I find the case of Alan Turing's development of the concept of computatibility to be an example. Before Turing, the logicians had no clear concept of what it means to say that a function is computable. Even Gödel had despaired to ever have a formal notion of computability, because he had expected that for any such formal notion of computability, we would be able to diagonalize against it and thereby find a function that was computable in an intuitive sense, but not with respect to the formal notion. This was true of the class of primitive recursive functions and other extensions of that idea.
Meanwhile, Turing proceeded on a mainly philosophical level to consider what it was that a human did when undertaking a computation, imagining a person with paper and pencil and plenty of time, following a rote computational procedure, and was thereby led to his notion of Turing machine, which led to the fields of computability theory, complexity theory and so on.