Integral

“ The following is a simple particular case of this refined problem: Let integral rational functions of one variable with integral rational coefficients, and a prime number be given. Consider the system of those integral rational functions of which can be expressed in the form
where is a rational integral function of the arguments and is any power of the prime number . ”

“ So, proceeding from the facts of experience and ideas of modern science, we inevitably come to the only integral, the only monistic understanding of the universe. It appears before us as an infinitely evolving fabric of forms of various types and levels of organisation, from elements of the ether unknown to us to human collectives and stellar systems. ”

“ Finally, of individuals, the living power will be most intense in that individual which, as a whole, has the greatest number of integral parts presupposed in it; when, moreover, these integral parts, together with a proportional increase of their interdependence, as parts, have themselves most the character of wholes in the sphere occupied by them. ”

“ If then we were to subtract the smaller area from the larger, we should have left as a remainder the area , which is what we want. Here we have the clue as to what to do; the definite integral between the two limits is the difference between the integral worked out for the superior limit and the integral worked out for the lower limit.
Let us then go ahead. First, find the general integral thus:
.
and, as is the equation to the curve (Fig. 52) ,
is the general integral which we must find. ”

“ As to time, which, it is to be presumed, would, by way of the counterpart to space, constitute the object-matter of the other division of pure mathematics, this is the notion itself in the form of existence. The principle of quantity, of difference which is not determined by the notion, and the principle of equality, of abstract, lifeless unity, are incapable of dealing with that sheer restlessness of life and its absolute and inherent process of differentiation. ”

“ So in the science of abstract number the fundamental assumptions, as that 1=1, x=x, etc., are contradicted by every fact of experience, for in the world as we know it, absolute equality is simply impossible to discover; and yet these fundamental conceptions are in their development most powerful instruments for the extension of man's command over his own experiences. ”

“ And so, as we saw, it is possible, at least by way of trial, to undertake to define Being wholly in terms of validity, to conceive that whoever says, of any object, It is, means only that a certain idea, — perhaps an idea suggested by passing experience, perhaps the thought of an empirically discovered law in a natural science, perhaps a free construction of an ideal object in mathematics, — but in any case an idea, is valid, has truth, defines an experience that, at least as a mathematical ideal, and perhaps as an empirical event, is determinately possible. ”

“ This principle is that the intrinsic value of a whole is neither identical with nor proportional to the sum of the values of its parts. It follows from this that, though in order to obtain the greatest possible sum of values in its parts, the Ideal would necessarily contain all the things which have all these parts might not be so valuable as some other whole, from which certain positive goods were omitted. ”

“ Reality, because systematic, must be the expression of a single principle in and through a multiplicity. The unity and multiplicity must both be real, and each must necessarily involve the other. § 5. If both are to be equally real, the whole system must be a single experience, and its constituents must also be experiences. A perfect systematic whole can be neither an aggregate, nor a mechanical whole of parts, nor an organism. The whole must exist for the parts, and they for it. ”

“ We are, indeed, justified in distinguishing between above and Below, right and left, beginning and end, gold and sheet metal, good and bad, but we must also enlighten ourselves as to how multiplicity can be a unity, the variable constant, and the constant variable. Formal logic has a wrong name. ”

“ All the uniformities which exist in the succession of phenomena, and most of the uniformities in their co-existence, are either, as we have seen, themselves laws of causation, or consequences resulting from, and corollaries capable of being deduced from, such laws. If we could determine what causes are correctly assigned to what effects, and what effects to what causes, we should be virtually acquainted with the whole course of nature. ”

“ The foundation of the whole is formed by the feeling of a reality, the perception of a fact, existing somewhere within the limits of that triple experience which takes in an inner and an outer and also a higher world. Now, the first step in this progressive intellectual development is formed by the notion or general term, which, as I formerly explained it, is a thought or conception that is mathematically determined and precisely limited, both inwardly and outwardly, according to the three dimensions of number, measure, and weight. ”

“ As has been already pointed out, this is all that art can do. To test a generalization, by showing that it either follows from, or conflicts with, some stronger induction, some generalization resting on a broader foundation of experience, is the beginning and end of the logic of induction. ”

“ All things strive for positiveness, for themselves, or for quasi-systems of which they are parts. Formality and the mathematic, the regular and the uniform are aspects of the positive state—but the Positive is the Universal—so all attempted positiveness that seems to satisfy in the aspects of formality and regularity, sooner or later disqualifies in the aspect of wideness or universalness. ”

“ It is obvious that if we ask the question ‘Why should the parts be such as they are?’ a proper answer may be ‘Because the whole they form has so much value.’ But it is equally obvious that the relation which we thus assert to exist between part and whole is quite different from that which we assert to exist between part and part when we say ‘This part exists, because that one could not exist without it.’ In the latter case we assert the two parts to be causally connected ”

“ But some “appearances” exhibit the structure of the whole more adequately than others, and have therefore a higher degree of reality. § 3. This conception of degree of reality may be illustrated by comparison with the successive orders of infinites and infinitesimals in Mathematics. It would be the task of a complete Philosophy to assign the contents of the world to their proper place in the series of “orders” of reality. § 4. In general any subordinate whole is real in proportion as it is a self-contained whole. ”

“ The unity of the Idea is thought, infinity, and subjectivity, and is in consequence to be essentially distinguished from the Idea as substance, just as this overlapping subjectivity, thought, or infinity is to be distinguished from the one-sided subjectivity, one-sided thought, one-sided infinity to which it descends in judging and defining. ”

