Vain are the thousand creeds That move men's hearts: unutterably vain; Worthless as withered weeds, Or idle froth amid the boundless main, To waken doubt in one Holding so fast by Thine infinity; So surely anchored on The stedfast rock of immortality.
Unendlichkeit kann nur das Wesen ahnen / Das zur Unendlichkeit erkoren ist=--Only that being can surmise the infinite who is chosen for infinity.
For it is of no avail to say it is uncertain that we gain, and certain that we risk, and that the infinite distance between the certainty of that which is staked and the uncertainty of what we shall gain, equals the finite good which is certainly staked against an uncertain infinite. This is not so. Every gambler stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty against a finite uncertainty without acting unreasonably. It is false to say there is infinite distance between the certain stake and the uncertain gain. There is in truth an infinity between the certainty of gain and the certainty of loss. But the uncertainty of gain is proportioned to the certainty of the stake, according to the proportion of chances of gain and loss, and if therefore there are as many chances on one side as on the other, the game is even. And thus the certainty of the venture is equal to the uncertainty of the winnings, so far is it from the truth that there is infinite distance between them. So that our argument is of infinite force, if we stake the finite in a game where there are equal chances of gain and loss, and the infinite is the winnings. This is demonstrable, and if men are capable of any truths, this is one.
To see a world in a grain of sand / And a heaven in a wild flower, / Hold infinity in the palm of your hand, / And eternity in an hour.
Be clearly aware of the stars and infinity on high. Then life seems almost enchanted after all.
When you will have thoroughly mastered perspective and have learnt by heart the parts and forms of objects, strive when you go about to observe. Note and consider the circumstances and the actions or men, as they talk, dispute, laugh or fight together, and not only the behaviour of the men themselves, but that of the bystanders who separate them or look on at these things; and make a note of them, in this way, with slight marks in your little note-book. And you should always carry this note-book with you, and it should be of coloured paper, so that what you {109} write may not be rubbed out; but (when it is used up) change the old for a new one, since these things should not be rubbed out, but preserved with great care, because such is the infinity of the forms and circumstances of objects, that the memory is incapable of retaining them; wherefore keep these sketches as your guides and masters.
Ad infinitum=--To infinity.
The power of the projecting force increases in proportion as the object projected is smaller; the acceleration of the motion increases to infinity proportionately to this diminution. It would follow that an atom would be almost as rapid as the imagination or the eye, which in a moment attains to the height of the stars, and consequently its voyage would be infinite, because the thing which can be infinitely diminished would have an infinite velocity and would travel on an infinite course (because every continuous quantity is divisible to infinity). And this opinion is {147} condemned by reason and consequently by experience.
The last process of reason is to recognise that there is an infinity of things which transcend it; it is but weak if it does not go so far as to know that.
A library is infinity under a roof.
He that would reckon up all the accidents preferments depend upon, may as well undertake to count the sands or sun up infinity.
Why should the thirst for knowledge be aroused, only to be disappointed and punished? My volition shrinks from the painful task of recalling my humiliation; yet, like a second Prometheus, I will endure this and worse, if by any means I may arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity.
That it was not one man who said it, but an infinity of men, and a whole nation lasting for four thousand years, prophesying, and created for that very purpose.... So I stretch out my arms to my Redeemer, who having been foretold for four thousand years, has come to suffer and to die for me on earth at the time and under all the circumstances which had been foretold, and by his grace I await death in peace, in the hope of being eternally united to him; yet I live with joy, whether in the good which it pleases him to bestow on me, or in the ill which he sends for my good, and which he has taught me to bear by his example.
Perhaps this effect of nature, which seems to you impossible beforehand, may teach you to know that there may be others also which you know not as yet. Do not then draw this conclusion from your apprenticeship, that nothing remains for you to know, but rather that an infinity remains for you to know.
I see first that they are a people wholly composed of brethren, and whereas all others are formed by the assemblage of an infinity of families, this, though so prodigiously fruitful, has sprung from one man only, and being thus all one flesh, and members one of another, they form a powerful state consisting of one family, a fact without example.
