INTRODUCTION

The ARROW PARADOX is a part of paradoxes set forth by Zeno Of Elea. Zeno was nearly 40 years old when Socrates was a young man, say 20. Since Socrates was born in 469 BC we can estimate a birth date for Zeno around 490 BC. Of all his work of paradoxes the ARROW PARADOX is considered to be complex. Well understanding and solving the more complex problems of universe is what mathematics aim at.

ZENO argues that an arrow in flight is always at rest. At any given instant, he claims, the arrow is where it is, occupying a portion of space equal to itself. During the instant it cannot move, for that would require the instant to have parts, and an instant is by definition a minimal and indivisible element of time. If the arrow did move during the instant it would have to be in one place at one part of the instant, and in a different place at another part of the instant. Moreover, for the arrow to move during the instant would require that during the instant it must occupy a space larger than itself, for otherwise it has no room to move. As Zeno says, "It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever".

ANALYTICAL SOLUTIONS

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 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.

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.

MATHEMATICAL FUNCTIONS

There is, however, still an underlying problem about instantaneous velocity. We have seen how such a concept can be defined intelligibly, but this definition makes essential reference to what is happening at neighboring instants. Instantaneous velocity is defined as a limit of a sequence of average velocities over finite time intervals; without some information about what happens in these intervals we can say nothing about the instantaneous velocity. If we know simply that the center of the arrow was at the point s_1 at time t_1 we can draw no conclusion whatever about its velocity at that instant. Unless we know what the arrow was doing at other times close to t_1 we cannot distinguish instantaneous motion from instantaneous rest.

In modern physics, motion is treated as a functional relationship between points of space and instants of time. The formula for the motion of a freely falling body, for example,

Such formulas make it possible, by employing the function f, to compute the position x given a value of time t. But to understand this treatment of motion fully, it is necessary to have a clear conception of mathematical functions. Before the nineteenth century there was no satisfactory treatment of functions; functions were widely regarded as things which moved or flowed. Such a conception is of no help in attempting to resolve Zeno's paradoxes; on the contrary, Zeno's paradoxes of motion constitute severe difficulties for any such notion of mathematical functions. The situation was dramatically improved when Cauchy defined a function as simply a pairing of numbers from one set with numbers from another set. The numbers of the first set are the values of the argument, sometimes called the independent variable; the numbers of the second set (which need not be a different set) are the values of the function, sometimes called the dependent variable. For example, the function

pairs real numbers with non-negative real numbers. With the number 2 it associates the number 4, with the number -1 it associates the number 1, with the number 1/2 it associates the number 1/4, and so forth.

Now according to Cauchy, the mathematical function F simply is the set of all such pairs of numbers.

Let us now apply this conception of a mathematical function to the motion of an arrow;

To keep the arithmetic simple, let it travel at the uniform speed of ten feet per second in a straight line, starting from x = 0 at t = 0. At any subsequent time t, its position x = 10t. Accordingly, part of what we mean by saying that the arrow moved from point A (x = 10) to point B (x = 30) is simply that it was at A when t = 1, and it was at B when t = 3. When we ask how it got from A to B, the answer is that it occupied each of the intervening points x (10 < x < 30) at suitable times t (1 < t < 3) — that is, satisfying the equation x = 10t. For example, when t = 2, the arrow was at the point C (x = 20). When we ask how it got from A to C, the answer is again: by occupying the intervening positions at suitable times. Notice that this answer is not: by zipping through the intervening points at ten feet per second. The requirement is that the arrow be at the appropriate point at the appropriate time — nothing is said about the instantaneous velocity of the arrow as it occupies each of these points. This approach has been appropriately dubbed "the at-at theory of motion". Once the motion has been described by a mathematical function that associates positions with times, it is then possible to differentiate the function and find its derivative, which in turn provides the instantaneous velocities for each moment of travel. But the motion itself is described by the pairing of positions with times alone. Thus, Russell was led to remark, "Weierstrass, by strictly banishing all infinitesimals, has at last shown that we live in an unchanging world, and that the arrow, at every moment of its flight, is truly at rest. The only point where Zeno probably erred was in inferring (if he did infer) that, because there is no change, therefore the world must be in the same state at one time as at another.

IMPLICATION

Solving the arrow paradox makes us equipped with the mathematical tools to solve various problems in today's social life. The Arrow Paradox proposed by Zeno is solved using the mathematical functions of instantaneous velocity there by solving any situations leading to any concern about the concepts of instantaneous rest or motion.

With enough tools from the Nineteenth century mathematics one can be ready to face the problems of basic object motions in physical world.

CONCLUSION

Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.

This assignment has been very helpful in bringing a clear understanding on the various ways how calculus is used in real life. I would like to take this opportunity to thank my calculus prof. Manish K. Gupta who not only assigned the project but also had been very supportive right to the end.

REFERENCES

After the relevant entries in this website, the place to begin any further investigation is Salmon (2001), which contains some of the most important articles on Zeno up to 1970, and an impressively comprehensive bibliography of works in English in the Twentieth Century. Few of them include :

1) Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 218-220, 1998.

2) Aristotle's Physics "Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye

3) Huggett, Nick (2010). "Zeno's Paradoxes: 3.3 The Arrow". Stanford Encyclopedia of Philosophy.