Even number

“ I am indeed well aware that there is a saying of wise men of old that an odd number is to be preferred to an even, and they declare that the source of increase is that which does not couple. For in a pair the one term being equal to the other remains of the same quality, but when there are two numbers the third produces oddness. ”

“ Or if an observer were stationed at a certain point, and 10,000 persons were to pass him in single file without his counting them as they passed, what sort of an estimate would he make of their number? The truth seems to be that our mental conception of number is much more limited than is commonly thought, and that we unconsciously adopt some new unit as a standard of comparison when we wish to render intelligible to our minds any number of considerable magnitude. ”

“ Nothing exists except by Number. Without it, all would be one and the same substance; for Number alone differentiates and qualifies substance. Number is to your Spirit what it is to Matter, an incomprehensible agent. ”

“ A fresh mental effort is required of the ignorant child of nature; and the result is “all the fingers and one more,” “both hands and one more,” “one on another count,” or some equivalent circumlocution. If he has an independent word for 10, the result will be simply ten-one. When this step has been taken, the base is established. The savage has, with entire unconsciousness, made all his subsequent progress dependent on the number 10, or, in other words, he has established 10 as the base of his number system. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ This distinction, the distinction by which number is assigned to reason and not to sense, is, I think, an important aid towards understanding the Pythagorean philosophy.
9. Number is a necessary form of thought under which we place or subsume whatever is presented to the mind. Hence form, which is another name for number, and not matter, is the essence of all things, at least of all intelligible things. It is the truth and substance of the universe—its truth and substance, not only in so far as it exists for us, but in so far as it exists for intelligence generally. ”

“ Without number they are absolutely incomprehensible to any intelligence. Take away number, that is to say, let the universe and its contents be neither one nor many, and chaos, or worse, is come again. We are involved in contradictory nonsense. Number, then, or form, and not matter, as the Ionic philosophers contended; number, and not the numberless, or ἄπειρον of Anaximander, is the true universal, the common ground, the ultimately real in all things. ”

“ 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. ”

“ Can a number that is not determinate, that is neither 1, nor 2, nor n, nor n±r, nor any other determinate number, be regarded as being really embodied in the actual world? Can there be in the external world a countless aggregate? ”

“ Evidently, then, these thinkers also consider that number is the principle both as matter for things and as forming their modifications and their permanent states, and hold that the elements of number are the even and the odd, and of these the former is unlimited, and the latter limited; ”

“ A most inviting field for research is, however, furnished by the primitive methods of counting and of giving visible expression to the idea of number. Our starting-point must, of course, be the sign language, which always precedes intelligible speech; and which is so convenient and so expressive a method of communication that the human family, even in its most highly developed branches, never wholly lays it aside. It may, indeed, be stated as a universal law, that some practical method of numeration has, in the childhood of every nation or tribe, preceded the formation of numeral words. ”

“ 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? ”

“ There is some distinction between number and the thought of number, between being and the thought of being; and on this ground it might be argued that number, being, and the others, might perhaps not be absolute truths. Whatever is different from thought is not necessarily true for all thought. Number, being, and other universals, are different from thought, and are therefore not true for all thought. ”

“ Number is more an object of reason, and less an object of sense, than either water or air; and therefore we say that, while the position of the Ionic school is more that of sense than that of reason, the position of the Pythagorean school is more that of reason than that of sense.
6. Number is a truer universal than either water or air, or any other sensible thing. It is possible that the conception of number may not be an adequate conception of the universal in all things ”

“ And, my dear Pastor, believers will tell you that God is Number endowed with motion, to be felt but not proved. He, like the unit, is the origin of number though having nothing in common with numbers. The existence of Number depends on that of the unit, which is not a number, but the parent of them all. And God, dear Pastor Becker, is a stupendous Unit, having nothing in common with His creations, but their Parent nevertheless. ”

“ His idea is merely of some very great amount, and our own is often but little clearer when we use the expressions which are so easily represented by a few digits. Among the uneducated portions of civilized communities the limit of clear comprehension of number is not only relatively, but absolutely, very low. ”

