Superiority of numbers

“ 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 duly weighed the vast advantages arising from mathematical and arithmetical knowledge in most occasions of human life, he frequently used it as an adage, that he who is ignorant of numbers is scarce half a man. ”

“ If we were fewer in numbers than you, I think that you could whip us; if we were equal it would likely be a drawn battle; but being inferior in numbers, you will make nothing by attempting to master us. ”

“ The difference, therefore, of our conduct in preferring the greater number depends not upon our passions, but upon custom, and general rules. We have found in a multitude of instances, that the augmenting the numbers of any sum augments the passion, where the numbers are precise and the difference sensible. ”

“ 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 ”

“ When, at last, the craft finished its labors and laid down its tools, its symbols, having served the faith of the worker, became a language for the thoughts of the thinker.
Few realize the service of the science of numbers to the faith of man in the morning of the world, when he sought to find some kind of key to the mighty maze of things. Living amidst change and seeming chance, he found in the laws of numbers a path by which to escape the awful sense of life as a series of accidents in the hands of a capricious [154] Power; and, when we think of it, his insight was not invalid. ”

“ And the second devoted her effort to the portrayal of arithmetic, and she strove hard to represent clearly how it wisely enumerates the days and the hours of time, and the water of the sea drop by drop, and then all the sand, and the stars one by one, knowing well how to tell the truth, and how many leaves there are in the woods: such is the skill of arithmetic that numbers have never deceived her, nor will she ever be in error when she wishes to apply her sense to them. ”

“ I have an immense respect for a man of talents plus “the mathematics.” But the calculating power alone should seem to be the least human of qualities, and to have the smallest amount of reason in it; since a machine can be made to do the work of three or four calculators, and better than any one of them. Sometimes I have been troubled that I had not a deeper intuitive apprehension of the relations of numbers. But the triumph of the ciphering hand-organ has consoled me. I always fancy I can hear the wheels clicking in a calculator’s brain. ”

“ The clearness and distinctness of each mode of number from all others, even those that approach nearest, makes me apt to think that demonstrations in numbers, if they are not more evident and exact than in extension, yet they are more general in their use, and more determinate in their application. ”

“ Then there is the phase when he is struggling with the development of number, when he begins to elaborate all sorts of ideas about numbers, until at last he develops complex superstitions about perfect numbers and imperfect numbers, about threes and sevens and the like. The same is the case with abstracted forms, and even to-day we are scarcely more than heads out of the vast subtle muddle of thinking about spheres and ideally perfect forms and so on, that was the price of this little necessary step to clear thinking. ”

“ 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. ”

“ But the grand agent for transforming experimental into deductive sciences, is the science of number. The properties of numbers, alone among all known phenomena, are, in the most rigorous sense, properties of all things whatever. All things are not coloured, or ponderable, or even extended; but all things are numerable. And if we consider this science in its whole extent, from common arithmetic up to the calculus of variations, the truths already ascertained seem all but infinite, and admit of indefinite extension. ”

“ When, therefore, we speak of particular numbers, and at the same time pretend to consider them in themselves, we fall into a contradiction, because we take the numbers both with and without particular conditions at the same time. ”

“ You believe in Number as the foundation on which rests the edifice of what you call the exact sciences. Without number mathematics are impossible. Well, then, what impossible being, to whom life everlasting should be granted, could ever [Pg 99] finish counting—and in what sufficiently concise language could he utter—the numbers contained in the infinite number of which the existence is demonstrated by your reason. Ask the greatest human genius, and suppose him to sit for a thousand years leaning on a table, his head in his hands, what would he answer? ”

“ Arithmetical numbers appealed to the Middle Ages just as they do to us, because they are difficult to deal with, and because the greatest masters of numbers, the Newtons and Leibnitzes, rank among the greatest men. ”

“ Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavor to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only ”

“ 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. ”

“ 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. ”

“ And moreover, seeing the conditions and ratios of what pertains to harmony to consist in numbers, since other things seemed in their entire nature to be formed in the likeness of numbers, and in all nature numbers are the first, they supposed the elements of numbers to be the elements of all things. ”

“ 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. ”

“ When we consider the importance of numbers to man, as an intellectual being, and compare the advancement he has made by this knowledge, beyond the [Pg 52] animals who have wanted the means of acquiring such information, the importance of investigating this curious subject will be fully acknowledged. Without numbers, by which the divisions of time, space, and value are characterised, man could have possessed no knowledge of the order and succession of events ”

