Infinity
Infinity (symbolzed: ∞) is a term derived from the Latin infinitas or "unboundedness" denoting concepts involving limitless quantity, numeration, extension or expansion. In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e. a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.
Quotes[edit]
A[edit]
 Ford, there’s an infinite number of monkeys outside who want to talk to us about this script for Hamlet they’ve worked out.
 Douglas Adams The Hitchhiker's Guide to the Galaxy (radio series) ep. 2 (1977)
 All things were together, infinite both in number and in smallness; for the small too was infinite.
 Anaxagoras, Frag. B 1 from Early Greek Philosophy, Chapter 6, John Burnet (1920).
 In Sorbière's day, European thinkers and intellectuals of widely divergent religious and political affiliations campaigned tirelessly to stamp out the doctrine of indivisibles and to eliminate it from philosophical and scientific consideration. In the very years that Hobbes was fighting Wallis over the indivisible line in England, the Society of Jesus was leading its own campaign against the infinitely small in Catholic lands. In France, Hobbes's acquaintance René Descartes, who had initially shown considerable interest in infinitesimals, changed his mind and banned the concept.. Even as late as the 1730s... George Berkeley mocked mathematicians for their use of infinitesimals... Lined up against these naysayers were some of the most prominent mathematicians and philosophers of that era, who championed the use of the infinitesimally small. These included, in addition to Wallis: Galileo and his followers, Bernard Le Bovier de Fontenelle, and Isaac Newton.
 Amir Alexander, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (2014)
 On the one side were ranged the forces of hierarchy and order—Jesuits, Hobbesians, French Royal Courtiers, and High Church Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesimals. On the other side were comparative "liberalizers" such as Galileo, Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesimals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come.
 Amir Alexander, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (2014)
 Mind is infinite and selfruled, and is mixed with nothing, but is alone itself by itself.
 Anaxagoras, Frag. B 12 from Early Greek Philosophy, Chapter 6, John Burnet (1920).
 Empedocles holds that the corporeal elements are four, while all the elementsincluding those which initiate movementare six in number; whereas Anaxagoras agrees with Leucippus and Democritus that the elements are infinite.
 Motion is supposed to belong to the class of things which are continuous; and the infinite presents itself first in the continuousthat is how it comes about that 'infinite' is often used in definitions of the continuous ('what is infinitely divisible is continuous'). Besides these, place, void, and time are thought to be necessary conditions of motion.
 Aristotle, Physics Bk III.1, Hardie and Gaye.
 The science of nature is concerned with spatial magnitudes and motion and time, and each of these at least is necessarily infinite or finite, even if some things dealt with by the science are not, e.g. a quality or a pointit is not necessary perhaps that such things should be put under either head. Hence it is incumbent on the person who specializes in physics to discuss the infinite and to inquire whether there is such a thing or not, and, if there is, what it is. The appropriateness to the science of this problem is clearly indicated. All who have touched on this kind of science in a way worth considering have formulated views about the infinite, and indeed, to a man, make it a principle of things.
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a selfsubsistent substance, and not as a mere attribute of some other thing. Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also.
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 The Pythagoreans identify the infinite with the even. For this, they say, when it is cut off and shut in by the odd, provides things with the element of infinity. An indication of this is what happens with numbers. If the gnomons are placed round the one, and without the one, in the one construction the figure that results is always different, in the other it is always the same. But Plato has two infinities, the Great and the Small.
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 The physicists... always regard the infinite as an attribute of a substance which is different from it and belongs to the class of the socalled elementswater or air or what is intermediate between them. Those who make them limited in number never make them infinite in amount. But those who make the elements infinite in number, as Anaxagoras and Democritus do, say that the infinite is continuous by contactcompounded of the homogeneous parts.
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 We cannot say that the infinite has no effect, and the only effectiveness which we can ascribe to it is that of a principle. Everything is either a source or derived from a source. But there cannot be a source of the infinite or limitless, for that would be a limit of it. Further, as it is a beginning, it is both uncreatable and indestructible. For there must be a point at which what has come to be reaches completion, and also a termination of all passing away. That is why, as we say, there is no principle of this, but it is this which is held to be the principle of other things, and to encompass all and to steer all, as those assert who do not recognize, alongside the infinite, other causes, such as Mind or Friendship. Further they identify it with the Divine, for it is 'deathless and imperishable' as Anaximander says, with the majority of the physicists.