If we are well informed, we understand that nature having graven her own image and that of her author on all things, they are almost all partakers of her double infinity. Thus we see that all the sciences are infinite in the extent of their researches, for none can doubt that geometry, for instance, has an infinite infinity of problems to propose. They are also infinite in the number and in the nicety of their premisses, for it is evident that those which are finally proposed are not self-supporting, but are based on others, which again having others as their support have no finality.
There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful.
The reverberation of the voice continues for a {173} great distance through the air; for a greater distance through fire. The mind travels for a still greater distance through the universe; but since it is finite it does not penetrate into infinity.
If the East loves infinity, the West delights in boundaries.
The fraction of life can be increased in value not so much by increasing your numerator as by lessening your denominator. Nay, unless my algebra deceives me, unity itself divided by zero will give infinity.
_The end of this argument._--Now what evil will happen to you in taking this side? You will be trustworthy, honourable, humble, grateful, generous, friendly, sincere, and true. In truth you will no longer have those poisoned pleasures, glory and luxury, but you will have other pleasures. I tell you that you will gain in this life, at each step you make in this path you will see so much certainty of gain, so much nothingness in what you stake, that you will know at last that you have wagered on a certainty, an infinity, for which you have risked nothing.
Science is that discourse of the mind which derives its origin from ultimate principles beyond which nothing in nature can be found which forms a part of that science: as in the continued quantity, that is to say, the science of geometry, which, starting from the surfaces of bodies, has its origin in the line, which is the end of the superficies; and we are not satisfied by this, because we know that the line terminates in the point, and the point is that which is the least of things. Therefore the point is the first principle of geometry, and nothing else can exist either {143} in nature or in the human mind from which the point can issue. Because if you say that the contact between a surface and the extreme point of an iron instrument is the creation of the point, it is not true; but let us say that this point of contact is a superficies which surrounds its centre, and in the centre the point dwells. And such a point is not a part of the substance of the superficies, neither it nor all the points of the universe can, even if combined,--it being granted that they could be combined,--compose any part of a superficies. And granted, as you imagined, a whole composed of a thousand points, if we divide any part of this quantity of a thousand, we can very well say that this part shall equal its whole; and this we can prove by zero, or naught, that is, the tenth figure of arithmetic, which is represented by a cipher as being nothing, and placed after unity it will signify 10, and if two ciphers are placed after unity it will signify 100, and thus the number will go on increasing by ten to infinity whenever a cipher is added, and the cipher in itself is worth nothing more than naught, and all the naughts in the universe are equal to one naught alone, in regard to their substance and value.
For a man can lose neither the past nor the future; for how can one take from him that which is not his? So remember these two points: first, that each thing is of like form from everlasting and comes round again in its cycle, and that it signifies not whether a man shall look upon the same things for a hundred years or two hundred, or for an infinity of time; second, that the longest lived and the shortest lived man, when they come to die, lose one and the same thing.
The philosopher forms his principles on an infinity of particular observations. … He does not confuse truth with plausibility; he takes for truth what is true, for false what is false, for doubtful what is doubtful, and probable what is probable. … The philosophical spirit is thus a spirit of observation and accuracy.
What is that thing which is not defined and would {16} not exist if it were defined? It is infinity, which if it could be defined would be limited and finite, because that which can be defined ends with the limits of its circumference, and that which cannot be defined has no limits.
There was a young fellow from Trinity, Who took the square root of infinity. But the number of digits, Gave him the fidgets; He dropped Math and took up Divinity.
Unity joined to infinity increases it not, any more than a foot measure added to infinite space. The finite is annihilated in presence of the infinite and becomes simply nought. Thus our intellect before God, thus our justice before the divine justice. There is not so great a disproportion between our justice and that of God, as between unity and infinity.
What is a man in the infinite? But to show him another prodigy no less astonishing, let him examine the most delicate things he knows. Let him take a mite which in its minute body presents him with parts incomparably more minute; limbs with their joints, veins in the limbs, blood in the veins, humours in the blood, drops in the humours, vapours in the drops; let him, again dividing these last, exhaust his power of thought; let the last point at which he arrives be that of which we speak, and he will perhaps think that here is the extremest diminutive in nature. Then I will open before him therein a new abyss. I will paint for him not only the visible universe, but all that he can conceive of nature's immensity in the enclosure of this diminished atom. Let him therein see an infinity of universes of which each has its firmament, its planets, its earth, in the same proportion as in the visible world; in each earth animals, and at the last the mites, in which he will come upon all that was in the first, and still find in these others the same without end and without cessation; let him lose himself in wonders as astonishing in their minuteness as the others in their immensity; for who will not be amazed at seeing that our body, which before was imperceptible in the universe, itself imperceptible in the bosom of the whole, is now a colossus, a world, a whole, in regard to the nothingness to which we cannot attain.
