Infinite divisibility

“ For in order to form a just notion of these animals, we must have a distinct idea representing every part of them; which, according to the system of infinite divisibility, is utterly impossible, and according to that of indivisible parts or atoms, is extremely difficult, by reason of the vast number and multiplicity of these parts. ”

“ This conclusion is fully warranted by the infinite divisibility of the continuum. But here again the advocates of material continuity contend that this divisibility is potential, and can never be reduced to act; whence they infer that the multitude of the parts is not actual, but potential. We, however, repeat that if the division is potential, the divisible matter is certainly actual; and therefore the potency of an infinite division presupposes an infinite multitude of distinct terms actually existing in the divisible matter. ”

“ Really to conceive the infinite divisibility of matter, is mentally to follow out the divisions to infinity; and to do this would require infinite time. On the other hand, to assert that matter is not infinitely divisible, is to assert that it is reducible to parts which no conceivable power can divide ”

“ The appearance seems to us superficial, it has to do with surfaces; but behind these there is the solid reality. What do we mean by this, he asks, in effect, but that we should, could we examine, find ever new surfaces, the process of infinite divisibility being only the possibility of an infinitude of surfaces? Thus, from whatever point we start, we find it impossible to get beyond the world of mental feelings and processes. It is impossible, because we cannot get out of ourselves. ”

“ The divisibility of a body rests upon the divisibility of space, which is the condition of the possibility of the body as an extended whole. A body is consequently divisible to infinity, though it does not, for that reason, consist of an infinite number of parts. ”

“ This receding limit of the infinitely extensible space involves the character of infinite divisibility, for if there are an infinite number of points from which straight lines can be drawn without intersecting one another to any fixed point there is therefore no smallest space that cannot be further divided. ”

“ Every thing capable of being infinitely divided contains an infinite number of parts; otherwise the division would be stopt short by the indivisible parts, which we should immediately arrive at. If therefore any finite extension be infinitely divisible, it can be no contradiction to suppose, that a finite extension contains an infinite number of parts: And vice versa, if it be a contradiction to suppose, that a finite extension contains an infinite number of parts, no finite extension can be infinitely divisible. ”

“ It requires scarce any induction to conclude from hence, that the idea, which we form of any finite quality, is not infinitely divisible, but that by proper distinctions and separations we may run up this idea to inferior ones, which will be perfectly simple and indivisible. In rejecting the infinite capacity of the mind, we suppose it may arrive at an end in the division of its ideas ”

“ We take a quantity, halve it, and again halve this half, and so on without end, but we shall never obtain the infinity of smallness; for the quantity indefinitely divided will always remain indefinitely divisible. At whatever degree of division we may have arrived, between what remains and nothingness there extends always the abyss of the infinite. So I seek for the object of infinite goodness: that object must be infinitely destitute. I diminish accordingly the existence of the universe: I extinguish all the rays of its beauty ”

“ Then slowly, caressingly, irresistibly, the witchery of the Infinite grows upon you: out of Time and Space you begin to dream with open eyes,—to drift into delicious oblivion of facts,—to forget the past, the present, the substantial,—to comprehend nothing but the existence of that infinite Blue Ghost as something into which you would wish to melt utterly away forever.... ”

“ But all the time one has to be living on the true plane as well, which is eternity and infinity. If you come to think of it, the one thing which the human mind cannot grasp is the finite, not the infinite, the temporary, not the eternal. ”

“ But why attempt to describe the infinite? It is a hopeless task; and as space is infinite, it is useless to attempt to describe it. Let imagination fancy it as vast as it can, but it is still vaster than that. ”

“ Is there an infinite beyond us? Is that infinite there, inherent, permanent; necessarily substantial, since it is infinite; and because, if it lacked matter it would be bounded; necessarily intelligent, since it is infinite, and because, if it lacked intelligence, it would end there? Does this infinite awaken in us the idea of essence, while we can attribute to ourselves only the idea of existence? ”

