Zero

“ It is true, in arithmetic, that zero is a very important member of the number series. But it gets its whole importance by its contrasts and its definite quantitative relations with the other numbers. Just so here, if the Absolute is not only zero, but also real, also the goal, also the valuable, it is so by contrast with the finite search for that goal. But to suppose, as the mystic does, that the finite search has of itself no Being at all, is illusory, is Mâya, is itself nothing, this is also to deprive the Absolute of even its poor value as a contrasting goal. ”

“ With zero the Eternal therefore originates directly, or both are only different expressions for one and the same act, according with the difference of the science wherein they are employed. Mathematics designates its primary act by the name of zero; Philosophy by that of Eternal. It is an error to believe that numbers were absolute nothings; they are acts and consequently realities. While numbers in a mathematical sense are positions and negations of Nothing, in the philosophical they are positions and negations of the Eternal. ”

“ If one hereupon replies that this paradox of the mystic, the passing to the limit, and undertaking to define it in terms of the vanishing series, deprives the Absolute of any value as a Being, by making the whole truth a mere zero, then the mystic assures you that just this zero has infinite value, because it is the goal of the series of states of finite consciousness. Do you not want peace? he says. Can anything be of more worth to you than attainment? ”

“ Human history moves upward in circles, like an aero. The circles are at times golden, sometimes they are bloody, but all have 360 degrees. They go from 0° to 10°, 20°, 200°, 360°—and then again 0°. Yes, we have returned to zero. But for a mathematically working mind it is obvious that this zero is different: it is a perfectly new zero. We started from zero to the right and came to zero on the left. Hence instead of plus zero we are at minus zero. Do you understand?
This zero appears to me now as a silent, immense, narrow rock, sharp as a blade. ”

“ The + - or, in other words, numbers are acts or functions. Zero is, consequently, the primary act. Zero is, therefore, no absolute nothing, but an act without substratum. Generally speaking there is, therefore, no nothing; the mathematical nothing is itself an act, consequently a something. The nothing is only postulate. ”

“ It is clear that by a continuous division of unity one is carried to a constantly growing number of constantly diminishing parts, but without ever reaching the nothing represented by the number zero. To-day we should say that in this way we can reach zero only by an infinite series of steps. Yet the idea of the infinite did not exist in this form for ancient man. On the other hand, in the arabic conception of number the steps necessary to reach zero are finite. ”

“ The highest mathematical idea, or the fundamental principle of all mathematics is the zero = 0.
The whole science of mathematics depends upon zero. Zero alone determines the value in mathematics.
32. Zero is in itself nothing. Mathematics is based upon nothing, and, consequently, arises out of nothing. ”

“ As Oken says, "The ideal zero is absolute unity; not a singularity, as the number one, but an indivisibility, a numberlessness, a homogeneity, a translucency, a pure identity. It is neither great nor small, quiescent nor moved ”

“ Life is moreover only a mathematical problem, which, the higher it ascends, approaches so much the nearer to absolute zero in its attainment of the infinity of numbers, becomes so much the more endowed with life.
105. Arithmetic is the science of the second idea, or that of time and motion, or of life; it is, therefore, the first science; mathematics not only begins with it, but creation also, with the becoming of time and of life. ”

“ At all events, may be defined independently of , and its definition may be regarded as more simple as not requiring the determination of the zero from which is measured, which sometimes involves questions of a delicate nature. ”

“ That all things may be resolved into their minima's, and what is beyond them, is nothing, and that there is one maximum, or biggest, which is the world, and what is beyond that, is Infinite. Truly, Madam, I must ingeniously confess, I am not so high learned, as to penetrate into the true sense of these words; for he says, they are both divisible, and indivisible, and yet no atomes, which surpasses my Understanding; for there is no such thing, as biggest and smallest in Nature, or in the Infinite matter; for who can know how far this World goes, or what is beyond it? ”

“ How we conceive the infinity of space. The same happens also in space, wherein, conceiving ourselves to be, as it were, in the centre, we do on all sides pursue those indeterminable lines of number ; and reckoning any way from ourselves, a yard, mile, diameter of the earth, or orbis magnus, – by the infinity of number, we add others to them, as often as we will. And having no more reason to set bounds to those repeated ideas than we have to set bounds to number, we have that indeterminable idea of immensity. ”

“ Is it a breath emanating from God to organize the material universe where nothing obtains form except by the Divinity which is an effect of Number? The least as well as the greatest of creations are distinguishable from each other by quantities, qualities, dimensions, forces,—all attributes created by Number. The infinitude of Numbers is a fact proved to your soul, but of which no material proof can be given. The mathematician himself tells you that the infinite of numbers exists, but cannot be proved. ”

“ Number is the negation of infinite; as black is the negation of white. The name infinite, in this case, is, in reality, nothing but a mark for that state of consciousness, in which the idea of one more is closely associated with every succeeding number. And Infinity, the abstract term, is the peculiar idea, without the connotation. ”

“ We know the conditions which must be fulfilled, but at the same time, we see the impossibility of fulfilling them. When any one tries to persuade us of the contrary, we reflect on the idea of the infinite, and say: "No; it is a contradiction of infinity; it is not infinite, but finite." We distinguish perfectly well between the absence of the perception of the limit and its non-existence. ”

“ Moreover, there is an infinite number of worlds, some like this world, others unlike it. [67] For the atoms being infinite in number, as has just been proved, are borne ever further in their course. For the atoms out of which a world might arise, or by which a world might be formed, have not all been expended on one world or a finite number of worlds, whether like or unlike this one. Hence there will be nothing to hinder an infinity of worlds. ”

