Cardinal number

“ One result is the formation of a number-series as a mere succession of names without any corresponding ideas of number; the series not being necessarily correct.
When numbering begins, the names of the successive numbers are attached to the individual objects; thus the numbers are originally ordinal, not cardinal.
The conception of number as cardinal, i.e. as something belonging to a group of objects as a whole, is a comparatively late one, and does not arise until the idea of a whole consisting of its parts has been formed. This is the quantitative aspect of number. ”

“ On the contrary, the premisses of arithmetic can be put in other forms, and, furthermore, an indefinite number of propositions of arithmetic can be proved directly from logical principles without mentioning them. Thus, while arithmetic may be defined as that branch of deductive reasoning concerning classes and relations which is concerned with the establishment of propositions concerning cardinal numbers, it must be added that the introduction of cardinal numbers makes no great break in this general science. It is no more than an interesting subdivision in a general theory. ”

“ The ordinal types of finite ordered aggregates offer no special interest. For we easily convince ourselves that, for one and the same finite cardinal number , all simply ordered aggregates are similar to one another, and thus have one and the same type. Thus the finite simple ordinal types are subject to the same laws as the finite cardinal numbers, and it is allowable to use the same signs for them, although they are conceptually different from the cardinal numbers. ”

“ That class which naturally presents itself first to us, and whose complete investigation must, accordingly, be the next special aim of the theory of transfinite aggregates, is the class of types which embraces all the types with the least transfinite cardinal number . From the cardinal number which determines the class of types we have to distinguish that cardinal number which for its part [499] is determined by the class of types . ”

“ The systems adopted for numeration and for notation do not always agree with one another; nor do they always correspond with the idea which the numbers subjectively present. This latter presentation may, in the absence of any accepted term, be called number-ideation; this word covering not only the perception or recognition of particular numbers, but also the formation of a number-concept.
13. Notation of Numbers.—The system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the Hindu, sometimes incorrectly called the Arabic, system. ”

“ From extant charters it is known to have been in use in England before the close of the 8th century. Before its adoption the usual practice in Latin countries was to distinguish the years by their number in the cycle of Indiction.
In the Christian era the years are simply distinguished by the cardinal numbers; those before Christ being marked B.C. (Before Christ) , or a.c. (Ante Christum) , and those after Christ A.D. (Anno Domini) . This method of reckoning time is more convenient than those which employ cycles or periods of any length whatever ”

“ It will suffice to mention here that Peano’s fourth premiss of arithmetic does not hold for infinite cardinals or for infinite ordinals. Contrasting the above definitions of number, cardinal and ordinals, with the alternative theory that number is an ultimate idea incapable of definition, we notice that our procedure exacts a greater attention, combined with a smaller credulity; for every idea, assumed as ultimate, demands a separate act of faith.
The Data of Analysis.—Rational numbers and real numbers in general can now be defined according to the same general method. ”

“ To that question you have replied by quoting a passage from the fourth book of the cardinal's great work. It is quite true that the words quoted by you occur at that place; it is quite as untrue that the "monstrous paradox" is there attempted to be maintained. A reference to the book will show at once that this paradox is simply used as an argument to enable the cardinal to prove his point by the common method of a reductio ad absurdum. If what I maintain, says the cardinal, is false, then it follows that "should the Pope," &c. ”

“ The system differs completely from the Hindu system. There are no single symbols for two, three, &c.; but numbers are represented by combinations of symbols for one, five, ten, fifty, one hundred, five hundred, &c., the numbers which have single symbols, viz. I, V, X, L, C, D, M, proceeding by multiples of five and two alternately. Thus 1878 is MDCCCLXXVIII, i.e. thousand five-hundred hundred hundred hundred fifty ten ten five one one one.
The system is therefore essentially a cardinal and grouping one, i.e. it represents a number as the sum of sets of other numbers. ”

“ B. Every one of these numbers is greater than the preceding ones and less than the following ones (§ 2) .
C. There are no cardinal numbers which, in magnitude, lie between two consecutive numbers and (§ 2) . ”

“ Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers. ”

“ While considering the extent of the savage's arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense? To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity? Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be. ”

“ Several nations in America have no other instruments of calculation. It was probably the same in the primitive ages. It is another strong presumption of the truth of what I now advance, that all civilized nations count by tens, tens of tens, or hundreds, tens of hundreds, thousands, and so on; still from ten to ten. We can discover no reason why the number ten should be chosen rather than any other for the term of numeration, except this primitive practice of counting by the fingers. ”

“ What applies to exponentiation applies also to the product of two cardinals. We might define "" as the sum of the numbers of classes each having terms, but we prefer to define it as the number of ordered couples to be formed consisting of a member of followed by a member of , where has terms and has terms. This definition, also, is designed to evade the necessity of assuming the multiplicative axiom. ”

“ That the first numbers of a numeral scale are usually derived from other sources, we have some reason to believe; but that all above 2, 3, or at most 4, are almost universally of digital origin we must admit. Exception should properly be made of higher units, say 1000 or anything greater, which could not be expected to conform to any law of derivation governing the first few units of a system. ”

