Speaker：
Yaroslav D. Sergeyev
University of Calabria
Time：
Fri., 16:00-17:00, Mar.7, 2025
Venue：
C548, Shuangqing Complex Building A
Online：
Zoom Meeting ID: 271 534 5558
Passcode: YMSC
Title:
Foundations and applications of numerical infinities and infinitesimals
Abstract:
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
