foundations, analysis, and probability

[Mikhail Katz]

Dear FoMers,

We just posted on the arxiv a pair of particles due to appear in
Philosophia Mathematica; see https://arxiv.org/abs/2008.11509 and
https://arxiv.org/abs/2008.11513 The articles situate themselves at
the crossroads of Foundations, Analysis, and Probability theory.

We analyze recent criticisms of the use of hyperreal probabilities as
expressed by Pruss, Easwaran, Parker, and Williamson.  We show that
the alleged arbitrariness of hyperreal fields can be avoided by
working in the Kanovei-Shelah model or in saturated models.  We argue
that some of the objections to hyperreal probabilities arise from
hidden biases that favor Archimedean models.  We discuss the advantage
of the hyperreals over transferless fields with infinitesimals.  We
analyze two underdetermination theorems by Pruss and show that they
hinge upon parasitic external hyperreal-valued measures, whereas
internal hyperfinite measures are not underdetermined.  A probability
model is underdetermined when there is no rational reason to assign a
particular infinitesimal value as the probability of single events.
Pruss claims that hyperreal probabilities are underdetermined.  The
claim is based upon external hyperreal-valued measures.  We show that
internal hyperfinite measures are not underdetermined. The importance
of internality stems from the fact that Robinson's transfer principle
only applies to internal entities.  We also evaluate the claim that
transferless ordered fields (surreals, Levi-Civita field, Laurent
series) may have advantages over hyperreals in probabilistic modeling.
We show that probabilities developed over such fields are less
expressive than hyperreal probabilities.

Looking forward to your comments.

M. Katz