“ The acquisition of a scale for which we had no other means of expression than that just described would be a matter of the extremest difficulty, and could never, save in the most exceptional circumstances, progress beyond the attainment of a limit of a few hundred. If the various numbers in question were designated by words instead of by symbols, the difficulty of the task would be still further increased. Hence, the establishment of some number as a base is not only a matter of the very highest convenience, but of absolute necessity, if any save the first few numbers are ever to be used. ”

“ That the mind distinguish carefully two ideas, which are different one from another only by the addition or subtraction of one unit : (2) That it retain in memory the names or marks of the several combinations, from an unit to that number ; and that not confusedly, and at random, but in that exact order that the numbers follow one another. In either of which, if it trips, the whole business of numbering will be disturbed, and there will remain only the confused idea of multitude, but the ideas necessary to distinct numeration will not be attained to. ”

“ It is the invention of general symbols serving to express the numerical relations of entities, whatever be their special characters. And what is the nature of the mental process by which numbers are found capable of having their relations expressed algebraically? It is the same. {21} It is the formation of certain abstract conceptions of numerical functions which are constant whatever be the magnitudes of the numbers. It is the invention of general symbols serving to express the relations between numbers, as numbers express the relations between things. ”

“ The merit of our actual system of numeration consists in including the expression of all numbers in a single idea, making the value of each figure ten times that to the right, and filling all intervals with zeros. ”

“ By the slow, and often painful, process incident to the extension and development of any mental conception in a mind wholly unused to abstractions, the savage gropes his way onward in his counting from 1, or more probably from 2, to the various higher numbers required to form his scale. The perception of unity offers no difficulty to his mind, though he is conscious at first of the object itself rather than of any idea of number associated with it. The concept of duality, also, is grasped with perfect readiness. ”

“ It is, besides, a good practical way of keeping up acquaintance with arithmetic, and you will soon find that the principles on which all military movements turn are arithmetical, and that though one may no doubt learn to do them by rote, yet to understand them, you must have recourse to numbers. ”

“ So that he that can add one to one, and so to two, and so go on with his tale, taking still with him the distinct names belonging to every progression ; and so again, by subtracting an unit from each collection, retreat and lessen them, is capable of all the ideas of numbers within the compass of his language, or for which he hath names, though not perhaps of more. ”

“ But when the number drawn is a multiple of the one they play, it is a sympathetic number, and is next door to winning; and if the number come reversed,—as if, having played 12, it come out 21,—he laughs with delight. "Eh, don't you see, you stupid fellow," said the speziale of a village one day to a dunce of a contadino, of whose infallible terno not a single number had been drawn,—"Don't you see, in substance all your three numbers have been drawn? and it's shameful in you to be discontented. ”

“ 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. ”

“ The Science of Numbers teaches that each letter, or number, possesses certain distinct characteristics, and that, in accordance with the manner of the occurance of these forces in the name and birth of the individual, such qualities manifest themselves for good or for evil. ”

“ To the contrary, every psychic law that is discovered only tends to emphasize more fully the importance of the Science of Numbers, and the necessity of taking advantage of the opportunities it offers to put ourselves more fully into harmony with the vibrations that shape the course of human experiences. ”

“ And some, through the default of their memories, who cannot retain the several combinations of numbers, with their names, annexed in their distinct orders, and the dependence of so long a train of numeral progressions, and their relation one to another, are not able all their lifetime to reckon, or regularly go over any moderate series of numbers. ”

“ It is concerned, not with visible objects, but with abstract truth; for numbers are pure abstractions—the true arithmetician indignantly denies that his unit is capable of division. When you divide, he insists that you are only multiplying; his ‘one’ is not material or resolvable into fractions, but an unvarying and absolute equality; and this proves the purely intellectual character of his study. Note also the great power which arithmetic has of sharpening the wits; no other discipline is equally severe, or an equal test of general ability, or equally improving to a stupid person. ”

“ For, the several simple modes of numbers being in our minds but so many combinations of units, which have no variety, nor are capable of any other difference but more or less, names or marks for each distinct combination seem more necessary than in any other sort of ideas. For, without such names or marks, we can hardly well make use of numbers in reckoning, especially where the combination is made up of any great multitude of units ”

“ Hence, if the statistics are large enough to be satisfactory, there has been some influence at work which has improved the chances of mere coincidence in the ratio of 20 to 1. We are in fact so little able to realise the meaning of very large numbers,—that is, to retain the ratios in the mind, where large numbers are concerned,—that unless we repeatedly check ourselves by arithmetical considerations we are too apt to treat and estimate all beyond certain limits as equally vast and vague. ”