Law of large numbers

“ There's nothing in the world I like better than to hear of such large numbers — millions, trillions — that gives one something to think about. — Upon my soul, that's quite a lot, a thousand million, more or less.
Leander
As the numbers increase, so human intelligence expands. ”

“ Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000, need more concrete aids than are commonly given. Guessing contests about the value in dollars of the school building and other buildings, the area of the schoolroom floor and other surfaces in square inches, the number of minutes in a week, and year, and the like, together with proper computations and measurements, are very useful to reënforce the concrete presentations and supply genuine problems in multiplication and subtraction with large numbers. ”

“ Assuming that number is the essence of the world, he deduced the idea that the world is regulated by numerical proportions, or by a system of laws which are regular and harmonious in their operations. ”

“ But when we speak of the units which go to form number, we no longer think of these units as sums, but as pure, simple, irreducible units, intended to yield the natural series of numbers by an indefinitely continued process of accumulation. It seems, then, that there are two kinds of units, the one ultimate, out of which a number is formed by a process of addition, and the other provisional, the number so formed, which is multiple in itself, and owes its unity to the simplicity of the act by which the mind perceives it. ”

“ Upon what Principles, and from what natural Causes do Mankind multiply? And what are the effects of Procreation in Countries where Numbers are not found to increase? The multiplication of mankind has been treated of in different ways; some have made out tables to shew the progression of multiplications, others have treated the question historically. The state of numbers in different ages of the world, or in different countries at different times, has been made the object of inquiry ”

“ Few tests of relative stages of civilisation are more trustworthy than that of the definite conception of high numbers. The prevalence of a decimal system of numerals among widely severed nations, alike in ancient and modern times, has been universally ascribed to the simple process of counting with the aid of the fingers. ”

“ It is in just this manner that, little by little, man has awakened to a deeper and truer appreciation of the nature of the laws that God has formulated for the government of human affairs, and there is no principle taught in the Science of Numbers that to any degree contradicts, or opposes any recognized manifestation of divine law. ”

“ Within them is embraced the whole of computation; but because every number, every aggregation of units, is also a new unit capable of being added or subtracted, there are also the operations of multiplication and division, which consists in one case of the addition of several equal numbers together, and in the other, of the subtraction of several equal numbers from a greater until that is exhausted. ”

“ Our ideas of numbers he found to be but the collected ideas of the impressions of the units of which the senses have received distinct impressions; and in confirmation of this he appealed to the distinctness of our notion of small numbers, which our mind has been accustomed to find represented by units, and our imperfect conception of those large numbers, which we have never had presented to us in detail. How readily we have a notion of six, but how imperfectly the mind receives the conception of six millions ”

“ To proceed: you have found for yourselves a place in the Infinite of number; you have fitted it to your stature by creating arithmetic—if you can be said to create anything—the basis on which everything is built up, even society. Arithmetic, or the use of number, has organized the moral world, just as number, the only thing in which your professing Atheists believe, organizes physical creation. This science of numbers ought to be absolute, like everything that is intrinsically [Pg 101] true ”

“ It is for practical reasons that man has reduced all other properties to numbers, for as counting can be accomplished only by classification, so properties can only be treated in mathematics when they are reduced to number by measure. ”

“ Having secured this control, they let loose upon you the astonishing doctrine that, if you have numbers, there you have your right cut and dried; that if you have not numbers your right (politically speaking) does not exist. ”

“ This, sir, is, I believe, about the true number. There is no occasion to exaggerate, where plain truth is of so much weight and importance. But whether I put the present numbers too high or too low, is a matter of little moment. Such is the strength with which population shoots in that part of the world, that, state the numbers as high as we will, while the dispute continues, the exaggeration ends. While we are discussing any given magnitude, they are grown to it. While we spend our time in deliberating on the mode of governing two millions, we shall find we have two millions more to manage. ”

“ Now a very simple operation in arithmetic will show that this means a population of fifteen hundred millions, or more than the population of the whole world at the present date. Another very simple operation in arithmetic will show that if we were to go on doubling our numbers, even once in every twenty-five years, we should reach that stupendous figure at about the close of the twentieth century,--that is, in the days of our great-greatgrandchildren. ”

“ When in every single instance a multitude, often an unknown multitude, of agencies, are clashing and combining, what security have we that in our computation a priori we have taken all these into our reckoning? How many must we not generally be ignorant of? Among those which we know, how probable that some have been overlooked; and, even were all included, how vain the pretense of summing up the effects of many causes, unless we know accurately the numerical law of each—a condition in most cases not to be fulfilled ”

“ But Turner counted his pines, and did all that could be done for them, and rested content with that.
So in all the affairs of life, the arithmetical part of the business is the dominant one. How many and how much have we? How many and how much do we want? How constantly does noble Arithmetic of the finite lose itself in base Avarice of the Infinite, and in blind imagination of it! In counting of minutes, is our arithmetic ever solicitous enough? In counting our days, is she ever severe enough? How we shrink from putting, in their decades, the diminished store of them! ”

“ II. But if it be thought that, either now, or at any future time, the citizens of the United States may be induced to undertake a thorough reformation of their whole system of measures, weights and coins, reducing every branch to the same decimal ratio already established in their coins, and thus bringing the calculation of the principal affairs of life within the arithmetic of every man who can multiply and divide plain numbers, greater changes will be necessary. ”