“ Although these arguments in favor of the existence of the idea of the infinite, if we examine our own mind, we cannot deny that we find there a certain vagueness and confusion which inspire strong doubts as to the reality of this idea. What is presented to our mind, when we think of the infinite? The imagination abandoned to itself, extends space, expands dimensions, multiplies numbers indefinitely, but it offers nothing to the intellect which has the marks of infinity. ”

“ No more it is, but still less is it mere reasoning.” “If it seems too bold to say that man actually sees the invisible, let us say that he suffers from the invisible, and the invisible is only a special name for the infinite.” ”

“ The same idea is here made clear that in this external world, which is full of finite things, it is impossible to see and find the Infinite. The Infinite must be sought in that alone which is infinite, and the only thing infinite about us is that which is within us, our own soul. ”

“ Reality exists now, but the infinite has disappeared. To describe a figure in space, we are obliged to limit infinite space; to represent to ourselves a change in time, we are obliged to divide the totality of time. Thus we only arrive at reality by limitation, at the positive, at a real position, by negation or exclusion; to determination, by the suppression of our free determinableness. ”

“ Because none is great and none is small. All things are interpenetrated by that infinite ocean; their reality is that infinite; and whatever there is on the surface is but that infinite. The tree is infinite; so is everything that you see or feel — every grain of sand, every thought, every soul, everything that exists, is infinite. Infinite is finite and finite infinite. This is our existence. ”

“ It is only by emphasising our finiteness that we ever become conscious of the infinite. The infinite can only be that which stretches far beyond the boundaries of our own personality. It is the charm of dreams that they introduce us into a new infinity. ”

“ It is the pang of separation that spreads throughout the world and gives birth to shapes innumerable in the infinite sky. 2
But the separation of the two is not the last word. The universe is not merely a going forth from God, but is also a coming towards Him. The union of the two is complete, but in the universe it is being made. The finite represents to us the tension of the self to become the infinite. ”

“ Logically speaking, the distance between two points, however near, may be said to be infinite, because it is infinitely divisible. But we do cross the infinite at every step, and meet the eternal in every second. Therefore some of our philosophers say there is no such thing as finitude; it is but a may a, an illusion. The real is the infinite, and it is only maya, the unreality, which causes the appearance of the finite. ”

“ Suddenly as thus they rode from infinite to infinite, suddenly as thus they tilted over abysmal worlds, a mighty cry arose—that systems more mysterious, that worlds more billowy,—other heights, and other depths,—were coming, were nearing, were at hand. ”

“ To deny the will of the infinite, that is to say, God, is impossible on any other conditions than a denial of the infinite. We have demonstrated this.
The negation of the infinite leads straight to nihilism. Everything becomes "a mental conception." ”

“ If we pass from these relatively near bodies to the thirty millions of remote suns and solar systems, where shall we find a reason for all this apparently unconscious existence, infinite in amount compared with the existence which is conscious—a waste Universe as it seems? Then behind these mysteries lies the all-embracing mystery—whence this universal transformation which has gone on unceasingly throughout a past eternity and will go on unceasingly throughout a future eternity? ”

“ For infinite is the distance that lies between truth and untruth, between death and deathlessness. Yet this measureless gulf is bridged in a moment when the self revealing one reveals himself in the soul. There the miracle happens, for there is the meeting-ground of the finite and infinite. ”

“ Infinity unification as an experiential reality is unthinkably remote, but an unqualified unity of infinity now dominates the present moment of universe existence and unites the divergencies of all reality with an existential majesty that is absolute.
When finite creatures attempt to conceive of infinite unification on the finality levels of consummated eternity, they are face to face with intellect limitations inherent in their finite existences. Time, space, and experience constitute barriers to creature concept ”

“ Shall we leave the finite wherein we dwell to be swallowed up in this or the other infinite? In other words, shall we end by absorption in the infinite which our reason conceives, or shall we remain eternally in that which our eyes behold, that is to say, in numberless changing and ephemeral worlds? ”