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 Belief in the existence of the infinite comes mainly from five considerations: 1) From the nature of timefor it is infinite. 2) From the division of magnitudesfor the mathematicians also use the notion of the infinite. 3) If coming to be and passing away do not give out, it is only because that from which things come to be is infinite. 4) Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself. 5) Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybodynot only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought.
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 That what is outside is infinite leads people to suppose that body also is infinite, and that there is an infinite number of worlds. Why should there be body in one part of the void rather than in another? Grant only that mass is anywhere and it follows that it must be everywhere. Also, if void and place are infinite, there must be infinite body too, for in the case of eternal things what may be must be.
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 The problem of the infinite is difficult: many contradictions result whether we suppose it to exist or not to exist. If it exists, we have still to ask how it exists; as a substance or as the essential attribute of some entity? Or in neither way, yet none the less is there something which is infinite or some things which are infinitely many?
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 The problem... which specially belongs to the physicist is to investigate whether there is a sensible magnitude which is infinite. We must begin by distinguishing the various senses in which the term 'infinite' is used. 1) What is incapable of being gone through, because it is not in its nature to be gone through (the sense in which the voice is 'invisible'). 2) What admits of being gone through, the process however having no termination, or what scarcely admits of being gone through. 3) What naturally admits of being gone through, but is not actually gone through or does not actually reach an end. Further, everything that is infinite may be so in respect of addition or division or both.
 Aristotle, Physics Bk III.4, Hardie and Gaye.
 If 'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible or sensible.
 Aristotle, Physics Bk III.5, Hardie and Gaye.
 Anaxagoras gives an absurd account of why the infinite is at rest. He says that the infinite itself is the cause of its being fixed. This because it is in itself, since nothing else contains iton the assumption that wherever anything is, it is there by its own nature. But this is not true: a thing could be somewhere by compulsion, and not where it is its nature to be.
 Aristotle, Physics Bk III.5, Hardie and Gaye.
 The view that there is an infinite body is plainly incompatible with the doctrine that there is necessarily a proper place for each kind of body, if every sensible body has either weight or lightness, and if a body has a natural locomotion towards the centre if it is heavy, and upwards if it is light. This would need to be true of the infinite also. But neither character can belong to it: it cannot be either as a whole, nor can it be half the one and half the other. For how should you divide it? Or how can the infinite have the one part up and the other down, or an extremity and a centre?
 Aristotle, Physics Bk III.5, Hardie and Gaye.
 To suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and an end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. ...clearly there is a sense in which the infinite exists and another in which it does not.
 Aristotle, Physics Bk III.6, Hardie and Gaye.
 The infinite turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it.
 Aristotle, Physics Bk III.6, Hardie and Gaye.
 Our definition then is as follows: A quantity is infinite if it is such that we can always take a part [or piece] outside what has been already taken. On the other hand, what has nothing outside it is complete and whole. For thus we define the wholethat from which nothing is wanting, as a whole man or a whole box. What is true of each particular is true of the whole as suchthe whole is that of which nothing is outside. On the other hand that from which something is absent and outside, however small that may be, is not 'all'. 'Whole' and 'complete' are either quite identical or closely akin. Nothing is complete (teleion) which has no end (telos); and the end is a limit.
 Aristotle, Physics Bk III.6, 207a7, Hardie and Gaye.
 Parmenides must be thought to have spoken better than Melissus. The latter says that the whole is infinite, but the former describes it as limited, 'equally balanced from the middle'. ...it is absurd and impossible to suppose that the unknowable and indeterminate should contain and determine.
 Aristotle, Physics Bk III.6, Hardie and Gaye.
 What is one is indivisible whatever it may be, e.g. a man is one man, not many. Number on the other hand is a plurality of 'ones' and a certain quantity of them. Hence number must stop at the indivisible: for 'two' and 'three' are merely derivative terms, and so with each of the other numbers.
 Aristotle, Physics Bk III.6, Hardie and Gaye.
 In the direction of largeness it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number. But this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be, like time and the number of time.
 Aristotle, Physics Bk III.7, 207b12.
 With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens.
 Aristotle, Physics Bk III.7, Hardie and Gaye.
 Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish.
 Aristotle, Physics Bk III.7, Hardie and Gaye.
