Initial series

“ Only the types of Series differ in these two worlds, the Well-Ordered Series being characteristic of the life of the Will just in so far as it is self-conscious, and consequently always knows what next to do, while the World of Description is characterized, in general, by another and less perfect type of serial order. ”

“ A series is continuous when any term divides the whole series unambiguously into two mutually exclusive parts which between them comprise all the terms of the series, and when every term which so divides the series is itself a term of the series. ”

“ In the world of our pure conceptions, we find that between successive whole numbers, such as 2 and 3, 3 and 4, it is impossible to conceive other whole numbers inserted, so long as one takes the whole numbers in their natural order. And the same holds true, in the world of empirical things, regarding any simple series of objects whose type is that of the whole number series, where every object of the series is followed by a next one, with nothing between that belongs to the series. ”

“ To illustrate empirically of what wide application the concept of Series is, and how it is present wherever the concept of Law is present, and vice versa, is useful in beginning a discussion of this category, although anything like present completeness in such illustrations is hopeless. You find serial order wherever you look in the world of definitely conceived or of exactly describable fact. Space and Time illustrate our principle in their every detail. ”

“ We distinguish what is called an actual series, each link of which has had an actual existence, from a potential series, in which the links have not as yet been projected into existence, but will be. Now, an actual series has an end—viz., the link marking the point of transition from the actual to the potential—and is susceptible of increase, since, indeed, it constantly receives fresh accessions from the potential. ”

“ Turning therefore to the hypothesis which regards one series as subordinate to the other, we assume in the first place, with Aristotle, Spinoza, Hegel, von Hartmann, and Wundt, that the mental series is the condition of the physical. ”

“ In the brain are male and female faculties; here is marriage, here is fruit. And there is no limit to this ascending scale, but series on series. Every thing, at the end of one use, is taken up into the next, each series punctually repeating every organ and process of the last. We are adapted to infinity. ”

“ In our Supplementary Essay we showed at length that a Self, as a real being, has a certain form of expression, which inevitably involves a serial structure, but that this serial structure, in its main outline, is most truly represented by the form of the series of whole numbers, rather than by the form of a series between any two of whose terms, further terms without end are interpolated. ”

“ Classification, logic, and mathematics all result, then, from the mere play of the mind comparing its conceptions, no matter whence the latter may have come. The essential condition for the formation of all these sciences is that we should have grown capable of apprehending series as such, and of distinguishing them as homogeneous or heterogeneous, and as possessing definite directions of what I have called 'increase.' This consciousness of series is a human perfection which has been gradually evolved, and which varies greatly from man to man. ”

“ In mathematics a set of quantities, real or complex, arranged in order so that each quantity is definitely and uniquely determined by its position, is said to form a series. Usually a series proceeds in one direction and the successive terms are denoted by ”

“ We call a series the more variable, the more frequently deviating types occur in it; so that the average amount of differences between the individuals constituting the series and the most common type may be used as a measure of the variability of the series. ”

“ And the rule of our discriminating intelligence is that, while the problem of the One and the Many is hopelessly baffling if we deal merely with two terms, and while it is equally hopeless so long as we deal with an indefinite number of objects not arranged in series, we begin to see the light so soon as we get one of our objects between the two others, and so begin to form a series possessing a definite character and direction. ”

“ This world of facts, arranged in these abstractly conceived series, is anybody’s world. All of us start from different points of view. We all, if we find this sort of truth, shall come in the end to define our results in terms of corresponding series. And this variety of possible points of view is not merely a chance accompaniment of description, but a necessary consequence of the way in which this series-forming-process of looking for what lies between any two objects proceeds. ”

“ To reflect that each one who enters imagines himself to be the first to enter whereas he is always the last term of a preceding series even if the first term of a succeeding one, each imagining himself to be first, last, only and alone whereas, he is neither first nor last nor only nor alone in a series originating in and repeated to infinity. ”

“ The words “silent, calm, and clear” qualify the word day and constitute a commencing series, because they require the word day to complete the thought.
What is a concluding series? A series is considered a concluding one when the series is complete with the close of the series. It generally concludes the sentence; as,
The remarkable people of this world are useful in their way; but the common people, after all, represent the nation, the age, and the civilization.
—Henry Ward Beecher
The series consists of “the nation,” “the age,” “the civilization” ”

“ Europe and America, compared geographically, or socially, or politically, lead us to attempt the formation of series of objects. Feelings, deeds, persons, lives, stellar spectra, chemical elements, processes of evolution, types of doctrine, modes of conduct, aesthetic values, in brief, beings of all grades, invite serial treatment as soon as they are compared. Various series, already conceived, can be combined in the most varied ways, so as to give us systems of objects that no longer can be arranged in any single serial order. ”

“ If, then, a B series without an A series can constitute time, change must be possible without an A series. Let us suppose that the distinction of past, present and future does not apply to reality. Can change apply to reality? ”

“ In a series thus secured there may be time-marks, though no time, and by these marks the series will be distinguished from other simultaneous series. To ask which is first among a number of simultaneous presentations is unmeaning; one might be logically prior to another, but in time they are together and priority is excluded. Nevertheless after each distinct representation a, b, c, d there probably follows, as we have supposed, some trace of that movement of attention of which we are aware in passing from one presentation to another. ”

