Regarding calculus, it is possible to accept infinitesimals and thus view dx/dy as meaningful in an absolute sense, and not just as the outcome of a limit process. This is what is done in nonstandard analysis, in which infinitesimals are the reciprocals of superreal numbers that are infinite (although not equal to or equivalent to the infinite cardinals). This is in fact how both Newton and Leibniz thought of the matter. For Leibniz, a monad was a point surrounded by infinitesimals.
However, while infinitesimals are smaller than any positive real number, they are not equal to zero, strictly speaking. Therefore, they are irrelevant to the discussion of the steps in the Heinlein exercise, which is dealing with the actual zero. Indeed, this exercise is a reminder as to why dividing by zero is ruled out. Allowing it allows absurdities such as this exercise. 0⁄0 can be anything.
My late father was once asked by a young woman in the audience at one of his public lectures, “Is zero a real number?” He replied, “one of the finests, my dear, one of the finest.”
Regarding calculus, it is possible to accept infinitesimals and thus view dx/dy as meaningful in an absolute sense, and not just as the outcome of a limit process. This is what is done in nonstandard analysis, in which infinitesimals are the reciprocals of superreal numbers that are infinite (although not equal to or equivalent to the infinite cardinals). This is in fact how both Newton and Leibniz thought of the matter. For Leibniz, a monad was a point surrounded by infinitesimals.
However, while infinitesimals are smaller than any positive real number, they are not equal to zero, strictly speaking. Therefore, they are irrelevant to the discussion of the steps in the Heinlein exercise, which is dealing with the actual zero. Indeed, this exercise is a reminder as to why dividing by zero is ruled out. Allowing it allows absurdities such as this exercise. 0⁄0 can be anything.
My late father was once asked by a young woman in the audience at one of his public lectures, “Is zero a real number?” He replied, “one of the finests, my dear, one of the finest.”