Skip to main content
SpringerLink
Log in

Infinitesimal Probabilities Based on Grossone

  • Original Research
  • Published:
SN Computer Science Aims and scope Submit manuscript

Abstract

In finite probability theory, the only probability zero event is the impossible one, but in standard Kolmogorov probability theory, probability zero events occur all the time. Prominent logicians, probability experts and philosophers of probability, including Carnap, Kemeny, Shimony, Savage, De Finetti, Jeffrey, have successfully argued that a sound probability should be regular, that is, only the impossible event should have zero probability. This intuition is shared by physicists too. Totality is another desideratum which means that every event should be assigned a probability. Regularity and totality are achievable in rigorous mathematical terms even for infinite events via hyper-reals valued probabilities. While the mathematics of these theories is not objectionable, some philosophical arguments purport to show that infinitesimal probabilities are inherently problematic. In this paper, we present a simpler and natural construction—based on Sergeyev’s calculus with Grossone (in a formalism inspired by Lolli) enriched with infinitesimals—of a regular, total, finitely additive, uniformly distributed probability on infinite sets of positive integers. These probability spaces—which are inspired by and parallels the construction of classical probability—will be briefly studied. In this framework, De Finetti fair lottery has the natural solution and Williamson’s objections against infinitesimal probabilities are mathematically refuted.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. This is a special case of the Euclidean principle requiring that any set should have a strictly larger probability than each of its strict subsets. It is easy to see that every regular, total and finitely additive probability is Euclidean: indeed, for all \(X,Y \subseteq \varOmega \), if \(X\subset Y\), then because \(Y= X \cup (Y{{\setminus}} X)\) and \(Y{{\setminus}} X \not = \emptyset \), then by finite additivity of the mutually disjoints events X and Y, we have: \(\Pr (Y) = \Pr (X) + \Pr (Y{\setminus} X)> \Pr (X)\) as \( \Pr (Y{\setminus} X)>0\).

  2. A more general, but less formal, form is to ask that the probability theory should allow for a mathematical representation of any probabilistic situation that is conceptually possible.

  3. Aristotles Principle is not satisfied by set theory. For other mathematical approaches to the notion of “size” of sets based on Aristotles Principle see [12, 13].

  4. According to Sergeyev [3,4,5,6], (1) and (2) represent the same set of numbers. In (1) we can “see” only the finite numbers, but in (2) we can “see” both finite and infinite numbers.

  5. The name “measure” will be justified by Theorem 1.

  6. An informal discussion of a probability on a set with more than \(\textcircled {1}\) elements is presented in Section 9.1 [6].

  7. In case \(\varOmega \) is contextually clear \({\Pr }_\varOmega \) will be simply written \(\Pr \).

References

  1. Benci V, Horsten L, Wenmackers S. Infinitesimal probabilities. Brit J Phil Sci. 2018;69:509–52.

    MathSciNet  MATH  Google Scholar 

  2. Olofsson P. Probability, statistics, and stochastic processes. New York: Wiley Interscience; 2005.

    Book  Google Scholar 

  3. Sergeyev YD. Arithmetic of infinity. Cosenza: Edizioni Orizzonti Meridionali; 2003.

    MATH  Google Scholar 

  4. Sergeyev YD. A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica. 2008;19(4):567–96.

    MathSciNet  MATH  Google Scholar 

  5. Sergeyev YD. Computations with grossone-based infinities. In: Calude CS, Dinneen MJ (eds) Unconventional Computation and Natural Computation—14th International Conference, UCNC 2015, Auckland, New Zealand, 2015, Proceedings, Springer, Lecture Notes in Computer Science, vol 9252, pp 89–106. 2015. https://doi.org/10.1007/978-3-319-21819-9_6

  6. Sergeyev YD. Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv Math Sci. 2017;4(2):219–320.

    Article  MathSciNet  Google Scholar 

  7. Lolli G. Metamathematical investigations on the theory of grossone. Appl Math Comput. 2015;255:3–14.

    MathSciNet  MATH  Google Scholar 

  8. Williamson T. How probable is an infinite sequence of heads? Analysis. 2007;67:173–80.

    Article  MathSciNet  Google Scholar 

  9. Nelson E. Radically elementarry probability theory. Princeton: Princeton University Press; 1987.

    Book  Google Scholar 

  10. Wenmackers S, Horsten L. Fair infinite lotteries. Synthese. 2013;190:37–61.

    Article  MathSciNet  Google Scholar 

  11. Benci V, Horsten L, Wenmackers S. Non-arhimedian probability. Milan J Math. 2013;81:121–51.

    Article  MathSciNet  Google Scholar 

  12. Gilbert T, Rouche N. Y a-t-il vraiment autant de nombres pairs que de naturels? Cahiers Centre Logique. 1996;9:99–139.

    MathSciNet  MATH  Google Scholar 

  13. Benci V, Nasso MD, Forti M. An aristotelian notion of size. Ann Pure Appl Logic. 2006;143(1):43–53. https://doi.org/10.1016/j.apal.2006.01.008, http://www.sciencedirect.com/science/article/pii/S016800720600042X.

  14. De Finetti B. Theory of Probability. London, New York, Sydney, Toronto: Wiley; 1975.

    MATH  Google Scholar 

  15. Truss J. Foundations of mathematical analysis. Oxford: Oxford University Press; 1997.

    MATH  Google Scholar 

  16. Rizza D. Numerical methods for infinite decision-making processes. Int J Unconvent Comput. 2019;14(2):139–58.

    Google Scholar 

  17. Lolli G. Infinitesimals and infinites in the history of mathematics: a brief survey. Appl Math Comput. 2012;218(16):7979–88.

    MathSciNet  MATH  Google Scholar 

  18. Parker MW. Symmetry arguments against regular probability: a reply to recent objections. Eur J Philos Sci. 2018;9(8):1–21. https://doi.org/10.1007/s13194-018-0229-1.

    Article  MathSciNet  Google Scholar 

  19. Mermin ND. Physics: Qbism puts the scientist back into science. Nature. 2014;507:421–3. https://doi.org/10.1038/507421a.

    Article  Google Scholar 

  20. Ellerman D. On classical finite probability theory as a quantum probability calculus. 2015. arXiv:1502.01048

  21. Khrennikov A. Quantum bayesianism as the basis of general theory of decision-making. Philos Trans A Math Phys Eng Sci. 2016;374(2068):20150245. https://doi.org/10.1098/rsta.2015.0245.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Yaro Sergeyev and Anatoly Zhigljavsky for comments and suggestions which improved this paper and Gabrielle Lolli for illuminating discussions on his paper [7], infinitesimals and Grossone. We also thank Elena Calude for asking challenging questions which improved our model.

Author information

Authors and Affiliations

  1. School of Computer Science, University of Auckland, Auckland, New Zealand

    Cristian S. Calude

  2. Faculty of Mathematics and Computer Science, Bucharest University, Bucharest, Romania

    Monica Dumitrescu

Authors
  1. Cristian S. Calude
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monica Dumitrescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristian S. Calude.

Ethics declarations

Conflict of interest

On behalf of all the authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Calude, C.S., Dumitrescu, M. Infinitesimal Probabilities Based on Grossone. SN COMPUT. SCI. 1, 36 (2020). https://doi.org/10.1007/s42979-019-0042-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42979-019-0042-8

Keywords

Mathematics Subject Classification

Search

Navigation