Introduction to Mathematical Philosophy

IN this chapter we shall be concerned with the definition of the limit of a function (if any) as the argument approaches a given value, and also with the definition of what is meant by a "continuous function." Both of these ideas are somewhat technical, and would hardly demand treatment in a mere introduction to mathematical philosophy but for the fact that, especially through the so-called infinitesimal calculus, wrong views upon our present topics have become so firmly embedded in the minds of professional philosophers that a prolonged and considerable effort is required for their uprooting. It has been thought ever since the time of Leibniz that the differential and integral calculus required infinitesimal quantities. Mathematicians (especially Weierstrass) proved that this is an error; but errors incorporated, e.g. in what Hegel has to say about mathematics, die hard, and philosophers have tended to ignore the work of such men as Weierstrass. Limits and continuity of functions, in works on ordinary mathematics, are defined in terms involving number. This is not essential, as Dr Whitehead has shown.[22] We will, however, begin with the definitions in the text-books, and proceed afterwards to show how these definitions can be generalised so as to apply to series in general, and not only to such as are numerical or numerically measurable. [22]See Principia Mathematica, vol. II. * 230-234. Let us consider any ordinary mathematical function , where [Pg 107] and are both real numbers, and is one-valued—i.e. when is given, there is only one value that can have. We call the "argument," and the "value for the argument ." When a function is what we call "continuous," the rough idea for which we are seeking a precise definition is that small differences in shall correspond to small differences in , and if we make the differences in small enough, we can make the differences in fall below any assigned amount. We do not want, if a function is to be continuous, that there shall be sudden jumps, so that, for some value of , any change, however small, will make a change in which exceeds some assigned finite amount. The ordinary simple functions of mathematics have this property: it belongs, for example, to , , ... , , and so on. But it is not at all difficult to define discontinuous functions. Take, as a non-mathematical example, "the place of birth of the youngest person living at time ." This is a function of ; its value is constant from the time of one person's birth to the time of the next birth, and then the value changes suddenly from one birthplace to the other. An analogous mathematical example would be "the integer next below ," where is a real number. This function remains constant from one integer to the next, and then gives a sudden jump. The actual fact is that, though continuous functions are more familiar, they are the exceptions: there are infinitely more discontinuous functions than continuous ones. Many functions are discontinuous for one or several values of the variable, but continuous for all other values. Take as an example . The function passes through all values from -1 to 1 every time that passes from to , or from to , or generally from to , where is any integer. Now if we consider when is very small, we see that as diminishes grows faster and faster, so that it passes more and more quickly through the cycle of values from one multiple of to another as becomes smaller and smaller. Consequently passes more and more quickly from -1 [Pg 108] to 1 and back again, as grows smaller. In fact, if we take any interval containing 0, say the interval from to where is some very small number, will go through an infinite number of oscillations in this interval, and we cannot diminish the oscillations by making the interval smaller. Thus round about the argument 0 the function is discontinuous. It is easy to manufacture functions which are discontinuous in several places, or in places, or everywhere. Examples will be found in any book on the theory of functions of a real variable. Proceeding now to seek a precise definition of what is meant by saying that a function is continuous for a given argument, when argument and value are both real numbers, let us first define a "neighbourhood" of a number as all the numbers from to , where is some number which, in important cases, will be very small. It is clear that continuity at a given point has to do with what happens in any neighbourhood of that point, however small. What we desire is this: If is the argument for which we wish our function to be continuous, let us first define a neighbourhood ( say) containing the value which the function has for the argument ; we desire that, if we take a sufficiently small neighbourhood containing , all values for arguments throughout this neighbourhood shall be contained in the neighbourhood , no matter how small we may have made . That is to say, if we decree that our function is not to differ from by more than some very tiny amount, we can always find a stretch of real numbers, having in the middle of it, such that throughout this stretch will not differ from by more than the prescribed tiny amount. And this is to remain true whatever tiny amount we may select. Hence we are led to the following definition:— The function is said to be "continuous" for the argument if, for every positive number , different from 0, but as small as we please, there exists a positive number , different from 0, such that, for all values of which are numerically [Pg 109] less[23] than , the difference is numerically less than . [23]A number is said to be "numerically less" than when it lies between and . In this definition, first defines a neighbourhood of , namely, the neighbourhood from to . The definition then proceeds to say that we can (by means of define