“ For, when we would consider eternity, a parte ante, what do we but, beginning from ourselves and the present time we are in, repeat in our minds the ideas of years, or ages, or any other assignable portion of duration past, with a prospect of proceeding in such addition with all the infinity of number : and when we would consider eternity, a parte post, we just after the same rate begin from ourselves, and reckon by multiplied periods yet to come, still extending that line of number as before. ”

“ Under those conditions—so barely escaped from—how wretchedly meagre and limited would have been our conception of the Universe! Aye, and, unless we foolishly imagine that our circumstances are such as to have already given us a clue to every kind of possible existence, I venture to say that "wretchedly meagre and limited" must be a true description of our conception of the Universe, even now,—even of the conception of those who have permitted themselves, with least hesitation, to follow whithersoever facts lead. ”

“ If we now cast a look on that universal nature which surrounds us, the endless multiplicity of its phænomena is indisputably manifest; and as it would be an absurdity to imagine a highest number to which another number cannot be added, we can fix the limits of nature nowhere, either in the great or in the small, because the infinite divisibility of each would lead again to infinity. ”

“ Well, you believe firmly in number and motion, inexplicable and incomprehensible as force and result, though I might apply to their existence the same logical dilemma as just now relieved you of the necessity of acknowledging that of God. You, a powerful reasoner, will surely relieve me of the necessity for proving that the Infinite must be everywhere the same, and that it is inevitably one? God alone is the Infinite, for there obviously cannot be two Infinites. ”

“ 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. ”

“ But when no more is said but this, number is infinite, it is to be understood as if it were said, this name number is an indefinite name.
What is infinite in power. Nothing infinite can be truly said to be either whole or one; nor infinite spaces or times, many. 12. Space or time is said to finite in power, or terminable, when there may be assigned a number of finite spaces or times, as of paces or hours, than which there can be no greater number of the same measure in that space or time ”

“ You must grant me that you are equally ignorant as to where number begins or ends, and as to where created eternity begins or ends? Why, then, if you believe in number, should you deny God? Does not creation hold a place between the infinite of inorganic substances and the infinite of the Divine spheres, as the unit stands between the infinite of fractions—lately termed decimals—and the infinite numbers you call whole numbers? Men alone on earth comprehend number, [Pg 100] the first step to the forecourt leading to God, and even there reason stumbles. ”

“ The various series of causes and effects are partly contingent relatively to each other, and are partly themselves continued into the so-called Infinite, and thus contain in their content nothing but those situations and forms of existence of which each is finite in itself; and what ought to give stability to the connection of the series, the Infinite namely, is not only something above and beyond this world, but is a mere negative, the very meaning of which is relative merely, and is conditioned by what is to be negated by it, and is consequently for this very reason not negated. ”

“ How could the one be like the other, which is infinite, being either below or above it, I mean larger or smaller in number? How could the infinite come to us? If an infinite number of things existed before us, how could the [idea of] number come to us? ”

“ And from hence it is manifest, that beginning and end depend upon the order in which we number them; and that to terminate or limit space and time, is the same thing with imagining their beginning and end; as also that every thing is FINITE or INFINITE, according as we imagine or not imagine it limited or terminated every way; and that the limits of any number are unities, and of these, that which is the first in our numbering is the beginning, and that which we number last, is the end. When we say number is infinite, we mean only that no number is expressed ”

“ At every step we measure motion, applying to it the idea of number; at every instant we count the forms of a portion of matter, as for example, a piece of wax, to which different forms have been given successively: whatever be [Pg 310] the number of the beings which we suppose to exist, every one of them will be susceptible of transformations which may be counted. Therefore, in the very nature of things, there is an intrinsic impossibility of the existence of an actual infinite number. ”

“ For as the absolute infinite number implies the absolute negation of all limit, when, in treating of the realization of this conception, I meet with things that are contradictory, I say that the realization of the conception is contradictory, because the general and indeterminate conception is more extended than all possible number. ”

“ But, again, the worlds also are infinite, whether they resemble this one of ours or whether they are different from it. For, as the atoms are, as to their number, infinite, as I have proved above, they necessarily move about at immense distances; for besides, this infinite multitude of atoms, of which the world is formed, or by which it is produced, could not be entirely absorbed by one single world, nor even by any worlds, the number of which was limited, whether we suppose them like this world of ours, or different from it. ”

“ But as soon as the logical reason dominates the situation, in place of this palpable plunge into a real concrete experience, with its accompanying sensation of appalling wonder and terrible freedom, we are offered nothing but a thin, dry, barren mathematical formula called "infinity," the mere mention of which freezes the imagination at its source. ”

“ The answer is that infinity is perfection without lack, and if you say that it has power without limit and no power with limit, you are detracting from its perfection. And if you say that the world, which is lacking in perfection, was the initial limit, you are lacking its power. Therefore, we must say that it is a limit without limit, and the initial limit is the spheres, which are the complete power and the lacking power. When they receive influence from its perfection, they are complete power, and when they are lacking influence, they are lacking power. ”

“ The general conception of the negation of limit gives us an idea of infinity in the abstract, but not any infinite thing. Without the intuitive cognition of an infinite object, and with only a very imperfect idea of it, we may speak of infinity without falling into contradiction, and determine the cases in which it may be applied to a being or to an order of beings, whether real or possible. Man has many ideas of this vague kind, which nevertheless answer his necessities. ”