Advertisement

Synthese

pp 1–23 | Cite as

Underdetermination of infinitesimal probabilities

  • Alexander R. PrussEmail author
Article

Abstract

A number of philosophers have attempted to solve the problem of null-probability possible events in Bayesian epistemology by proposing that there are infinitesimal probabilities. Hájek (Synthese 137:273–323, 2003) (more tentatively) and Easwaran (Philos Rev 123:1–41, 2014) (more definitively) have argued that because there is no way to specify a particular hyperreal extension of the real numbers, solutions to the regularity problem involving infinitesimals, or at least hyperreal infinitesimals, involve an unsatisfactory ineffability or arbitrariness. The arguments depend on the alleged impossibility of picking out a particular hyperreal extension of the real numbers and/or of a particular value within such an extension due to the use of the Axiom of Choice. However, it is false that the Axiom of Choice precludes a specification of a hyperreal extension—such an extension can indeed be specified. Moreover, for all we know, it is possible to explicitly specify particular infinitesimals within such an extension. Nonetheless, I prove that because any regular probability measure that has infinitesimal values can be replaced by one that has all the same intuitive features but other infinitesimal values, the heart of the arbitrariness objection remains.

Keywords

Infinitesimals Hyperreals Bayesianism Probability Regularity Set theory Axiom of Choice Saturated models Underdetermination 

Notes

References

  1. Benacerraf, P. (1962). Tasks, super-tasks, and the modern eleatics. Journal of Philosophy, 59, 765–784.CrossRefGoogle Scholar
  2. Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74, 47–73.CrossRefGoogle Scholar
  3. Benci, V., Horsten, L., & Wenmackers, S. (2013). Non-archimedean probability. Milan Journal of Mathematics, 81, 121–151.CrossRefGoogle Scholar
  4. Benci, V., Horsten, L., & Wenmackers, S. (2018). Infinitesimal probabilities. British Journal for the Philosophy of Science, 69, 509–552.Google Scholar
  5. Brickhill, H., & Horsten, L. (2016). Popper functions, lexicographical probability, and non-Archimedean probability. arXiv:1608.02850.
  6. DiBella, N. (2018). The qualitative paradox of non-conglomerability. Synthese, 195, 1181–1210.CrossRefGoogle Scholar
  7. Easwaran, K. (2014). Regularity and hyperreal credences. Philosophical Review, 123, 1–41.CrossRefGoogle Scholar
  8. Ehrlich, P. (2012). The absolute arithmetic continuum and the unification of all numbers great and small. Bulletin of Symbolic Logic, 18, 1–45.CrossRefGoogle Scholar
  9. Fitelson, B., & Hájek, A. (2017). Declarations of independence. Synthese, 194, 3979–3995.CrossRefGoogle Scholar
  10. Hájek, A. (2003). What conditional probability could not be. Synthese, 137, 273–323.CrossRefGoogle Scholar
  11. Hofweber, T. (2014). Infinitesimal chances. Philosophers’ Imprint, 14, 1–14.Google Scholar
  12. Jech, T. (1997). Set Theory (2nd ed.). Berlin: Springer.CrossRefGoogle Scholar
  13. Kanovei, V., & Shelah, S. (2004). A definable nonstandard model of the reals. Journal of Symbolic Logic, 69, 159–164.CrossRefGoogle Scholar
  14. Keisler, H. J. (1994). The hyperreal line. In P. Erlich (Ed.), Real numbers, generalizations of reals, and theories of continua (pp. 207–237). Dordrecht: Kluwer.CrossRefGoogle Scholar
  15. Keisler, H. J. (2007). Foundations of infinitesimal calculus, Online Edition. 2007. https://www.math.wisc.edu/~keisler/foundations.html. Accessed 17 Dec 2018.
  16. Krauss, P. H. (1968). Representation of conditional probability measures on Boolean algebras. Acta Mathematica Academiae Scientiarum Hungarica, 19, 229–241.CrossRefGoogle Scholar
  17. Laugwitz, D. (1968). Eine nichtarchimedische Erweiterung angeordneter Körper. Mathematische Nachrichten, 37, 225–236.CrossRefGoogle Scholar
  18. McGee, V. (1994). Learning the impossible. In E. Eells & B. Skyrms (Eds.), Probability and conditionals: Belief revision and rational decision. Cambridge: Cambridge University Press.Google Scholar
  19. Norton, J. D. (2018). How to build an infinite lottery machine. European Journal for the Philosophy of Science, 8, 71–95.CrossRefGoogle Scholar
  20. Norton, J. D., & Pruss, A. R. (2018). Correction to John D. Norton ‘How to build an infinite lottery machine’. European Journal for the Philosophy of Science, 8, 143–144.CrossRefGoogle Scholar
  21. Pedersen, A., & Paul, M. S. (2013) Strictly coherent preferences, no holds barred.Google Scholar
  22. Pincus, D., & Solovay, R. M. (1977). Definability of measures and ultrafilters. Journal of Symbolic Logic, 42, 179–190.CrossRefGoogle Scholar
  23. Pruss, A. R. (2013a). Probability, regularity, and cardinality. Philosophy of Science, 80, 231–240.CrossRefGoogle Scholar
  24. Pruss, A. R. (2013b). Null probability, dominance and rotation. Analysis, 73, 682–685.CrossRefGoogle Scholar
  25. Pruss, A. R. (2014). Infinitesimals are too small for countably infinite fair lotteries. Synthese, 191, 1051–1057.CrossRefGoogle Scholar
  26. Pruss, A. R. (2015). Popper functions, uniform distributions and infinite sequences of heads. Journal of Philosophical Logic, 44, 259–271.CrossRefGoogle Scholar
  27. Pruss, A. R. (2018). Infinity, causation and paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
  28. Robert, A. M. (2000). A course in p-adic analysis. New York: Springer.CrossRefGoogle Scholar
  29. Sacks, G. E. (2010). Saturated model theory (2nd ed.). Singapore: World Scientific Publishing.Google Scholar
  30. Shamseddine, K., & Berz, M. (2010). Analysis on the Levi–Civita field, a brief overview. Contemporary Mathematics, 508, 215–237.CrossRefGoogle Scholar
  31. Thomson, J. F. (1954). Tasks and super-tasks. Analysis, 15, 1–13.CrossRefGoogle Scholar
  32. Weintraub, R. (2008). How probable is an infinite sequence of heads? A reply to Williamson. Analysis, 68, 247–250.CrossRefGoogle Scholar
  33. Williamson, T. (2007). How probable is an infinite sequence of heads? Analysis, 67, 173–80.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Baylor UniversityWacoUSA

Personalised recommendations