Aggregation

“ It seems at first sight a probable view that aggregation is due to the glands being excited to secrete more copiously, so that sufficient fluid is not left in their cells, and in the cells of the pedicels, to hold the protoplasm in solution. In favour of this view is the fact that aggregation follows the inflection of the tentacles, and during the movement the glands generally, or, as I believe, always, secrete more copiously than they did before. ”

“ Much, if not all, of this open contention and misery would be avoided if only the natural aggregating causes produced effects. It is the artificial "aggregating" force of corporations that has so overloaded the national stomach with its "aggregations," combines, and trusts ”

“ An aggregate whose elements are real numbers is said to “extend to infinite values” if, after any number , however great, is specified, it is possible to find in the aggregate numbers which exceed in absolute value. Such an aggregate is always infinite. The “neighbourhood of a number (or point) for a positive number ” is the aggregate of all numbers (or points) for which the absolute value of denoted by , does not exceed .
5. General Notion of Functionality.—A function of one variable was for a long time commonly regarded as the ordinate of a curve ”

“ The totality of entities which lend themselves to a given type of order, form a collection, or an aggregate; and if the order of a given aggregate is to be unambiguous, the members of this aggregate must be uniquely determined, and must be mutually exclusive: they must be what we shall later denote by the term elements of the aggregate. Externality and ‘elementarity’ are only different consequences of the same relation, which we denote by the term extension, and which thus together with the relation of order, will constitute the basis of our definition of space and time. ”

“ A domain which consists of all the real numbers which exceed some fixed number may be described as an “interval unlimited towards the right”; similarly we may have an interval” unlimited towards the left.”
In more complicated cases we must have some rule or process for assigning the aggregate of numbers which constitute the domain of a variable. The methods of definition of particular types of aggregates, and the theorems relating to them, form a branch of analysis called the “theory of aggregates” (Mengenlehre, Théorie des ensembles, Theory of sets of points) . ”

“ If a quantity being given, first one quantity be both added to it and subtracted from it, and then another greater or less, the proportion of the remainder to the aggregate, is greater where the less quantity is added and substracted, than where the greater quantity is added and substracted. ”

“ Subtract from the idea of an aggregate the idea of quality or kind, and order or arrangement, what is left is its cardinal number. If you subtract quality and not order, the result is an ordinal number. This reasoning applies both to finite and infinite aggregates; in fact, the Infinite may be said to possess most of the properties which we attach to the Finite. Two infinite aggregates, for example, can have an ordinal correspondence, and infinite aggregates submit, like finite ones, to arithmetical processes. ”

“ A thousand cases may be imagined to prove it. Take one; suppose eight of the states had 45,000 inhabitants each, and the other seven 44,999 each, that is to say, each one less than each of the others, the aggregate would be 674,993, and the number of representatives, at one for 30,000 of the aggregate, would be 22. Then, after giving one member to each state, distribute the seven residuary members among the seven highest fractions; and, though the difference of population be only an unit, the representation would be the double. Here a single inhabitant the more would count as 30,000. ”

“ It needs but to recall the truth exemplified some chapters back, that the properties of the aggregate are determined by the properties of the units, to see at once that, so long as the characters of citizens remain substantially unchanged, there can be no substantial change in the political organization which has slowly been evolved by them. ”

“ The aggregate of the wants of the entire public economy etc., is satisfied by the aggregate activity of the people. Every individual who employs his lands, labor or capital for the whole, receives his share of the aggregate produce, whether he contributed or not to the creation of the kind of produce in which he is paid. Thus, in a pin-manufactory, the workman who is occupied solely in making the heads of pins is not paid in pins or pin-heads, but in a part of the aggregate result of the manufacture, in money. ”

“ It has to apportion labor among different occupations for the sake of collective production, which is a grand synthetic operation whereby each man puts something into a common total which is the income of all society. It has, further, to divide the grand total into shares for its different members—an analytical operation in which each man takes something out of the aggregate for his personal use. This is distribution in the narrower sense of that term—the apportionment among the members of a civilized society of the fruits of production. ”

“ Their merits, when every consideration is taken into view, are nearly in the inverse ratio of the sums denoting their aggregate dissonance. That of Mr. Hawkes is the best, and, in many respects, has a remarkable analogy to the one derived from the preceding investigations.
Cor. 2. As the aggregate dissonance of the changeable scale is calculated on the same principles, in Prop. VIII., as that of the Douzeave in this, a comparison of the results in Table VIII. with those in Table XIII., will furnish us with the relative dissonance of different systems for these different scales. ”

“ Other experts, testifying to an aggregate, without assigning a proportion to the contributory factors, give estimates so vague as to be little more than guesses, one of them, for illustration, holding to the opinion that ten would be a fair percentage, yet unable to give a reason why the amount should not be less or greater. ”

“ I must premise that I shall call some of the groups political communities, meaning by a political community a number of persons living under one government and also having much else in common besides government: the rest I shall call political aggregates, meaning by a political aggregate a number of persons or bodies of persons living under one government and having nothing else or very little else in common. ”

“ Spencer’s, • 39 we shall find them to be all conclusions about aggregates as wholes, not about parts of aggregates • 39 The only differences recognised by him between men are differences between one homogeneous aggregate and another, • 40 and differences between similar men who happen to be occupied differently • 41 But, as has already been said, the social problems of to-day arise out of a conflict between different parts of the same aggregate ”