“ The problem that confronts you, then, in either kind of analysis, however formal or informal it may be, is, How shall I go to work? The first necessity is the choosing of a basis for division of the subject, whether it be in classification or partition. The necessity for this arises from the demand of the human mind for logical consistency. Life seems often wildly inconsistent, but we demand that explanation of it or any phase of it be arranged according to what seems to us some logical law of progression, some consistent point of view. ”

“ The sciences here grouped together as normative all agree in that they seek 'ideal' truth, as distinct from physical truth or from historical fact—are concerned with the consequences, implications and interrelations of ideas or of ideals, rather than with the order of phenomena or events. The mathematician is concerned with the exact expression and abstract logical development of ideas, the meaning of which in terms of their ultimate relations is sought by the philosopher. ”

“ The absolute and the world are one fact, I said, when materially considered. Our philosophy, for example, is not numerically distinct from the absolute's own knowledge of itself, not a duplicate and copy of it, it is part of that very knowledge, is numerically identical with as much of it as our thought covers. ”

“ But, by the constitution of my nature, my belief is irresistibly carried along by my apprehension of the axiom; and by the constitution of my nature, my belief is no less irresistibly carried along by my perception of the tree. All reasoning is from principles. The first principles of mathematical reasoning are mathematical axioms and definitions; and the first principles of all our reasoning about existences, are our perceptions. ”

“ This last principle, which we may call the axiom of constant result, holds good throughout all our mental operations, not only when we compare, but when we add, divide, class, or infer a given matter in any conceivable way. ”

“ In all our knowledge of general principles, what actually happens is that first of all we realise some particular application of the principle, and then we realise that the particularity is irrelevant, and that there is a generality which may equally truly be affirmed. ”

“ Has some prodigious philosopher devised a new system of calculus satisfactory to all for determining the exact and comparative value of all sorts of service, whether by brawn or brain, by hand or voice, by ear or eye? Or has human nature itself changed, so that no man looks upon his own things but 'every man on the things of his neighbor'? ”

“ But when we are studying objects not for any special practical end, but for the sake of extending our knowledge of the whole of their properties and relations, we must consider as the most important attributes those which contribute most, either by themselves or by their effects, to render the things like one another, and unlike other things; which give to the class composed of them the most marked individuality ”

“ In so far as they are genuine systems at all, they are not mere “parts” of a whole, but each is the expression, in a concrete conscious life, of the nature of a larger whole from a special “point of view.” The whole, if not equally in every part, is yet as a whole present in every part, and precisely for that reason the category of part and whole is inadequate to express their relation. Somewhat after this fashion we must conceive the structure of any individual whole of lesser individuals. ”

“ The first point, therefore, to be noted in regard to what is called the uniformity of the course of nature, is, that it is itself a complex fact, compounded of all the separate uniformities which exist in respect to single phenomena. These various uniformities, when ascertained by what is regarded as a sufficient induction, we call, in common parlance, Laws of Nature. ”

“ This doctrine of the regular subordination of beings, the scale of existence, and the chain of nature, I have often considered, but always left the inquiry in doubt and uncertainty.
That every being not infinite, compared with infinity, must be imperfect, is evident to intuition; that, whatever is imperfect must have a certain line which it cannot pass, is equally certain. ”

“ That there must be an indefinite number of such terms is obvious, on reflection; since we cannot define anything except by an analysis, which, when carried as far as it will go, refers us to something, which is simply different from anything else, and which by that ultimate difference explains the peculiarity of the whole which we are defining: for every whole contains some parts which are common to other wholes also. ”

“ But with the enlargement of mind and its more perfect organization, there arises a new method of appraising values, and so a new type of value altogether. I mean of course the so-called absolute values. Absolute values are never absolute in the sense of absolute completeness; they are relative to two things—to external reality and to our mental powers and organization. [48] They are abstractions; we generalize the value in our minds, and at the same time raise it to the highest pitch of intensity we can. ”

“ Further, we must discuss the essential attributes of these themselves; and we must ask not only what each of these is, but also whether one thing always has one contrary. [7] Again, whether are the principles and elements of things the classes, or the parts present in each thing, into which it is divided; [8] and if they are the classes, whether are they the classes that are predicated proximately of the individuals, or the highest classes, e. g. whether is animal or man the first principle and the more independent of the individual instance? ”

“ And if we carry our observation outward, we also find that light and darkness, sound and silence, motion and quiescence, equality and inequality, preceding and following, succession and simultaneousness, any positive phenomenon whatever and its negative, are distinct phenomena, pointedly contrasted, and the one always absent where the other is present. I consider the maxim in question to be a generalization from all these facts. ”

“ Why rob one to feed the other?
“If you adopt, as absolute, a standard of evenness which is so only relatively, your results will not be absolutely even. If you adopt, as absolute, a criterion of right which is so only relatively, your results will not be absolutely right. ”

“ And thus we see the foundation upon which the whole doctrine of quantity is built; for all quantity is expounded either by number or extension, and their common and sole exponent is algebra. The coincidence of ideas is the foundation of the rational assent in simple cases; and that of ideas and terms together, or of terms alone, in complex ones. ”

“ In order to arrive at a correct decision on the first part of this question, it is necessary to consider what things are such that, if they existed by themselves, in absolute isolation, we should yet judge their existence to be good; and, in order to decide upon the relative degrees of value of different things, we must similarly consider what comparative value seems to attach to the isolated existence of each. ”