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.
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,
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
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.
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.
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;
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.
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.