a neighbourhood, namely, that from to , such that, for all arguments within this neighbourhood, the value of the function lies within the neighbourhood horn to . If this can be done, however may be chosen, the function is "continuous" for the argument . So far we have not defined the "limit" of a function for a given argument. If we had done so, we could have defined the continuity of a function differently: a function is continuous at a point where its value is the same as the limit of its value for approaches either from above or from below. But it is only the exceptionally "tame" function that has a definite limit as the argument approaches a given point. The general rule is that a function oscillates, and that, given any neighbourhood of a given argument, however small, a whole stretch of values will occur for arguments within this neighbourhood. As this is the general rule, let us consider it first. Let us consider what may happen as the argument approaches some value from below. That is to say, we wish to consider what happens for arguments contained in the interval from to , where is some number which, in important cases, will be very small. The values of the function for arguments from to ( excluded) will be a set of real numbers which will define a certain section of the set of real numbers, namely, the section consisting of those numbers that are not greater than all the values for arguments from to . Given any number in this section, there are values at least as great as this number for arguments between and , i.e. for arguments that fall very little short [Pg 110] of (if is very small). Let us take all possible 's and all possible corresponding sections. The common part of all these sections we will call the "ultimate section" as the argument approaches . To say that a number belongs to the ultimate section is to say that, however small we may make , there are arguments between and for which the value of the function is not less than . We may apply exactly the same process to upper sections, i.e. to sections that go from some point up to the top, instead of from the bottom up to some point. Here we take those numbers that are not less than all the values for arguments from to ; this defines an upper section which will vary as varies. Taking the common part of all such sections for all possible 's, we obtain the "ultimate upper section." To say that a number belongs to the ultimate upper section is to say that, however small we make , there are arguments between and for which the value of the function is not greater than . If a term belongs both to the ultimate section and to the ultimate upper section, we shall say that it belongs to the "ultimate oscillation." We may illustrate the matter by considering once more the function as approaches the value 0. We shall assume, in order to fit in with the above definitions, that this value is approached from below. Let us begin with the "ultimate section." Between and 0, whatever may be, the function will assume the value 1 for certain arguments, but will never assume any greater value. Hence the ultimate section consists of all real numbers, positive and negative, up to and including 1; i.e. it consists of all negative numbers together with 0, together with the positive numbers up to and including 1. Similarly the "ultimate upper section" consists of all positive numbers together with 0, together with the negative numbers down to and including -1. Thus the "ultimate oscillation" consists of all real numbers from -1 to 1, both included. [Pg 111] We may say generally that the "ultimate oscillation" of a function as the argument approaches from below consists of all those numbers which are such that, however near we come to , we shall still find values as great as and values as small as . The ultimate oscillation may contain no terms, or one term, or many terms. In the first two cases the function has a definite limit for approaches from below. If the ultimate oscillation has one term, this is fairly obvious. It is equally true if it has none; for it is not difficult to prove that, if the ultimate oscillation is null, the boundary of the ultimate section is the same as that of the ultimate upper section, and may be defined as the limit of the function for approaches from below. But if the ultimate oscillation has many terms, there is no definite limit to the function for approaches from below. In this case we can take the lower and upper boundaries of the ultimate oscillation (i.e. the lower boundary of the ultimate upper section and the upper boundary of the ultimate section) as the lower and upper limits of its "ultimate" values for approaches from below. Similarly we obtain lower and upper limits of the "ultimate" values for approaches from above. Thus we have, in the general case, four limits to a function for approaches to a given argument. The limit for a given argument only exists when all these four are equal, and is then their common value. If it is also the value for the argument , the function is continuous for this argument. This may be taken as defining continuity: it is equivalent to our former definition. We can define the limit of a function for a given argument (if it exists) without passing through the ultimate oscillation and the four limits of the general case. The definition proceeds, in that case, just as the earlier definition of continuity proceeded. Let us define the limit for approaches from below. If there is to be a definite limit for approaches to from below, it is necessary and sufficient that, given any small number , two values for arguments sufficiently near to (but both less than ) will differ [Pg 112] by less than ; i.e. if is sufficiently small, and our arguments both lie between and ( excluded), then the difference between the values for these arguments will be less