Sergeyev, Yaroslav (2017) Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems. EMS Surveys in Mathematical Sciences, 4. pp. 219-320.
![[img]](https://philsci-archive.pitt.edu/style/images/fileicons/text.png) |
Text
EMSS_Survey.pdf Download (1MB) |
Abstract
In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one 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 methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that 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. It is argued 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. (recall that traditionally different numerals are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. 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.
| Export/Citation: | EndNote | BibTeX | Dublin Core | ASCII/Text Citation (Chicago) | HTML Citation | OpenURL |
| Social Networking: |
| Item Type: | Published Article or Volume | ||||||
|---|---|---|---|---|---|---|---|
| Creators: |
|
||||||
| Keywords: | Numerical infinities and infinitesimals, numbers and numerals, grossone, Infinity Computer, numerical analysis, infinite sets, divergent series, Continuum hypothesis, Riemann zeta function, numerical differentiation, ODEs, optimization, probability, fractals. | ||||||
| Subjects: | Specific Sciences > Computation/Information Specific Sciences > Cognitive Science > Concepts and Representations Specific Sciences > Mathematics Specific Sciences > Cognitive Science > Perception Specific Sciences > Probability/Statistics |
||||||
| Depositing User: | Prof. Yaroslav Sergeyev | ||||||
| Date Deposited: | 23 Sep 2024 18:28 | ||||||
| Last Modified: | 23 Sep 2024 18:28 | ||||||
| Item ID: | 23929 | ||||||
| Journal or Publication Title: | EMS Surveys in Mathematical Sciences | ||||||
| Publisher: | European Mathematical Society | ||||||
| Official URL: | https://ems.press/content/serial-article-files/369... | ||||||
| DOI or Unique Handle: | 10.4171/EMSS/4-2-3 | ||||||
| Subjects: | Specific Sciences > Computation/Information Specific Sciences > Cognitive Science > Concepts and Representations Specific Sciences > Mathematics Specific Sciences > Cognitive Science > Perception Specific Sciences > Probability/Statistics |
||||||
| Date: | 2017 | ||||||
| Page Range: | pp. 219-320 | ||||||
| Volume: | 4 | ||||||
| URI: | https://philsci-archive.pitt.edu/id/eprint/23929 |
Monthly Views for the past 3 years
Monthly Downloads for the past 3 years
Plum Analytics
Altmetric.com
Actions (login required)
 |
View Item |
![[feed]](http://philsci-archive.pitt.edu/style/images/feed-icon-32x32.png){kind=icon}