 It is plain that the infinite is a cause in the sense of matter, and that its essence is privation, the subject as such being what is continuous and sensible. All the other thinkers, too, evidently treat the infinite as matterthat is why it is inconsistent in them to make it what contains, and not what is contained.
 Aristotle, Physics Bk III.7, Hardie and Gaye.
 It remains to dispose of the arguments which are supposed to support the view that the infinite exists not only potentially but as a separate thing. Some have no cogency; others can be met by fresh objections that are valid. 1) In order that coming to be should not fail, it is not necessary that there should be a sensible body which is actually infinite. The passing away of one thing may be the coming to be of another, the All being limited. 2) There is a difference between touching and being limited. The former is relative to something and is the touching of something (for everything that touches touches something), and latter is an attribute of some one of the things which are limited. On the other hand, what is limited is not limited in relation to anything. Again, contact is not necessarily possible between any two things taken at random. 3) To rely on mere thinking is absurd, for then the excess or defect is not in the thing but in the thought. One might think that one of us is bigger than he is and magnify him ad infinitum. But it does not follow that he is bigger than the size we are, just because some one thinks he is, but only because he is the size he is. The thought is an accident. a) Time indeed and movement are infinite, and also thinking, in the sense that each part that is taken passes in succession out of existence. b) Magnitude is not infinite either in the way of reduction or of magnification in thought. This concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is.
 Aristotle, Physics Bk III.8, Hardie and Gaye.
 If, then, there is some end of the things we do, which we desire for its own sake (everything else being desired for the sake of this), and if we do not choose everything for the sake of something else (for at that rate the process would go on to infinity, so that our desire would be empty and vain), clearly this must be the good and the chief good.
 Aristotle, The Nicomachean Ethics David Ross, 1961.
 SWA Magazine: How about orbital space colonies? Do you see these facilities being built or is the government going to cut back on projects like this?
 Asimov: Well, now you've put your finger right on it. In order to have all of these wonderful things in space, we don't have to wait for technology  we've got the technology, and we don't have to wait for the knowhow  we've got that too. All we need is the political goahead and the economic willingness to spend the money that is necessary. It is a little frustrating to think that if people concentrate on how much it is going to cost they will realize the great amount of profit they will get for their investment. Although they are reluctant to spend a few billions of dollars to get back an infinite quantity of money, the world doesn't mind spending $400 billion every years on arms and armaments, never getting anything back from it except a chance to commit suicide.
 Isaac Asimov "Southwest Airlines Magazine." (1979) [1]
 For if they imagine infinite spaces of time before the world, during which God could not have been idle, in like manner they may conceive outside the world infinite realms of space, in which, if any one says that the Omnipotent cannot hold His hand from working, will it not follow that they must adopt Epicurus’ dream of innumerable worlds? with this difference only, that he asserts that they are formed and destroyed by the fortuitous movements of atoms, while they will hold that they are made by God’s hand, if they maintain that, throughout the boundless immensity of space, stretching interminably in every direction round the world, God cannot rest, and that the worlds which they suppose Him to make cannot be destroyed. ... there is no place beside the world ...no time before the world.
 Augustine of Hippo, The City of God, Book XI, Ch. 5 "That We Ought Not to Seek to Comprehend the Infinite Ages of Time Before the World, Nor the Infinite Realms of Space"
 Every material body has some natural movement, and can change its place. But an infinite body would occupy every place, and every place would be its own place. ...Every mathematical body must be imagined as having some shape. But shape is defined by some term or boundary, and nothing infinite can have a boundary.
 Thomas Aquinas, Summa Theologica (I Q.7) Book I, Question vii (ca. 1270).
B[edit]
 Frank: I was bored. I'd done everything. I'd gone to the limits. There was nothing left to experience. At least nothing I could buy on earth.
 In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered.
 Eric Temple Bell, The Development of Mathematics (1940)
 Galileo observed as early as 1638 that there are precisely as many squares 1, 4, 9, 16, 25,... as are positive integers all together. This is evident from the sequences
1, 2, 3, 4, 5, 6, ... , n, ... He thus recognized the fundamental distinction between finite and infinite classes that became current in the late nineteenth century. An infinite class is one in which there is a onetoone correspondence between the whole class and a subclass of the whole. Or, what is equivalent, there are as many things in one part of an infinite class as there are in the whole class.
