The instrumentalist position is a pragmatic one. It says that probability is simply a mathematical tool that can help us navigate the world. As stated by Freedman and Stark in What is the Chance of an Earthquake?:
Probability is just a property of a mathematical model intended to describe some features of the natural world. For the model to be useful, it must be shown to be in good correspondence with the system it describes.
This denies that there is a true nature or essence, of probability. It is not fundamentally about limiting frequencies or subjective belief. It’s about usefulness in various problems.
Instrumentalism fits well in a deterministic universe. Tossing a coin is a deterministic procedure: whether it lands heads or tails is determined by the initial conditions of the environment. But we model toin cossing as a Bernoulli with bias 0.5 because it’s useful in many applications.
The instrumentalist/pragmatist position is arguably that used by most practitioners of statistics. Nobody thinks the Mercer kernel specifying the Gaussian prior in Gaussian process regression encodes actual beliefs. We use priors because they are mathematically convenient and lead to good results.
Despite its (implicit) popularity, there doesn’t seem to be much literature on the instrumentalist theory of probability. I’m taking the name instrumentalist view from Vaden Masrani’s blog post. See also Ben Recht discussing A Dappled View of Probability.