These are some of the causes which render man so totally unable to know nature. For nature has a twofold infinity, he is finite and limited. Nature is permanent, and continues in one stay; he is fleeting and mortal. All things fail and change each instant, he sees them only as they pass, they have their beginning and end, he conceives neither the one nor the other. They are simple, he is composed of two different natures. And to complete the proof of our weakness, I will finish by this reflection on our natural condition. In a word, to complete the proof of our weakness, I will end with these two considerations....
We recognize a noble image, a marvellous conception, a supreme figure, an exalted shadow, an infinite representation of the represented infinity, a spectacle worthy of the excellence and supremacy of Him who transcendeth understanding, comprehension or grasp. Thus is the excellence of God magnified and the greatness of his kingdom made manifest; He is glorified not in one, but in countless suns; not in a single earth, a single world, but in a thousand thousand, I say in an infinity of worlds.
Are the mystics and sages insane? Because they all tell variations on the same story, don't they? The story of awakening one morning and discovering you are one with the All, in a timeless and eternal and infinite fashion. Yes, maybe they are crazy, these divine fools. Maybe they are mumbling idiots in the face of the Abyss. Maybe they need a nice, understanding therapist. Yes, I'm sure that would help. But then, I wonder. Maybe the evolutionary sequence really is from matter to body to mind to soul to spirit, each transcending and including, each with a greater depth and greater consciousness and wider embrace. And in the highest reaches of evolution, maybe, just maybe, an individual's consciousness does indeed touch infinity a total embrace of the entire Kosmos a Kosmic consciousness that is Spirit awakened to its own true nature. It's at least plausible. And tell me: is that story, sung by mystics and sages the world over, any crazier than the scientific materialism story, which is that the entire sequence is a tale told by an idiot, full of sound and fury, signifying absolutely nothing? Listen very carefully: just which of those two stories actually sounds totally insane?
A sine curve goes off to infinity, or at least the end of the blackboard.
We will say, therefore, that poetry is an art which is supremely potent for the blind, and the painting has the same result on the deaf. Painting, therefore, excels poetry in proportion as the sense to which it ministers is the nobler. The only true function of the poet is to represent the words of people who talk among each other, and these alone he represents to the hearing as if they were natural, because they are natural in themselves and created by the human voice; and in all other respects he is surpassed by the painter. Still more, incomparably greater is the width of range of painting than that of speech, because the painter can accomplish an infinity of things which speech will not be able to name for want of the appropriate terms. And seest thou not that if the painter wishes to depict animals and devils in Hell with what richness of invention he proceeds?
Five is a sufficiently close approximation to infinity.
Nature always begins the same things again, years, days, and hours, and in like manner spaces and numbers follow each other, end without end. So is made a sort of infinity and eternity, not that any thing of these is infinite and eternal, but these finite entities are infinitely multiplied.
I am incapable of conceiving infinity, and yet I do not accept finity. I want this adventure that is the context of my life to go on without end.
Imagination is the queen of truth, and possibility is one of the regions of truth. She is positively akin to infinity.
There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man's fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call the Twilight Zone.
Why is my knowledge so restricted, or my height, or my life to a hundred years rather than to a thousand? What was nature's reason for giving me such length of days, and for choosing this number rather than another, in that infinity where there is no reason to choose one more than another, since none is preferable to another?
We know that there is an infinite, but are ignorant of its nature. As we know it to be false that numbers are finite, it must therefore be true that there is an infinity in number, but what this is we know not. It can neither be odd nor even, for the addition of an unit can make no change in the nature of number; yet it is a number, and every number is either odd or even, at least this is understood of every finite number.
To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour.
