Mathematical Analysis
The Standard Solution to THE ARROW PARADOX requires the reasoning to use our contemporary theory of speed from calculus. This theory defines instantaneous motion, that is, motion at an instant, without defining motion during an instant. This new treatment of motion originated with Newton and Leibniz in the sixteenth century, and it employs what is called the "AT-AT" theory of motion, which says motion is being at different places at different times. Motion isn't some feature that reveals itself only within a moment.The modern difference between rest and motion, as opposed to the difference in antiquity, has to do with what is happening at nearby moments and- contra ZENO - has nothing to do with what is happening during a moment.
The Standard Solution to THE ARROW PARADOX requires the reasoning to use our contemporary theory of speed from calculus. This theory defines instantaneous motion, that is, motion at an instant, without defining motion during an instant. This new treatment of motion originated with Newton and Leibniz in the sixteenth century, and it employs what is called the "AT-AT" theory of motion, which says motion is being at different places at different times. Motion isn't some feature that reveals itself only within a moment.The modern difference between rest and motion, as opposed to the difference in antiquity, has to do with what is happening at nearby moments and- contra ZENO - has nothing to do with what is happening during a moment.
Instantaneous Velocity :
Instantaneous Velocity :
Instantaneous Velocity is the derivative of the displacement with respect to time at a particular instant. :
Instantaneous Velocity is the derivative of the displacement with respect to time at a particular instant. :
An initial reaction to the paradox of the Arrow might be the suspicion that it hinges on a confusion between the concepts of instantaneous motion and instantaneous rest, Perhaps Zeno did feel that the only way for an arrow to be at a particular place was to be at rest — that the notion of instantaneous non-zero velocity was illegitimate. If Zeno argued — we have no way of knowing whether he did or not — that at every moment of its flight the arrow is at some place in its trajectory, and hence at every moment of its flight it has velocity zero, then he would have been correct in concluding that its velocity during the whole course of its flight would be ZERO, rendering the arrow motionless. Nineteenth-century mathematics showed, however, that one of these assumptions is incorrect. It is entirely intelligible to attribute non-zero instantaneous velocities to moving objects when an instantaneous velocity is understood as a derivative — namely, the rate of change of position with respect to time. This derivative is defined as the limit of the average velocity during decreasing non-zero intervals of time. Suppose, for example, that the arrow flies at a uniform speed. We find that in one second it covers ten feet, in one-tenth of a second it covers one foot, in one-hundredth of a second it covers one-tenth of a foot, and so on. As we take these average velocities over decreasing finite time intervals which converge to an instant t_1, the average velocities approach a limit of ten feet per second, and this is, by definition, the instantaneous velocity of the arrow at t_1. The same can be said for every moment during its flight; it travels its whole course at ten feet per second, and its velocity at each moment is ten feet per second. If Zeno felt that the only intelligible instantaneous velocity is zero, nineteenth-century mathematics proved him wrong.
An initial reaction to the paradox of the Arrow might be the suspicion that it hinges on a confusion between the concepts of instantaneous motion and instantaneous rest, Perhaps Zeno did feel that the only way for an arrow to be at a particular place was to be at rest — that the notion of instantaneous non-zero velocity was illegitimate. If Zeno argued — we have no way of knowing whether he did or not — that at every moment of its flight the arrow is at some place in its trajectory, and hence at every moment of its flight it has velocity zero, then he would have been correct in concluding that its velocity during the whole course of its flight would be ZERO, rendering the arrow motionless. Nineteenth-century mathematics showed, however, that one of these assumptions is incorrect. It is entirely intelligible to attribute non-zero instantaneous velocities to moving objects when an instantaneous velocity is understood as a derivative — namely, the rate of change of position with respect to time. This derivative is defined as the limit of the average velocity during decreasing non-zero intervals of time. Suppose, for example, that the arrow flies at a uniform speed. We find that in one second it covers ten feet, in one-tenth of a second it covers one foot, in one-hundredth of a second it covers one-tenth of a foot, and so on. As we take these average velocities over decreasing finite time intervals which converge to an instant t_1, the average velocities approach a limit of ten feet per second, and this is, by definition, the instantaneous velocity of the arrow at t_1. The same can be said for every moment during its flight; it travels its whole course at ten feet per second, and its velocity at each moment is ten feet per second. If Zeno felt that the only intelligible instantaneous velocity is zero, nineteenth-century mathematics proved him wrong.
The INFINITESIMAL CALCULUS made use of instantaneous velocities. These were, unfortunately, considered to be infinitesimal distances covered in infinitesimal times.It is possible that Zeno's Arrow paradox was also directed against just such a conception. If we try to conceive of finite motion over a finite distance during a finite time as being composed of a large number of motions over infinitesimal distances during infinitesimal times, enormous confusion is likely to ensure. How much space does an arrow occupy during an infinitesimal time? Is it just as large as the arrow, or is it a wee big larger? If it is larger, then how does the arrow get from one part of that space to another? And if not, then how can the arrow be moving at all? And how long is an infinitesimal time span? Does it have parts or not? If so, how can we characterize motion during its parts? If not, how can motion occur during this infinitesimal time? These are questions that Zeno and his fellow Greeks could not answer, and to which modern calculus prior to Cauchy had no satisfactory answer either. This is why I remarked earlier that nineteenth-century — not seventeenth-century — mathematics held an important key, in the concept of the derivative, to the resolution of Zeno's Arrow paradox.
The INFINITESIMAL CALCULUS made use of instantaneous velocities. These were, unfortunately, considered to be infinitesimal distances covered in infinitesimal times.It is possible that Zeno's Arrow paradox was also directed against just such a conception. If we try to conceive of finite motion over a finite distance during a finite time as being composed of a large number of motions over infinitesimal distances during infinitesimal times, enormous confusion is likely to ensure. How much space does an arrow occupy during an infinitesimal time? Is it just as large as the arrow, or is it a wee big larger? If it is larger, then how does the arrow get from one part of that space to another? And if not, then how can the arrow be moving at all? And how long is an infinitesimal time span? Does it have parts or not? If so, how can we characterize motion during its parts? If not, how can motion occur during this infinitesimal time? These are questions that Zeno and his fellow Greeks could not answer, and to which modern calculus prior to Cauchy had no satisfactory answer either. This is why I remarked earlier that nineteenth-century — not seventeenth-century — mathematics held an important key, in the concept of the derivative, to the resolution of Zeno's Arrow paradox.