Title

Methodological and philosophical aspects of a new approach to deal with infinities and infinitesimals

Abstract Course

The lecture presents a recent methodology allowing one to execute numerical computations with finite, infinite, and infinitesimal numbers on a new type of a computer – the Infinity Computer – patented in EU, USA, and Russia (see [20]). The new approach is based on the principle ‘The whole is greater than the part’ (Euclid’s Common Notion 5) 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 new infinite integers that possess both cardinal and ordinal properties as usual finite numbers. Its relations to traditional approaches are discussed. It is emphasized that the philosophical triad – researcher, object of investigation, and tools used to observe the object – existing in such natural sciences as Physics and Chemistry, exists in Mathematics, too. In natural sciences, the instrument used to observe the object influences the results of observations. The same happens in Mathematics where numeral systems used to express numbers are among the instruments of observations used by mathematicians. The usage of powerful numeral systems gives the possibility to obtain more precise results in Mathematics, in the same way as the usage of a good microscope gives the possibility to obtain more precise results in Physics. A numeral system using a new numeral called grossone is described. It allows one to express easily infinities and infinitesimals offering rich capabilities for describing mathematical objects, mathematical modeling, and practical computations. The concept of the accuracy of numeral systems is introduced. The accuracy of the new numeral system is compared with traditional numeral systems used to work with infinity. Numerous examples are given. The Infinity Calculator using the Infinity Computer technology is presented during the talk.

Selected References

• De Cosmis S, De Leone R (2012) The use of grossone in mathematical programming and operations research. Applied Mathematics and Computation, 218(16):8029–8038
• D’Alotto L (2013) A classification of two-dimensional cellular automata using infinite computations. Indian Journal of Mathematics, 55:143–158
• Iudin DI, Sergeyev YaD,  Hayakawa M (2012) Interpretation of percolation in terms of infinity computations. Applied Mathematics and Computation, 218(16):8099–8111
• Iudin DI, Sergeyev YaD,  Hayakawa M (2015) Infinity computations in cellular automaton forest-fire model. Communications in Nonlinear Science and Numerical Simulation, 20(3):861–870.
• Lolli G (2015) Metamathematical investigations on the theory of grossone. Applied Mathematics and Computation, 255:3–14
• Margenstern M (2011) Using grossone to count the number of elements of infinite sets and the connection with bijections. p-Adic Numbers, Ultrametric Analysis and Applications, 3(3):196–204
• Margenstern M (2015) Fibonacci words, hyperbolic tilings and grossone. Communications in Nonlinear Science and Numerical Simulation, 21(1–3):3–11
• Sergeyev YaD (2003) Arithmetic of Infinity. Edizioni Orizzonti Meridionali, CS, 2^nd electronic edition 2013.
• Sergeyev YaD (2009) Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains. Nonlinear Analysis Series A: Theory, Methods & Applications, 71(12):e1688–e1707
• Sergeyev YaD (2010) Counting systems and the First Hilbert problem. Nonlinear Analysis Series A: Theory, Methods & Applications, 72(3-4):1701–1708
• Sergeyev YaD (2010) Lagrange Lecture: Methodology of numerical computations with infinities and infinitesimals. Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino, 68(2):95–113
• Sergeyev YaD (2011) Higher order numerical differentiation on the infinity computer. Optimization Letters, 5(4):575–585
• Sergeyev YaD (2011) On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function. p-Adic Numbers, Ultrametric Analysis and Applications, 3(2):129–148
• Sergeyev YaD (2011) Using blinking fractals for mathematical modelling of processes of growth in biological systems. Informatica, 22(4):559–576
• Sergeyev YaD (2013) Solving ordinary differential equations by working with infinitesimals numerically on the infinity computer. Applied Mathematics and Computation, 219(22):10668–10681
• Sergeyev YaD, Garro A (2010) Observability of Turing machines: A refinement of the theory of computation. Informatica, 21(3):425–454
• Sergeyev YaD, Garro A (2013)  Single-tape and multi-tape Turing machines through the lens of the Grossone methodology. Journal of Supercomputing, 65(2):645–663
• Zhigljavsky AA (2012) Computing sums of conditionally convergent and divergent series using the concept of grossone. Applied Mathematics and Computation, 218(16):8064–8076

Contact

yaro@si.dimes.unical.it