In the exchange over Dennis' wonderful post on infinitesimals, Dennis writes, "Casting doubt on someone’s pronouncements is very far from devising a consistent theory to show them false (consistent because it has models in the category of sets). A mathematician would regard that as the difference between gilt and gold." I call Dennis' move here (and it is one that Russell also was frequently attracted to), an instance of "Newton's Challenge to Philosophy." That is, a philosopher appeals to natural science (or mathematics) to settle a dispute within philosophy. Let me grant Dennis' claim about the "mathematician." But within philosophy burden-shifting is no small achievement. Note, in particular, that Russell appeals to mathematics in order to condemn Leibniz's wrong turn to "speculation." That is to say, it is one thing to get the math wrong or to be unable to provide a mathematical proof for a claim within mathematics. It is another thing to make a claim to the effect that metaphysics of mathematics has been settled. I suspect it was inevitable that Russell would be wrong about the latter.
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