Hacking's distinction between inference to the truth of hypotheses and the classical statistician's evaluation of procedures shows these are fundamentally incompatible epistemic projects.

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Classical statistician(the alternative approach to statistics mentioned in the statement): A person who uses the traditional approach to statistics, which focuses on testing whether procedures and methods work reliably rather than asking whether specific theories are true.
Epistemic projects(what the statement says these two statistical approaches are): Different goals or approaches people use when trying to gain knowledge and understand how we know things.
Evaluation of procedures(what classical statisticians focus on): Judging whether a method or technique gives consistent and reliable results, without necessarily claiming that any particular explanation is correct.
Fundamentally incompatible(describing the relationship between the two statistical approaches): So different in their basic nature that they cannot work together or be made to fit with each other.
Hacking(as the philosopher referenced in this statement): Ian Hacking is a philosopher who studies how science works in practice. He's famous for arguing that what makes science real and trustworthy isn't just whether we can see something directly, but whether we can actually use it to do things.
Inference to the truth of hypotheses(one approach to statistics discussed in the statement): A way of reasoning where you look at evidence and decide which explanation is most likely to be true—like being a detective who picks the best theory based on clues.

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