“ An infinite series may contain both positive and negative terms. The terms may be positive and negative alternately or they may occur in groups which without altering the order of the terms of the series may each be collected into a single term; thus all series may be regarded as belonging to one of two types, ul-i-u2+u3+ in which the terms are all positive, or u1-u2-|-u3- in which the terms are alternately positive and negative. 8. It is clear that if a series is convergent un must tend to the limit zero as n is increased indefinitely. ”

“ We may omit, for the time, from our notice, the before-mentioned possibility of a knowledge of the world that would reduce it to a single serial order of the general type of the Well-Ordered Series of whole numbers, where every term has one coming next after it. If we abstract from that possibility, we are left to the conception that Between any two facts there are to be found various series of intermediaries of the type now defined. The world thus regarded will consist for us of all these interwoven series, and will constitute a single System. ”

“ That which has been present is Past, that which will be present is Future.{3} Thus to our previous conclusion that there can be no time unless the A series is true of reality, we can add the further conclusion that no other elements are required to constitute a time-series except an A series and a C series. ”

“ Moreover, it was difficult to find a scale of judgment by which to decide whether a series was really alternating or plain repetition. From one point of view, every repetition is alternating, that is, the repeated unit always alternates with an empty space. Although such repetitions bear out the theory still further, and emphasize yet more strongly the invariability of alternates, and the possibility of variations in the principal units, I have used the term in a stricter sense, and only given illustrations of repeated objects, when one unit actually alternated with another definite unit. ”

“ Two series may have the same average value and yet differ considerably in range. Now it is obvious that, where the number of individuals considered is small, excessive values are less likely to occur than in a larger series. ”

“ Construct the series
When , series (H) is a geometric series with the common ration less than unity, and is therefore convergent. But the sum of (G) is less than the sum of (H) , as shown in the above inequalities; therefore (G) is also convergent.
When series (G) becomes the harmonic series which we saw was divergent, and neither of the above tests apply. ”

“ For ascertaining the relation of dependence between the increase in the number of repetitions of a series and the ever deeper impression of it which result, I have formulated the problem as follows: If homogeneous series are impressed to different extents as a result of different numbers of repetitions, and then 24 hours later are learned to the point of the first possible reproduction by heart, how are the resulting savings in work related to each other and to the corresponding number of former repetitions? ”

“ One point, I fancy, there will be very little difficulty in proving, which is, that the whole form one continuous group, extending in an unbroken series, from the earliest to the latest. There is no hiatus or break anywhere; and if some can be proved to belong to the 10th century, it is only a question how far you can, by extenuating the thread, extend it backwards. ”

“ Then, because the series of real numbers between 0 and 1 is continuous, there must be a least real number, greater than every number of that endless series. This property, which may be called the Aristotelicity of the series, together with Kant’s property, or its Kanticity, completes the definition of a continuous series. ”

“ Therefore the actual repeated series conforms to the given conditions and is made to do so in every case by its unchanging alternate in the same plane; while human figures with values of their own never can be considered quite as alternates, but are really felt to be a series by themselves. ”

“ If, in what you choose to call the present moment of the world’s history, deeds are accomplished, suns actually move from place to place, light waves traverse the ether, and men’s lives pass from stage to stage, then within what you thus call the present there are distinguishable and more elementary events, arranged in series, such that when any conceived element, or mere elementary portion of any series is taken in relation to its predecessors and successors, it is not yet when its antecedents are taken as temporally present, and is past and gone when its successors are viewed as present. ”

“ Great as seems to be the difference between the two extremes of this series when considered by themselves, it is possible to pass from one to the other through a series of intermediate actions, without the necessity for any great jump in any part of the series ”

“ Here, then, we meet the difficulty that diverse environments often shade into each other insensibly and form a continuous series, whereas the specific forms of life which are subject to them on the whole form a discontinuous series. ”

“ Of course, if the series were left entirely to themselves and if they were always produced in precisely the same order, for each term there would appear only one association, the relatively strongest—namely, that with the immediately succeeding term. But series of ideas are never left to themselves. The rich and quickly changing order of events brings them into the most manifold relations. ”

“ That, in considering the phenomena of nature, we should confine our attention to the mere antecedents and consequents, which succeed each other in regular series, is unquestionably the soundest advice that can be given. But it is sound advice, for this reason more than any other, that the regular series is, in truth, all that constitutes the phenomena, and that to search for any thing more, is not to have an unattainable object in view, but to have no conceivable object whatever. ”

“ He warns his readers occasionally against the use of divergent series, but is nevertheless very careless himself. The rigid treatment to which infinite series are subjected now was then undreamed of. No clear notions existed as to what constitutes a convergent series. ”

“ The simplest way to introduce the element of hesitance and wavering, and thereby make the story more truly suggestive of the intricate variety of life, is to interrupt the series by the introduction of events whose apparent tendency is to hinder its progress, and in this way emphasize the ultimate triumph of the series in attaining its predestined culmination. ”