1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, 6^{2}, ..., n, ...
...A class whose elements can be put in a onetoone correspondence with the integers 1, 2, 3, ... is said to be denumerable. All the points in any line segment, finite or infinite in length, form a nondenumerable set. A basic course in calculus starts from the theory of point sets. The distinction between denumerable and nondenumerable classes was not started by Galileo; it was observed about 1840 by Bolzano and in 1878 by Cantor. But Galileo's recognition of the cardinal property of all infinite classes makes him one of the genuine anticipators in the history of calculus. The other was Archimedes. Eric Temple Bell, The Development of Mathematics (1940)
 If a thing loves, it is infinite.
 William Blake, in Annotations to Swedenborg (1788).
 If the doors of perception were cleansed everything would appear to man as it is, infinite. For man has closed himself up, till he sees all things thro' narrow chinks of his cavern.
 William Blake, A Memorable Fancy in The Marriage of Heaven and Hell (1790–1793).
 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. William Blake, in Auguries of Innocence (c. 1805)  Full text online
 There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.
 Jorge Luis Borges, "Avatars of the Tortoise" or "Avatares de la Tortuga" (1939).
 There are no moral or intellectual merits. Homer composed the Odyssey; if we postulate an infinite period of time, with infinite circumstances and changes, the impossible thing is not to compose the Odyssey, at least once.
 Jorge Luis Borges, "The Immortal" (1949).
 Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist.
 Attributed to Kenneth Boulding in United States Congress, House (1973), Energy reorganization act of 1973: Hearings, Ninetythird Congress, first session, on H.R. 11510. p. 248
C[edit]
 There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.
 Georg Cantor, in "Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"]  Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886).
 Each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.
 Georg Cantor, in "Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"]  Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886).
 The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.
 Georg Cantor, in "Mitteilungen" (18878).
 I believe that there is no part of matter which is not — I do not say divisible — but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.
 Georg Cantor, as quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by Rosemary Schmalz.
 The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
 Georg Cantor, as quoted in Infinity and the Mind (1995) by Rudy Rucker
 For science, the invention of the differential calculus was a giant step. For the first time in human history the concept of the infinite, which had intrigued philosophers and poets from time immemorial, was given a precise mathematical definition, which opened countless new possibilities for the analysis of natural phenomena. ...According to Zeno, the great athlete Achilles can never catch up with a tortoise... The flaw in Zeno's argument lies in the fact that even though it will take Achilles an infinite number of [procedural] steps to reach the tortoise, this does not take an infinite time. With the tools of Newton's calculus it is easy to show that a moving body will run through an infinite number of infinitely small intervals in a finite time.
 Fritjof Capra, The Web of Life (1996).
 And in that moment, I swear we were infinite.
 The primary Imagination I hold to be the living power and prime agent of all human perception, and as a repetition in the finite mind of the eternal act of creation in the infinite I AM.
D[edit]
 Although velocity was a relative in Newtonian science, yet there did not exist one definite velocity which was assumed to be absolute. This was the infinite velocity. It was assumed that a velocity that was infinite or instantaneous for one observer would remain infinite or instantaneous for all other observers. So far... as velocity was concerned, the sole difference between Einstein and Newton is that with Einstein the absolute or invariant velocity is no longer infinite. ...It is this difference between the invariant velocities of Newton and Einstein which is responsible for all the major differences between classical and Einsteinian science... it is this finiteness of the invariant velocity which precludes us from attaching any absolute value to shape and distance. Einstein's theory proves that molar matter can never move with the absolute speed of light. We are therefore perfectly justified in saying that the velocity of matter remains essentially relative, since it can never attain that critical velocity (i.e., that of light) which is absolute.
 A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) pp. 154155.
 Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum,
And the great fleas themselves, in turn, have greater fleas to go on,
While these again have greater still, and greater still, and so on. Augustus De Morgan, A Budget of Paradoxes (c. 1850)
EG[edit]
 If I should ask... how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. ...
But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square. This being granted we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers. ...
So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former, and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. Or if I had replied to him that the points in one line were equal in number to the squares; in another, greater than the totality of numbers; and in the little one, as many as the number of cubes, might I not, indeed, have satisfied him by thus placing more points in one line than in another and yet maintaining an infinite number in each. So much for the first difficulty. Galileo Galilei as Salviati, Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze or Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) and as Dialogues Concerning Two New Sciences (1914) Tr. Henry Crew, Alfonso de Salvio
 As to the query whether the finite parts of a limited continuum [continuo terminato] are finite infinite in number I will... answer that they are neither finite or infinite.
