An Introduction to Mathematics · Alfred North Whitehead
The invention of the differential calculus
marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level. These contrasted periods in the progress of the history of thought are compared by Shelley to the formation of an avalanche.
The sun-awakened avalanche! whose mass,
Thrice sifted by the storm, had gathered there
Flake after flake,--in heaven-defying minds
As thought by thought is piled, till some great truth
Is loosened, and the nations echo round,
The comparison will bear some pressing. The final burst of sunshine which awakens the avalanche is not necessarily beyond comparison in magnitude with the other powers of nature which have presided over its slow
formation. The same is true in science. The genius who has the good fortune to produce the final idea which transforms a whole region of thought, does not necessarily excel all his predecessors who have worked at the preliminary formation of ideas. In considering the history of science, it is both silly and ungrateful to confine our admiration with a gaping wonder to those men who have made the final advances towards a new epoch.
In the particular instance before us, the
subject had a long history before it assumed its final form at the hands of its two inventors. There are some traces of its methods even among the Greek mathematicians, and finally, just before the actual production of the subject, Fermat (born 1601 A.D.,
and died 1665 A.D.), a distinguished French mathematician, had so improved on previous ideas that the subject was all but created by him. Fermat, also, may lay claim to be the joint inventor of coordinate geometry in company with his contemporary and countryman, Descartes. It was, in fact,
Descartes from whom the world of science received the new ideas, but Fermat had certainly arrived at them independently.
We need not, however, stint our admiration either for Newton or for Leibniz. Newton was a mathematician and a student of physical science, Leibniz was a mathematician
and a philosopher, and each of them in his own department of thought was one of the greatest men of genius that the world has known. The joint invention was the occasion of an unfortunate and not very creditable dispute. Newton was using the methods of Fluxions, as he called the subject,
in 1666, and employed it in the composition of his Principia, although in the work as printed any special algebraic notation is avoided. But he did not print a direct statement of his method till 1693. Leibniz published his first statement in 1684. He was accused by Newton's friends of having got it from a MS. by Newton, which he had been shown privately. Leibniz also accused Newton of having plagiarized from him. There is now not very much doubt but that both should have the credit of being independent discoverers. The subject had arrived at a stage in which it was ripe for discovery, and there is nothing surprising in the fact that two such able men should have independently hit upon it.
These joint discoveries are quite common in science. Discoveries are not in general made before they have been led up to by the previous trend of thought, and by that time many minds are in hot pursuit of the important idea. If we merely keep to discoveries in which Englishmen are
concerned, the simultaneous enunciation of the law of natural selection by Darwin and
Wallace, and the simultaneous discovery of
Neptune by Adams and the French astronomer,
Leverrier, at once occur to the mind.
The disputes, as to whom the credit ought to be given, are often influenced by an unworthy spirit of nationalism. The really inspiring reflection suggested by the history of mathematics is the unity of thought and interest among men of so many epochs, so many nations, and so many races. Indians, Egyptians, Assyrians, Greeks, Arabs, Italians, Frenchmen, Germans, Englishmen, and Russians, have all made essential contributions to the progress of the science. Assuredly the jealous exaltation of the contribution of one particular nation is not to show the larger spirit.
The importance of the differential calculus
arises from the very nature of the subject, which is the systematic consideration of the rates of increase of functions. This idea is immediately presented to us by the study of nature; velocity is the rate of increase of the distance travelled, and acceleration is the rate of increase of velocity. Thus the fundamental idea of change, which is at the basis of our whole perception of phenomena, immediately suggests the enquiry as to the rate of change. The familiar terms of "quickly" and "slowly" gain their meaning from a tacit
reference to rates of change. Thus the differential calculus is concerned with the very key of the position from which mathematics can be successfully applied to the explanation of the course of nature.
This idea of the rate of change was certainly in Newton's mind, and was embodied in the
language in which he explained the subject. It may be doubted, however, whether this point of view, derived from natural phenomena, was ever much in the minds of the preceding mathematicians who prepared the subject for its birth. They were concerned with the more abstract problems of drawing tangents
to curves, of finding the lengths of curves, and of finding the areas enclosed by curves. The
last two problems, of the rectification of curves and the quadrature of curves as they are named, belong to the Integral Calculus, which
is however involved in the same general subject as the Differential Calculus.