“ Is it a breath of God sent forth to organize the material universe, wherein nothing takes shape but as a result of divisibility which is an effect of number? The most minute as well as the most immense objects in creation are distinguished from each other by quantity, quality, dimension, and force,—are not these all conditions of number? That number is infinite is a fact proved to your intellect, but of which no material proof is obtainable. A mathematician will tell you that infinity of number is certain, but cannot be demonstrated. ”

“ We know of no language in which the suggestion of number does not appear, and we must admit that the words which give expression to the number sense would be among the early words to be formed in any language. They express ideas which are, at first, wholly concrete, which are of the greatest possible simplicity, and which seem in many ways to be clearly understood, even by the higher orders of the brute creation. The origin of number would in itself, then, appear to lie beyond the proper limits of inquiry; and the primitive conception of number to be fundamental with human thought. ”

“ Numbers surely depend on a generative principle, and can carry on consecutive increase through the whole series. But I hold that it is far more just to assign the cause of that increase to the even than to the odd number. For the number one is not odd, when it has no number in respect to which it were odd. ”

“ Ten, according to some, is a perfect number, not in its composition like six, but because it contains inclosed in itself all the simple numbers and all the properties, as of even and odd, perfect and imperfect, and it is the beginning of all numbers composed of tens, and also for the foundation of our law. ”

“ But we do not always, when we perceive a thing, observe its negation with respect to others, and consequently do not always perceive number. The idea of number originates in comparison, when we see an object which is not another. ”

“ Innumerable as are the true propositions which can be formed concerning particular numbers, no adequate conception could be gained, from these alone, of the extent of the truths composing the science of number. Such propositions as we have spoken of are the least general of all numerical truths. It is true that even these are co-extensive with all nature; the properties of the number four are true of all objects that are divisible into four equal parts, and all objects are either actually or ideally so divisible. ”

“ But among races whose intelligence is of a low order, or—if it be permissible to express it in this way—among races whose number sense is feeble, progression from one number to the next is not always in accordance with any well-defined law. After one or two distinct numerals the count may, as in the case of the Veddas and the Andamans, proceed by finger pantomime and by the repetition of the same word. Occasionally the same word is used for two successive numbers, some gesture undoubtedly serving to distinguish the one from the other in the savage's mind. ”

“ Therefore, in future, on this part of the subject I shall cease to talk of any particular units, such as pebbles or acres, and speak of numbers only. A number, considered without intending to allude to any particular things, is called an abstract number: and it then merely signifies repetitions of a unit, or the number of times a unit is repeated. ”

“ 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. ”

“ In the third place, the real desideratum about such a definition as that of number is not that it should represent as nearly as possible the ideas of those who have not gone through the analysis required in order to reach a definition, but that it should give us objects having the requisite properties. Numbers, in fact, must satisfy the formulae of arithmetic ”

“ First the actual hand picture fades away, and the number is recognized without the original assistance furnished by the derivation of the word. But the number is still for a long time a certain number of objects, and not an independent concept. It is only when the savage ceases to be wholly an animal, and becomes a thinking human being, that number in the abstract can come within the grasp of his mind. It is at this point that mere reckoning ceases, and arithmetic begins. ”

“ In savage languages it is sometimes possible to detect these meanings, and thus to obtain possession of the clue that leads to the development, in the barbarian's rude mind, of a count scale—a number system. But in languages like those of modern Europe, the pedigree claimed by numerals is so long that, in the successive changes through which they have passed, all trace of their origin seems to have been lost. ”

“ But races as fond of metaphor and of linguistic embellishment as are those of the East, or as are our American Indians even, might readily resort to some other source than that furnished by the members of the human body, when in want of a term with which to describe the 5, 10, or any other number of the numeral scale they were unconsciously forming. ”

“ Such a child may count correctly from the point of view of a grown-up listener, without having any idea of numbers at all. The operation of counting, in fact, can only be intelligently performed by a person who already has some idea what the numbers are; and from this it follows that counting does not give the logical basis of number. ”

“ If a savage expresses the number 3 by the combination 2-1, it will at once be suspected that his numerals will, by the time he reaches 10 or 20, become so complex and confused that numbers as high as these will be expressed by finger pantomime rather than by words. Such is often the case; and the comment is frequently made by explorers that the tribes they have visited have no words for numbers higher than 3, 4, 5, 10, or 20, but that counting is carried beyond that point by the aid of fingers or other objects. ”