1. Amodio, P., Iavernaro, F., Mazzia, F., Mukhametzhanov, M.S., Sergeyev, Y.D. (2017) A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic, Math. & Comput. in Simulation, 141, 24–39.
2. Falcone, A., Garro, A., Mukhametzhanov, M.S., Sergeyev, Y.D. (2022) Simulation of hybrid systems under Zeno behavior using numerical infinitesimals, Commun. in Nonlin. Sci. & Numer. Simul., 111, 106443.
3. Lolli G. (2015) Metamathematical investigations on the theory of Grossone, Appl. Math. & Comput., 255, 3-14.
4. Nasr L. (2023) Students’ resolutions of some paradoxes of infinity in the lens of the grossone methodology, Informatics and Education, 38(1), 83–91.
5. Rizza D. (2023) Primi Passi nell’Aritmetica dell’Infinito, Bonomo Editore, Bologna.
6. Sergeyev Y.D. (2003, 2nd ed. 2013) Arithmetic of Infinity, Orizzonti Meridionali, CS.
7. Sergeyev Y.D. (2017) Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Math. Sciences, 4(2), 219–320.
8. Sergeyev, Y.D. (2022) Some paradoxes of infinity revisited, Mediterranean J. of Math., 19(3), 143.
9. Sergeyev Y.D., De Leone R., eds. (2022) Numerical Infinities and Infinitesimals in Optimization, Springer, Cham.
10. Sergeyev Y.D., Garro A. (2010) Observability of Turing Machines: a refinement of the theory of computation, Informatica, 21(3), 425–454.
Yaroslav D. Sergeyev is Distinguished Professor at the University of Calabria, Italy (chiamata diretta per chiara fama) and Head of Numerical Calculus Laboratory at the same university. Several decades he was also Affiliated Researcher at the Institute of High-Performance Computing and Networking of the Italian National Research Council, and is Affiliated Faculty at the Center for Applied Optimization, University of Florida, Gainesville, USA.
He was awarded his Ph.D. (1990) from Lobachevski Gorky State University and his D.Sc. degree (1996) from Lomonosov State University, Moscow (this degree is Habilitation for the Full Professorship in Russian universities). In 2013, he was awarded Degree of Honorary Doctor from Glushkov Institute of Cybernetics of The National Academy of Sciences of Ukraine, Kiev.
His research interests include global optimization (he was President of the International Society of Global Optimization, 2017-2021), infinity computing and calculus (the field he has founded), numerical computations, scientific computing, philosophy of computations, set theory, number theory, fractals, parallel computing, and interval analysis.
He was awarded several research prizes (International Constantin Carathéodory Prize, International ICNAAM Research Excellence Award, International Prize of the city of Gioacchino da Fiore, all in 2023; Khwarizmi International Award, 2017; Pythagoras International Prize in Mathematics, 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.). In 2020, he was elected corresponding member of Accademia Peloritana dei Pericolanti in Messina, Italy. Since 2020 he is included in the rating “Top 2% highly cited authors in Scopus” produced by Stanford University, the list “Top Italian Scientists. Mathematics”, the list of top researchers produced by Research.com, etc. In 2022, his biography has been published in Chinese by the journal Mathematical Culture. In 2023, the book “Primi Passi nell’Aritmetica dell’Infinito” authored by Prof. Davide Rizza from the University of East Anglia has been published. The book is dedicated to teaching the Infinity Computing methodology developed by Prof. Sergeyev.
His list of publications contains more than 300 items (among them 6 authored and 11 edited books and more than 130 articles in international journals). He is a member of editorial boards of one book series (Springer), 12 international and 3 national journals and co-editor of 14 special issues. He delivered more than 90 plenary and keynote lectures and tutorials at prestigious international congresses. He was Chairman of 7 and Co-Chairman of 8 international conferences and a member of Scientific Committees of more than 110 international congresses. He is Coordinator of numerous national and international research and educational projects, supervises master and Ph.D. theses, takes part of Ph.D. commissions in Italy and abroad. Numerous magazines, newspapers, TV and radio channels have dedicated a lot of space to his research. In 2023, the 21st International Conference of Numerical Analysis and Applied Mathematics, Crete (Greece) has been dedicated to the achievements of Prof. Sergeyev and his 60th birthday.
Personal Website： https://www.yaroslavsergeyev.com/