 Galileo Galilei as Salviati, Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze or Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) and as Dialogues Concerning Two New Sciences (1914) Tr. Henry Crew, Alfonso de Salvio
 I will now say something which may perhaps astonish you; it refers to the possibility of dividing a line into its infinitely small elements by following the same order which one employs in dividing the same line into forty, sixty, or a hundred parts, that is, by dividing it into two, four, etc. He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken; for if this process were followed to eternity there would still remain finite parts which were undivided. ...
Indeed by such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is... getting farther and farther away from it. My reason is this. In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [tutti i numeri], because both of these are as numerous as their roots which constitute the totality of the natural numbers. Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows that since this process carries us farther and farther from the end sought, if on turning back we shall find that any number can be said to be infinite, it must be unity. Here indeed are satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as there are cubes and natural numbers [tutti i numeri]. ...
There is no difficulty in the matter because unity is at once a square, a cube, a square of a square, and all the other powers [dignitā]; nor is there any essential peculiarity in squares or cubes which does not belong to unity; as, for example, the property of two square numbers that they have between them a mean proportional; take any square number you please as the first term and unity for the other, then you will always find a number which is a mean proportional. Consider the two square numbers, 9 and 4; then 3 is the mean proportional between 9 and 1 []; while 2 is a mean proportional between 4 and 1 []; between 9 and 4 we have 6 as a mean proportional []. A property of cubes is that they must have between them two mean proportional numbers; take 8 and 27; between them lie 12 and 18 []; while between 1 and 8 we have 2 and 4 intervening []; and between 1 and 27 there lie 3 and 9 []. Therefore we conclude that unity is the only infinite number. These are some of the marvels which our imagination cannot grasp and which should warn us against the serious error of those who attempt to discuss the infinite by assigning to it the same properties which we employ for the finite, the natures of the two having nothing in common. Galileo Galilei as Salviati, Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze or Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) and as Dialogues Concerning Two New Sciences (1914) Tr. Henry Crew, Alfonso de Salvio.
 Neither one nor the other doth follow, for that both the assertions may be true. The Oracle adjudged Socrates the wisest of all men, whose knowledg is limited; Socrates acknowledgeth that he knew nothing in relation to absolute wisdome, which is infinite; and because of infinite, much is the same part as is little, and as is nothing (for to arrive... to the infinite number, it is all one to accumulate thousands, tens, or ciphers,) therefore Socrates well perceived his wisdom to be nothing, in comparison of the infinite knowledg which he wanted. But yet, because there is some knowledg found amongst men, and this not equally shared to all, Socrates might have a greater share thereof than others, and therefore verified the answer of the Oracle.
 Galileo Galilei, Dialogo sopra i Due Massi Sistemi del Mondo (1632) as quoted in Salusbury tr. The Systeme of the World: in Four Dialogues (1661)
 Nature doth, and in it alone is discovered an infinite wisdom, so that Divine Wisdom may be concluded to be infinitely infinite.
 Galileo Galilei, Dialogo sopra i Due Massi Sistemi del Mondo (1632) as quoted in Salusbury tr. The Systeme of the World: in Four Dialogues (1661)
 I must have recourse to a Philosophical distinction, and say that the understanding is to be taken two ways, that is intensivè, or extensivè; and that extensive, that is, as to the multitude of intelligibles, which are infinite, the understanding of man is as nothing, though he should understand a thousand propositions; for that a thousand, in respect of infinity is but as a cypher: but taking the understanding intensive, (in as much as that term imports) intensively, that is, perfectly some propositions, I say, that humane wisdom understandeth some propositions so perfectly, and is as absolutely certain thereof, as Nature herself; and such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few comprehended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the necessity thereof, than which there can be no greater certainty.
 Galileo Galilei, Dialogo sopra i Due Massi Sistemi del Mondo (1632) as quoted in Salusbury tr. The Systeme of the World: in Four Dialogues (1661)
 Although I might very rationally put it in dispute, whether there be any such centre in nature, or no; being that neither you nor any one else hath ever proved, whether the World be finite and figurate, or else infinite and interminate; yet nevertheless granting you, for the present, that it is finite, and of a terminate Spherical Figure, and that thereupon it hath its centre; it will be requisite to see how credible it is that the Earth, and not rather some other body, doth possesse the said centre.