The introduction of coordinate geometry
makes the two points of view coalesce. For (cf. fig. 32) let AQP be any curved line and let PT be the tangent at the point P on it. Let the axes of coordinates be OX and OY ; and let y = f(x) be the equation to the curve, so that OM = x , and PM = y. Now let Q be any moving point on the curve, with coordinates x_1 , y_1 ; then y_1 = f(x_1). And let Q' be the point on the tangent with the same abscissa x_1 ; suppose that the coordinates of Q' are x_1 and y'. Now suppose that N moves along the axis OX from left to right with a uniform velocity; then it is easy to see that the ordinate y' of the point Q' on the tangent TP also increases uniformly as Q' moves along the tangent in a corresponding way. In fact it is easy to see that the ratio of the rate of increase of Q'N to the rate of increase of ON is in the ratio of Q'N to TN , which is the same at all points of the straight line. But the rate of increase of QN , which is the rate of increase of f(x_1) , varies from point to point of the curve so long as it is not straight. As Q passes through the point P , the rate of increase of f(x_1) (where x_1 coincides with x for the moment)
is the same as the rate of increase of y' on the tangent at P. Hence, if we have a general method of determining the rate of increase of a function f(x) of a variable x , we can determine the slope of the tangent at any point (x, y,) on a curve, and thence can draw it. Thus the problems of drawing tangents to a curve, and of determining the rates of increase of a function are really identical.
It will be noticed that, as in the cases of Conic Sections and Trigonometry, the more artificial of the two points of view is the one in which the subject took its rise. The really fundamental aspect of the science only rose into prominence comparatively late in the day. It is a well-founded historical generalization, that the last thing to be discovered in any science is what the science is really about. Men go on groping for centuries, guided merely by a dim instinct and a puzzled curiosity, till at last "some great truth is loosened."
Let us take some special cases in order to familiarize ourselves with the sort of ideas which we want to make precise. A train is in motion--how shall we determine its velocity at some instant, let us say, at noon? We can take an interval of five minutes which includes noon, and measure how far the train has gone in that period. Suppose we find it to be five
miles, we may then conclude that the train was running at the rate of 60 miles per hour. But five miles is a long distance, and we cannot be sure that just at noon the train was moving at this pace. At noon it may have been running 70 miles per hour, and afterwards the brake may have been put on. It will be safer to work with a smaller interval, say one minute, which includes noon, and to measure the space traversed during that period. But for some purposes greater accuracy may be required, and one minute may be too long. In practice, the necessary inaccuracy of our measurements makes it useless to take too small a period for measurement. But in theory the smaller the period the better, and we are tempted to say that for ideal accuracy an infinitely small period is required. The older mathematicians, in particular Leibniz, were not only tempted, but yielded to the temptation, and did say it. Even now it is a useful fashion of speech, provided that we know how to interpret it into the language of common sense. It is curious that, in his exposition of the foundations of the calculus, Newton, the natural scientist, is much more philosophical than Leibniz, the philosopher, and on the other hand, Leibniz provided the admirable notation which has been so essential for the progress of the subject.
Now take another example within the region of pure mathematics. Let us proceed to find the rate of increase of the function x^2 for any value x of its argument. We have not yet really defined what we mean by rate of increase. We will try and grasp its meaning in relation to this particular case. When x increases to x + h , the function x^2 increases to (x + h)^2 ; so that the total increase has been (x + h)^2 - x^2 , due to an increase h in the argument. Hence throughout the interval x to (x + h) the average increase of the function per unit increase of the argument is (x + h)^2 - x^2/h. But (x + h)^2 = x^2 + 2hx + h^2, and therefore (x + h)^2 - x^2/h = 2hx + h^2/h = 2x + h. Thus 2x + h is the average increase of the function x^2 per unit increase in the argument, the average being taken over by the interval x to x + h. But 2x + h depends on h , the size of the interval. We shall evidently get what we want, namely the rate of increase at the value x of the argument, by diminishing h more and more. Hence in the limit when h has
decreased indefinitely, we say that 2x is the rate of increase of x^2 at the value x of the argument.