“ In numbers from a hundred and one upwards, the larger number comes first, either with or without et; but with distributives et is not used. With cardinals and ordinals the smaller number sometimes comes first with et ”

“ The division of the fingers into two sets of five each is still in his mind, but it is no longer the leading idea. As the count proceeds further, the quinary base may be retained, or it may be supplanted by a decimal or a vigesimal base. How readily the one or the other may predominate is seen by a glance at the following numerals:
By the time 20 is reached the savage has probably allowed his conception of any aggregate to be so far modified that this number does not present itself to his mind as 4 fives. ”

“ The process just indicated may be gone through with at 5, or at 20, thus giving us a quinary or a vigesimal, or, more probably, a mixed system; and, in rare instances, some other number may serve as the point of departure from simple into compound numeral terms. But the general idea is always the same, and only the details of formation are found to differ.
Without the establishment of some base any system of numbers is impossible. The savage has no means of keeping track of his count unless he can at each step refer himself to some well-defined milestone in his course. ”

“ You have appropriated to yourself a place in the Infinite of Number; you have fitted it to your own proportions by creating (if indeed you did create) arithmetic, the basis on which all things rest, even your societies. Just as Number—the only thing in which your self-styled atheists believe—organized physical creations, so arithmetic, in the employ of Number, organized the moral world. This numeration must be absolute, like all else that is true in itself ”

“ Any other origin is difficult to reconcile with observed facts, and still more difficult to reconcile with any rational theory of number system development. Dismissing from consideration the quinary scale, let us briefly examine once more the natural process of evolution through which the decimal and the vigesimal scales come into being. After the completion of one count of the fingers the savage announces his result in some form which definitely states to his mind the fact that the end of a well-marked series has been reached. ”

“ In the selection of a perfect name, therefore, these three forces must be taken into consideration. Thus, more than ordinary care should be exerted in the choice of the Cornerstone, for the influence of this force will be felt throughout the entire life of the individual, or object, under its vibrations. If it is in opposition to the Triad in which the BIRTH NUMBER has its place, it is certain to interfere seriously with the successful culmination of every undertaking over which that discordant number can assert an influence. ”

“ On the contrary, monks, some arithmetician or master of arithmetic might, indeed, be able by calculation to find the end or limit of the worlds, both those where the atoms have been deposited and where they have not, but it is impossible by applying the rules of arithmetic to find the limit of those hundred thousands of myriads of Æons; so long, so inconceivable, so immense is the number of Æons which have elapsed since the expiration of that Lord, the Tathâgata Mahâbhigñâgñânâbhibhû. ”

“ It is not likely that Alcuin was familiar with the apices of Boethius or with the Roman method of reckoning on the abacus. He belongs to that long list of scholars who dragged the theory of numbers into theology. Thus the number of beings created by God, who created all things well, is 6, because 6 is a perfect number (the sum of its divisors being ) ”

“ In the selection of a base,—of a number from which he makes a fresh start, and to which he refers the next steps in his count,—the savage simply follows nature when he chooses 10, or perhaps 5 or 20. But it is a matter of the greatest interest to find that other numbers have, in exceptional cases, been used for this purpose. Two centuries ago the distinguished philosopher and mathematician, Leibnitz, proposed a binary system of numeration. ”

“ In 1624 appeared in London a curious work on the subject of numbers, bearing the following title: The Secrets of Numbers, according to Theological, Arithmetical, Geometrical, and Harmonical Computation; drawn, for the better part, out of those Ancients, as well as Neoteriques. Pleasing to read, profitable to understand, opening themselves to the capacities of both learned and unlearned; being no other than a key to lead men to any doctrinal knowledge whatsoever. In the ninth chapter the author has given many notable opinions from learned men, to prove the excellency of the number seven. ”

“ 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. ”

“ According to Ball's 'Short Account of the History of Mathematics,' the arabic arithmetic was practically unknown in Europe until the end of the thirteenth century. At that date numbers were written in Roman numerals. When we see a date inscribed in Roman numerals upon the portal of a public building, we witness the persistence, in art, of the system which was in universal use for all the arithmetical purposes of the civilized world only a few centuries ago.
How simple to our minds to-day seems such a numerical problem as a determination of the number of seconds in a mean solar day. ”

“ Seven, or the heptad, is a number very powerful for good or for evil. It belongs especially to sacred things.
“The number Eight, or the octad, is the first cube, that is to say, squared in all senses, as a die, proceeding from its base two, an even number; so is man four-square, or perfect.
“The number Nine, or the ennead, being the multiple of three, should be regarded as sacred. ”

“ But the beginning of arithmetic, which includes by combination philosophy, is [208] a number which is boundless and incomprehensible, containing within itself all the numbers capable of coming to infinity by multitude. But the beginning of the numbers becomes by hypostasis the first monad, which is a male unit begetting as does a father all the other numbers. Second comes the dyad, a female number, and the same is called even by the arithmeticians. Third comes the triad, a male number ”