Prof. Dr. Yaroslav Sergeyev,
University of Calabria, Italy
Numerical infinities and infinitesimals How primitive counting systems can change our perception of Infinity
In this lecture, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, and the rectangle paradox of Torricelli are considered. It is shown that the surprising counting systems of Amazonian and Australian tribes, working with only three numerals (one, two, many) can help us to change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described (see [1]). This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching (see, e.g., [1-3]). This methodology gives the possibility to consider the paradoxes listed above in a new constructive light (see, e.g., [4,5]) showing so that even primitive cultures can give rise to very interesting developments in the modern sophisticated cultural and scientific life.
The Infinity Calculator using the Infinity Computer technology (patented in several countries) is presented during the talk. Reviews, videos, more than 60 papers of authors from several research areas using this methodology in their applications can be downloaded from http://www.theinfinitycomputer.com. The web page https://www.numericalinfinities.com developed at the University of East Anglia contains materials related to teaching this methodology in colleges in UK and Italy.
Selected references
Sergeyev Ya.D. (2017). Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences, 4(2), 219–320.
Sergeyev Ya.D., De Leone R. (eds.). (2022). Numerical Infinities and Infinitesimals in Optimization, Springer.
Nasr L. (2022). The effect of Arithmetic of Infinity Methodology on students’ beliefs of infinity, Mediterranean Journal for Research in Mathematics Education, 19, 5-19.
Rizza D. (2018). A study of mathematical determination through Bertrand’s Paradox, Philosophia Mathematica, 26(3), 375–395.
Sergeyev Ya.D. (2022). Some paradoxes of infinity revisited, Mediterranean Journal of Mathematics, 19, article 143.