“ We mathematically demonstrate
Infinite God as infinitely false
To infinite impossibility.
Henceforth a grievous shadow quits the earth,
While man, the fruitage and the flower of things,
Walks fetterless and free. ”

“ And by an infinite decomposition we should know nothing more of what a thing really is, for, the moment we decompose it, it ceases to be, and all its meaning is vanished. Infinitely more than astronomy even, which destroys nothing, can do for us, is done by the mere aspect and changes of the vault over our heads. Think for a moment what would be our idea of greatness, of God, of infinitude, of aspiration, if, instead of a blue, far withdrawn, light-spangled firmament, we were born and reared under a flat white ceiling! ”

“ What is meant by a part of the Infinite? The Infinite is indivisible; there cannot be parts of the Infinite. The Absolute cannot be divided. What is meant therefore that all these sparks are from Him? ”

“ The tremendous disintegrating force slackens in exact relation to the time which passes, so that the smaller the morsel the less the relative loss of mass. Here, then, is the Indefinite Regress. In the world of fact as well as of ideas we are dealing with aspects of Infinity. ”

“ Let us not say to ourselves that it would require an unimaginable concourse of circumstances to reproduce a globe like unto our earth in every respect. We must remember that we are in the infinite and that this unimaginable concourse must necessarily take place in the innumerousness which we are unable to imagine. ”

“ In the second place, though matter, considered as the occult cause of our sensations, do really exist, yet what we call divisibility may be an attribute only of our sensations of sight and touch, and not of their uncognizable cause. Divisibility may not be predicable at all, in any intelligible sense, of Things in themselves, nor therefore of Matter in itself ”

“ But is not the infinite itself an absurdity for our finite appreciation? We feel all the same that it is; the infinite invades us, overflows us, renders us dizzy at its abysses and crushes us by its awful height.
Scientifically probable hypotheses are one and all the last half-lights or shadows of science; faith begins where reason falls exhausted. Beyond human reason there is that Reason which is Divine—for my weakness a supreme absurdity, but an infinite absurdity which confounds me, and in which I believe. ”

“ The universe begins by being an aggregate of accidents, pluralistic, discontinuous, irrational, and, of itself, cannot become otherwise. There is nothing actual within to change its character.... God is himself finite, helpless to bring about this great change, a part, and no more, of a universe broken in fragments. ”

“ First, of Infinity, or the type of the Divine Incomprehensibility.—A boundless space will occur directly to the reader as a type of the infinite; perhaps it should be rather described as itself the infinite under one form. But Mr Ruskin finds the infinite in everything. That idea which he justly describes as the incomprehensible, and which is so profound and baffling a mystery to the finite being, is supposed to be thrust upon the mind on every occasion. Every instance of variety is made the type of the infinite, as well as every indication of space. ”

“ With regard to the idea of Infinite, comprehended also within the idea of Space, I say that we owe this idea of Infinite only to the power which a man standing on a plain has of continually extending its limits, the boundary of his imagination not being determinable: the absence of limits is therefore the only idea we can form of Infinite. ”

“ If He, the Infinite one, who, veiled from thy sight, sits surrounded by the gleaming abysses, without bounds save such as Himself creates, were to lay bare to thy sight the immeasurability of infinity, and let Himself be seen of thee as he distributes the suns, the great spirits, the little human hearts, and our days, and a tear or two therein ”

“ The natural Necessity of the ancient Greeks, the trinitarian God of the mediaeval school-man, the great First Cause of the eighteenth-century deist, the primordial Life-Force of the modern man of science, are all on common ground here in regard to the unfathomableness of the ultimate mystery.
But the revelation of the complex vision saves us from the logical boredom of the word "infinite." The idea of the infinite is merely a tedious mathematical formula, marking the psychological point where the mind finds its stopping-place. ”