Proof techniques #2: Proof by Oddity. SAMPLE: To prove that horses have an infinite number of legs. (1) Horses have an even number of legs. (2) They have two legs in back and fore legs in front. (3) This makes a total of six legs, which certainly is an odd number of legs for a horse. (4) But the only number that is both odd and even is infinity. (5) Therefore, horses must have an infinite number of legs. Topics is be covered in future issues include proof by: Intimidation Gesticulation (handwaving) "Try it; it works" Constipation (I was just sitting there and ...) Blatant assertion Changing all the 2's to _n's Mutual consent Lack of a counterexample, and "It stands to reason"
(1) Alexander the Great was a great general. (2) Great generals are forewarned. (3) Forewarned is forearmed. (4) Four is an even number. (5) Four is certainly an odd number of arms for a man to have. (6) The only number that is both even and odd is infinity. Therefore, Alexander the Great had an infinite number of arms.
Every Horse has an Infinite Number of Legs (proof by intimidation): Horses have an even number of legs. Behind they have two legs, and in front they have fore-legs. This makes six legs, which is certainly an odd number of legs for a horse. But the only number that is both even and odd is infinity. Therefore, horses have an infinite number of legs. Now to show this for the general case, suppose that somewhere, there is a horse that has a finite number of legs. But that is a horse of another color, and by the lemma ["All horses are the same color"], that does not exist.
(1) Alexander the Great was a great general. (2) Great generals are forewarned. (3) Forewarned is forearmed. (4) Four is an even number. (5) Four is certainly an odd number of arms for a man to have. (6) The only number that is both even and odd is infinity. Therefore, all horses are black.
I had a feeling once about mathematics -- that I saw it all. Depth beyond depth was revealed to me -- the Byss and the Abyss. I saw -- as one might see the transit of Venus or even the Lord Mayor's Show -- a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why tergiversation was inevitable -- but it was after dinner and I let it go. -- Winston Churchill
Lemma: All horses are the same color. Proof (by induction): Case n = 1: In a set with only one horse, it is obvious that all horses in that set are the same color. Case n = k: Suppose you have a set of k+1 horses. Pull one of these horses out of the set, so that you have k horses. Suppose that all of these horses are the same color. Now put back the horse that you took out, and pull out a different one. Suppose that all of the k horses now in the set are the same color. Then the set of k+1 horses are all the same color. We have k true => k+1 true; therefore all horses are the same color. Theorem: All horses have an infinite number of legs. Proof (by intimidation): Everyone would agree that all horses have an even number of legs. It is also well-known that horses have forelegs in front and two legs in back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a horse to have! Now the only number that is both even and odd is infinity; therefore all horses have an infinite number of legs. However, suppose that there is a horse somewhere that does not have an infinite number of legs. Well, that would be a horse of a different color; and by the Lemma, it doesn't exist.
God is the tangential point between zero and infinity. -- Alfred Jarry
Five is a sufficiently close approximation to infinity. -- Robert Firth "One, two, five." -- Monty Python and the Holy Grail
"Lines that are parallel meet at Infinity!" Euclid repeatedly, heatedly, urged. Until he died, and so reached that vicinity: in it he found that the damned things diverged. -- Piet Hein
A sine curve goes off to infinity, or at least the end of the blackboard. -- Prof. Steiner
It would seem that Our Lord finds our desires not too strong, but too weak. We are half-hearted creatures, fooling about with drink and sex and ambition when infinite joy is offered us, like an ignorant child who wants to go on making mud pies in a slum because he cannot imagine what is meant by the offer of a holiday at the sea. We are far too easily pleased.
My opinion, my conviction, gains infinitely in strength and sureness the moment a second mind has adopted it.
There is nothing good or godlike in this world but has in it something of "infinite sadness."
There is in this world infinitely more joy than pain to be shared, if you will only take your share when it is set before you.
But the infinitely little is far less evident. Philosophers have much more frequently asserted they have attained it, yet in that very point they have all stumbled. This has given occasion to such common titles as _The Origin of Creation_, _The Principles of Philosophy_, and the like, as presumptuous in fact though not in appearance as that dazzling one, _De omni scibili_.
"The first love, which is infinite," can be followed by no second like it.
Alone with Jesus! What a sweet and holy spot! What a blessed refuge to which the soul may betake itself from the charges of Satan, the accusations of the world, and the sorrows of life! Sweet spot for the heart to unfold itself, to tell its hidden tale in the ear of Infinite love, tenderness, and compassion!