Motion
The nature of motion and the philosophical problems surrounding it have been perennial issues in Western philosophy. Motion is a special case of change, and much discussion relevant to motion extends naturally to change in general (see Mortensen 2002).
Notable among the problems of motion are those provided by Zeno's paradoxes. Perhaps the hardest of these is the Arrow paradox. Consider an object in motion. At any instant of that motion, since it is an instant, the object makes no advance on its journey. But if it makes no advance in any instant of its journey, how can it make advance in all of them? The sum of a collection of nothings—even an infinite collection—is nothing. It would seem that it cannot move at all.
Motion and the Calculus
Substantial progress concerning the topic of motion was made with the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the seventeenth century. The velocity of an object at time t, v(t) (with respect to a frame of reference), is given by the derivative of its spatial location, x(t), with respect to time. That is, v(t0) is dx(t)/dt, evaluated at t0. An object is in motion at an instant if its velocity at that instant is nonzero; it is at rest if its velocity is zero.
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