  Note: Published now as a chapter in "Handbook of the History and Philosophy
of Mathematical Practice" (Springer Nature, editor B. Sriraman,
https://doi.org/10.1007/978-3-030-19071-2_105-1).
  The application of mathematical probability theory in statistics is quite
controversial. Controversies regard both the interpretation of probability, and
approaches to statistical inference. After having given an overview of the main
approaches, I will propose a re-interpretation of frequentist probability. Most
statisticians are aware that probability models interpreted in a frequentist
manner are not really true in objective reality, but only idealisations. I
argue that this is often ignored when actually applying frequentist methods and
interpreting the results, and that keeping up the awareness for the essential
difference between reality and models can lead to a more appropriate use and
interpretation of frequentist models and methods, called
"frequentism-as-model". This is elaborated showing connections to existing
work, appreciating the special role of independently and identically
distributed observations and subject matter knowledge, giving an account of how
and under what conditions models that are not true can be useful, giving
detailed interpretations of tests and confidence intervals, confronting their
implicit compatibility logic with the inverse probability logic of Bayesian
inference, re-interpreting the role of model assumptions, appreciating
robustness, and the role of "interpretative equivalence" of models. Epistemic
probability shares the issue that its models are only idealisations, and an
analogous "epistemic-probability-as-model" can also be developed.