Here again we are apparently driven up against the idea of infinitely small quantities in the use of the words "in the limit when h has decreased indefinitely." Leibniz held that, mysterious as it may sound, there were actually existing such things as infinitely small quantities, and of course infinitely small numbers corresponding to them. Newton's language and ideas were more on the modern lines; but he did not succeed in explaining the matter with such explicitness so as to be evidently doing more than explain Leibniz's ideas in rather indirect language. The real explanation of the subject was first given by Weierstrass and the Berlin School of mathematicians
about the middle of the nineteenth century. But between Leibniz and Weierstrass a copious literature, both mathematical and philosophical, had grown up round these mysterious infinitely small quantities which mathematics had discovered and philosophy proceeded to explain. Some philosophers,
Bishop Berkeley, for instance, correctly denied the validity of the whole idea, though for reasons other than those indicated here. But the curious fact remained that, despite all criticisms of the foundations of the subject, there could be no doubt but that the mathematical
procedure was substantially right. In fact, the subject was right, though the explanations were wrong. It is this possibility of being right, albeit with entirely wrong explanations as to what is being done, that so often makes external criticism--that is so far as it is meant to stop the pursuit of a method--singularly barren and futile in the progress of science. The instinct of trained observers, and their sense of curiosity, due to the fact that they are obviously getting at something, are far safer guides. Anyhow the general effect of the success of the Differential Calculus was to generate a large amount of bad philosophy, centring round the idea of the infinitely small. The relics of this verbiage may still be found in the explanations of many elementary mathematical text-books on the Differential Calculus. It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.
Newton would have phrased the question
by saying that, as h approaches zero, in the limit 2x + h becomes 2x. It is our task so to explain this statement as to show that it does not in reality covertly assume the existence of Leibniz's infinitely small quantities. In reading over the Newtonian method of statement, it is tempting to seek simplicity by
saying that 2x + h is 2x , when h is zero. But this will not do; for it thereby abolishes the interval from x to x + h , over which the average increase was calculated. The problem is, how to keep an interval of length h over which to calculate the average increase, and at the same time to treat h as if it were zero. Newton did this by the conception of a limit, and we now
proceed to give Weierstrass's explanation of its real meaning.
In the first place notice that, in discussing 2x + h , we have been considering x as fixed in value and h as varying. In other words x has been treated as a "constant" variable, or parameter, as explained in Chapter IX.; and we have really been considering 2x + h as a function of the argument h. Hence we can generalize the question on hand, and ask what we mean by saying that the function f(h) tends to the limit l , say, as its argument h tends to the value zero. But again we shall see that the special value zero for the argument does not belong to the essence of the subject; and again we generalize still further, and ask, what we mean by saying that the function f(h) tends to the limit l as h tends to the value a.
Now, according to the Weierstrassian explanation the whole idea of h tending to the value a , though it gives a sort of metaphorical picture of what we are driving at, is really off the point entirely. Indeed it is fairly obvious
that, as long as we retain anything like " h tending to a ," as a fundamental idea, we are really in the clutches of the infinitely small; for we imply the notion of h being infinitely near to a. This is just what we want to get rid of.
Accordingly, we shall yet again restate our phrase to be explained, and ask what we mean by saying that the limit of the function f(h) at a is l.
The limit of f(h) at a is a property of the
neighbourhood of a , where "neighbourhood" is used in the sense defined in Chapter XI. during the discussion of the continuity of functions. The value of the function f(h) at a is f(a) ; but the limit is distinct in idea from the value, and may be different from it, and may exist when the value has not been defined. We shall also use the term "standard of approximation" in the sense in which it is defined in Chapter XI. In fact, in the definition of "continuity" given towards the end of that chapter we have practically defined a limit. The definition of a limit is:--
A function f(x) has the limit l at a value a of its argument x , when in the neighbourhood of a its values approximate to l within every standard of approximation.
Compare this definition with that already given for continuity, namely:--
A function f(x) is continuous at a value a of its argument, when in the neighbourhood of a its values approximate to its value at a within every standard of approximation.