“ So far from being true, it is, in all the aspects in which it can be regarded, preeminently a government of the concurrent majority: with an organization, more complex and refined, indeed, but far better calculated to express the sense of the whole (in the only mode by which this can be fully and truly done — to wit, by ascertaining the sense of all its parts) than any government ever formed, ancient or modern. Instead of population, mere numbers, being the sole element, the numerical majority is, strictly speaking, excluded, even as one of its elements ”

“ For the unit is the originator of every number, since all multitude is made up by the composition and addition of units; nor is it possible without the unit to conceive the existence of number at all. But the unit itself is independent of multitude, apart from and superior to all number; forming, indeed, and making all, but receiving no increase from any.
13. Kindred to this is the triad; equally indivisible and perfect, the first of those sums which are formed of even and uneven numbers. ”

“ To a great extent proficiency in applying the the principles of the Science of Numbers is a question of practice, for, while the rules are exact, the phenomena of their manifestation varies so materially in each individual case that it would require a volume of huge proportions to describe all the operations of each rule, to say nothing of the many exceptions that affect this play of lights and shadows. ”

“ The Law of Multiplication in Geometrical Progression.—All organized beings have enormous powers of multiplication. Even man, who increases slower than all other animals, could under the most favourable circumstances double his numbers every fifteen years, or a hundred-fold in a century. Many animals and plants could increase their numbers from ten to a thousand-fold every year.
2. The Law of Limited Populations.—The number of living individuals of each species in any country, or in the whole globe, is practically stationary ”

“ While measure is thus conventional there is still another conventional usage in the science of mathematics. In natural units bodies are the higher units of particles, the particle and the body are units of different orders, and the different orders of units in nature are thus coextensive with all the bodies of the universe. Thus there is an infinite system of orders of numbers; but man devises a numerical system where a definite plurality is considered as a higher unity, and such a system serves him a valuable purpose as a labor-saving device for the mental faculties. ”

“ But we soon perceive that, while all multiplication implies the possibility of treating any number whatever as a provisional unit which can be added to itself, inversely the units in their turn are true numbers which are as big as we like, but are regarded as provisionally indivisible for the purpose of compounding them with one another. ”

“ In so small a volume, however, it would be folly to attempt so pretentious a task, especially in view of the fact that, for all present purposes, it Is sufficient that we should realize that the Science of Numbers has been known and credited since very early in the world’s history, and that, having been cherished with most tender solicitude through the turmoil and storms of the centuries, it has come to us with the signs of truth and authenticity indelibly fixed by the stamp of years. ”

“ Were plentiful years more common, mankind would be more numerous; were scarcity more frequent, numbers would be less. Numbers therefore must ever be, in my humble opinion, in the ratio of food, and multiplication will never stop until the balance comes to be nearly even. ”

“ In the last analysis, division is only the obverse side of multiplication, and any one who has not learned his multiplication table is reduced to some such expedient as that of the Egyptian. Indeed, whenever we pass beyond the range of our memorized multiplication table-which for most of us ends with the twelves—the experimental character of the trial multiplication through which division is finally effected does not so greatly differ from the experimental efforts which the Egyptian was obliged to apply to smaller numbers. ”

“ Thus, in seeking our definition of number we have arrived so far at the result that numbers are properties of general terms or general descriptions, not of physical things or of mental occurrences.
Instead of speaking of a general term, such as “man,” as the subject of which a number can be asserted, we may, without making any serious change, take the subject as the class or collection of objects—i.e. “mankind” in the above instance—to which the general term in question is applicable. ”

“ He had the power of simplifying mathematical reasoning, and produced to that end the unitary method in arithmetic and a simple and ingenious plan for the conversion into l.s.d. of money expressed in decimals, a development of which simplifies the process of long division in a large class of cases (Brit. Assoc. Report, 1902, p. 529; Caius College Magazine, Michs. Term, 1902) . ”

“ Without sufficient numbers, it would be useless to ask for admission. Hence, in the concentration of numbers, we are blest, as well as having power to preach to millions in all parts of the earth at the same time. ”

“ So for the incommensurable number, the vague idea of continuity, which we owe to intuition, has resolved itself into a complicated system of inequalities bearing on whole numbers. Thus have finally vanished all those difficulties which frightened our fathers when they reflected upon the foundations of the infinitesimal calculus. To-day only whole numbers are left in analysis, or systems finite or infinite of whole numbers, bound by a plexus of equalities and inequalities. Mathematics we say is arithmetized. ”

“ Moreover, each number is so defined by its own properties, that no two numbers are equal. They are therefore both unequal and different from one another; and while they are simply finite, collectively they are infinite. ”

“ The science of number is natural, for units and pluralities are found in nature, but measure is conventional, for conventional units of measure are used in order that undiscovered numbers may be represented by their equivalents in computation, for while we may not be able to discover the number of natural units in a body we may be able to measure its form in conventional units of extension, and for some purposes of computation these units serve the desired purpose. ”

“ It is within the power of the human mind to count thousands or millions multiplied without end, at least in theory, but this cannot be done in reality. For that which becomes actual and can be counted as one, is like the number which is both actual and finite without doubt. How can the infinite become actual? The world has, therefore, a beginning, and the revolutions of the spheres are subject to a finite number. ”