Many philosophers have discussed the ability of thinkers to think thoughts that the thinker cannot justify because the thoughts involve concepts that the thinker incompletely understands. A standard example of this phenomenon involves the concept of the derivative in the early days of the calculus: Newton and Leibniz incompletely understood the derivative concept and, hence, as Berkeley noted, they could not justify their thoughts involving it. Later, Weierstrass justified their thoughts by giving a correct explication of the derivative concept. This paper discusses various accounts of how a thinker manages to think with a concept that they incompletely understand and finds them wanting in the case at hand. Part of the overlooked complexity is that there are many derivative concepts, and it is unclear in virtue of what a thinker would be thinking with one of them rather than another. After critical evaluation of standard accounts, this paper suggests a novel account of how one should think about the derivative concepts with which Newton and Leibniz thought and how Weierstrass could have managed to justify their thoughts even if their thoughts did not involve the same derivative concept as Weierstrass’s.

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