“ And although it may be said that the amount in each case of these concessions is in itself but small, yet, when we come to add together all that have been made from time to time by every different denomination, and by every different benevolent organization, the aggregate is truly appalling ”

“ An aggregate of men is a mob, and not ‘Humanity’; and an aggregate of sheep is a flock, and not Ovility; and an aggregate of geese is——perhaps you had better consult Mr. Herbert Spencer and the late Mr. John Stuart Mill for the best modern expression,—but if you want to know the proper names for aggregates, in good old English, go and read Lady Juliana’s list in the book of St. Albans. ”

“ This numerical excess increases rapidly with each successive increase in the original aggregate, so that if we apply the same reasoning to an infinite aggregate, the aggregate of partial aggregates becomes an infinity of a higher order. This is expressed by saying that the infinite aggregate of partial aggregates has a greater potentiality than the infinity of the elements of the original aggregate.
So we see that the one infinity is, in popular language, much more comprehensive, more powerful than the other. ”

“ As social integration advances, the increasing aggregates exercise increasing restraints over their units—a truth which is the obverse of the one just set forth, that the maintenance of its integrity by a larger aggregate implies greater cohesion. ”

“ Let us imagine that one gives to each voter an urn which contains an infinity of balls by means of which he is able to shade all the degrees of merit of the candidates; let us conceive again that he draws from his urn a number of balls proportional to the merit of each candidate, and let us suppose this number written upon a ticket at the side of the name of the candidate. It is clear that by making a sum of all the numbers relative to each candidate upon each ticket, that one of all the candidates who shall have the largest sum will be the candidate whom the assembly prefers ”

“ Neither shall we make it a part of our present task to study the equally interesting and more important problem of measurement, which is concerned with quantitative determinations of the variation of the ordinal characteristic; it will here suffice if we say that the possibility of measurement of a given continuous aggregate is dependent upon the existence of a certain relation between every two elements of the continuum, which relation it is possible to express numerically, and to compare as to equality or inequality between various pairs of elements ”

“ The endeavours of the individuals engaged count cumulatively, to such effect that doubling their forces will more than double the aggregate efficiency; and conversely, reducing the number will reduce the effectiveness of their work by something more than the simple numerical proportion. ”

“ If, ridiculing this system, any one were to say that, in order to find the mean proportional and give form to the body of the government, it is only necessary, according to me, to find the square root of the number of the people, I should answer that I am here taking this number only as an instance; that the relations of which I am speaking are not measured by the number of men alone, but generally by the amount of action, which is a combination of a multitude of causes ”

“ The whole food of a country divided by the sum of its population, constitutes the share of each person. Here, the food is the numerator, and the population the denominator of a fraction. In order that this fraction shall diminish, it is not necessary that the numerator shall continue stationary while the denominator increases: it is sufficient that it shall not increase as fast; and this is the case with food, which, we know, cannot increase as rapidly as an unchecked population.
I have also stated that the channel of employment can receive as many labourers as can possibly be maintained. ”

“ If, for example, the assembly is poorly informed about the subject submitted to its decision, if this subject requires delicate considerations, or if the truth on this point is contrary to established prejudices, so that it would be a bet of more than one against one that each voter will err; then the decision of the majority will be probably wrong, and the fear of it will be the better based as the assembly is more numerous. It is important then, in public affairs, that assemblies should have to pass upon subjects within reach of the greatest number ”

“ D. If is an aggregate such that it is of equal power with none of its parts, then the aggregate which arises from by the addition of a single new element , has the same property of being of equal power with none of its parts.
E. If is an aggregate with the finite cardinal number , and is any part of , the cardinal number of is equal to one of the preceding numbers . ”

“ To add or unite, as, a person, to an association.
It is many times hard to discern to which of the two sorts, the good or the bad, a man ought to be aggregated. Wollaston.
3. To amount in the aggregate to; as, ten loads, aggregating five hundred bushels. [Colloq.]
Syn. — To heap up; accumulate; pile; collect.
AGGREGATE
Ag"gre*gate, n.
1. A mass, assemblage, or sum of particulars; as, a house is an aggregate of stone, brick, timber, etc.
In an aggregate the particulars are less intimately mixed than in a
compound. ”

“ It is the obvious one that when the cases which exist are shared among several kinds, it is impossible that each of those kinds should be a majority of the whole: on the contrary, there must be a majority against each kind, except one at most; and if any kind has more than its share in proportion to the total number, the others collectively must have less. ”

“ Indeed, the numerical majority often fails to accomplish that at which it professes to aim — to take truly the sense of the majority. It assumes, that by assigning to every part of the State a representative in every department of its government, in proportion to its population, it secures to each a weight in the government, in exact proportion to its population, under all circumstances. But such is not the fact. The relative weight of population depends as much on circumstances, as on numbers. ”

“ In a House of 670 Members, or any other even number, if divided into two parties, the majority (in the sense he uses the word—viz., the difference) must always be an even number. It is true that the division lists sometimes show a majority which is an odd number, but in such a case an odd number of Members must have been absent from the division. ”

“ Such an aggregate it conceives as a new unit, a new number of higher order, and subjecting it to criteria of new and hitherto quite unsuspected kinds such as “potency,” “order,” “equivalence,” “countableness,” and devising laws and operative methods for it in respect of these criteria. Thus is being actualized a last 427extension of the function-theory. ”