It is at once evident that a function is continuous at a when (i) it possesses a limit at a , and (ii) that limit is equal to its value at a. Thus the illustrations of continuity which have been given at the end of Chapter XI. are illustrations of the idea of a limit, namely, they were all directed to proving that f(a) was the limit of f(x) at a for the functions considered and the value of a considered. It is really more instructive to consider the limit at a point where a function is not continuous. For example, consider the function of which the graph is given in fig. 20 of Chapter XI. This function f(x) is defined to have the value 1 for all values of the argument except the integers 0 , 1 , 2 , 3 , etc., and for these integral values it has the value 0. Now let us think of its limit when x = 3. We notice that in the definition of the limit the value of the function at a (in this case, a = 3 ) is excluded. But, excluding f(3) , the values of f(x) , when x lies within any interval which (i) contains 3 not as an end-point, and (ii) does not extend so far as 2 and 4 , are all equal to 1 ; and hence these values approximate to 1 within every standard of approximation. Hence 1 is the limit of f(x) at the
value 3 of the argument x , but by definition f(3) = 0.
This is an instance of a function which possesses both a value and a limit at the value 3 of the argument, but the value is not equal to the limit. At the end of Chapter XI. the function x^2 was considered at the value 2 of the argument. Its value at 2 is 2^2 , i.e. 4 , and it was proved that its limit is also 4. Thus here we have a function with a value and a limit which are equal.
Finally we come to the case which is essentially important for our purposes, namely, to a function which possesses a limit, but no defined value at a certain value of its argument. We need not go far to look for such a function, 2x/x will serve our purpose. Now in any mathematical book, we might find the equation, 2x/x = 2 , written without hesitation or comment. But there is a difficulty in this; for when x is zero, 2x/x = 0/0 ; and 0/0 has no defined meaning. Thus the value of the function 2x/x at x = 0 has no defined
meaning. But for every other value of x , the value of the function 2x/x is 2. Thus the limit of 2x/x at x = 0 is 2 , and it has no value at x = 0. Similarly the limit of x^2/x at x = a is a whatever a may be, so that the limit of x^2/x at x = 0 is 0. But the value of x^2/x at x = 0 takes the form 0/0 , which has no defined meaning. Thus the function x^2/x has a limit but no value at 0.
We now come back to the problem from which we started this discussion on the nature of a limit. How are we going to define the rate of increase of the function x^2 at any value x of its argument. Our answer is that this rate of increase is the limit of the function (x + h)^2 - x^2/h at the value zero for its argument h. (Note that x is here a "constant.") Let us see how this answer works
in the light of our definition of a limit. We have (x + h)^2 - x^2/h = 2hx + h^2/h = h(2x + h)/h.
Now in finding the limit of h(2x + h)/h at the value 0 of the argument h , the value (if any) of the function at h = 0 is excluded. But for all values of h , except h = 0 , we can divide through by h. Thus the limit of h(2x + h)/h at h = 0 is the same as that of 2x + h at h = 0. Now, whatever standard of approximation k we choose to take, by considering the interval from -1/2k to +1/2k we see that, for values of h which fall within it, 2x + h differs from 2x by less than 1/2k , that is by less than k. This is true for any standard k. Hence in the neighbourhood of the value 0 for h , 2x + h approximates to 2x within every standard of approximation, and therefore 2x is the limit of 2x + h at h = 0. Hence by what has been said above 2x is the limit of (x + h)^2 - x^2/h at the value 0 for h. It follows, therefore, that 2x is what we have called the rate of increase of x^2 at the value x of the argument. Thus this method conducts us to the same rate of increase
for x^2 as did the Leibnizian way of making h grow "infinitely small."
The more abstract terms "differential coefficient,"
or "derived function," are generally
used for what we have hitherto called the "rate of increase" of a function. The general definition is as follows: the differential coefficient of the function f(x) is the limit, if it exist, of the function f(x + h) - f(x)/h of the argument h at the value 0 of its argument.
How have we, by this definition and the subsidiary definition of a limit, really managed to avoid the notion of "infinitely small numbers" which so worried our mathematical forefathers? For them the difficulty arose because on the one hand they had to use an interval x to x + h over which to calculate the average increase, and, on the other hand, they finally wanted to put h = 0. The result was they seemed to be landed into the notion of an existent interval of zero size. Now how do we avoid this difficulty? In this way--we use the notion that corresponding to any standard of approximation, some interval with such and such properties can be found. The difference is that we have
grasped the importance of the notion of "the variable," and they had not done so. Thus,
at the end of our exposition of the essential notions of mathematical analysis, we are led back to the ideas with which in Chapter II. we commenced our enquiry--that in mathematics the fundamentally important ideas are those of "some things" and "any things."
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End of part 15 of 19
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