 Galileo Galilei, Dialogo sopra i Due Massi Sistemi del Mondo (1632) as quoted in Salusbury tr. The Systeme of the World: in Four Dialogues (1661)
H[edit]
 The Infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.
 David Hilbert, "Über das Unendliche" ["On the Infinite"] (1925) address to der Westfälischen Mathematischen Gesellschaft in honour to the memory of Karl Weierstrass, Mathematische Annalen, 95 (1926) pp. 161190, as quoted by Tobias Dantzig, Number The Language of Science (1930)
 When we turn to the question, what is the essence of the infinite, we must first give ourselves an account as to the meaning the infinite has for reality: let us then see what physics teaches us about it.
 David Hilbert, "Über das Unendliche" ["On the Infinite"] (1925) ibid.
 The first naive impression of nature and matter is that of continuity. Be it a piece of metal or a fluid volume, we cannot escape the conviction that it is divisible into infinity, and that any of its parts, however small, will have the properties of the whole. But wherever the method of investigation into the physics of matter has been carried sufficiently far, we have invariably struck a limit of divisibility, and this was not due to a lack of experimental refinement but resided in the very nature of the phenomenon. One can indeed regard this emancipation from the infinite as a tendency of modern science and substitute for the old adage natura non facit saltus its opposite: Nature does make jumps.
 David Hilbert, "Über das Unendliche" ["On the Infinite"] (1925) ibid.
 It is well known that matter consists of small particles, the atoms, and that the macroscopic phenomena are but manifestations of combinations and interactions among these atoms. But physics did not stop there: at the end of that last century it discovered atomic electricity of a still stranger behavior. Although up to then it had been held that electricity was a fluid and acted as a kind of continuous eye, it became clear then that electricity too, is built up of positive and negative electrons.
Now besides matter and electricity there exists in physics another reality, for which the law of conservation holds; namely energy. But even energy, it is found, does not admit of simple and unlimited divisibility. Planck discovered the energyquanta.
And the verdict is that nowhere in reality does there exist a homogeneous continuum in which unlimited divisibility is possible, in which the infinitely small can be realized. The infinite divisibility of a continuum is an operation which exists in thought only, is just an idea which is refuted by our observations of nature, as well as by physical and chemical experiments. David Hilbert, "Über das Unendliche" ["On the Infinite"] (1925) ibid.
 The second place in which we encounter the problem of the infinite in nature is when we regard the universe as a whole. Let us then examine the extension of this universe to ascertain whether there exists there an infinitely great. The opinion that the world was infinite was a dominant idea for a long time. Up to Kant and even afterward, few expressed any doubt in the infinitude of the universe.
Here too modern science, particularly astronomy, raised the issue anew and endeavored to decide it not by means of inadequate metaphysical speculations, but on grounds which rest on experience and on the application of the laws of nature. There arose weighty objections against the infinitude of the universe. It is Euclidean geometry which leads to infinite space as a necessity. ...Einstein showed that Euclidean geometry must be given up. He considered this cosmological question too from the standpoint of his gravitational theory and demonstrated the possibility of a finite world; and all the results discovered by the astronomers are consistent with this hypothesis of an elliptic universe. David Hilbert, "Über das Unendliche" ["On the Infinite"] (1925) ibid.
 No priestly dogmas, invented on purpose to tame and subdue the rebellious reason of mankind, ever shocked common sense more than the doctrine of the infinitive divisibility of extension, with its consequences; as they are pompously displayed by all geometricians and metaphysicians, with a kind of triumph and exultation. A real quantity, infinitely less than any finite quantity, containing quantities infinitely less than itself, and so on in infinitum; this is an edifice so bold and prodigious, that it is too weighty for any pretended demonstration to support, because it shocks the clearest and most natural principles of human reason.
 David Hume, An Enquiry Concerning Human Understanding (1748) Ch.XII, Part II.
IK[edit]
 Even if there were exceedingly few things in a finite space in an infinite time, they would not have to repeat in the same configurations. Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial lineup would never recur.
 Walter Kaufmann, Nietzsche: Philosopher, Psychologist, Antichrist p. 327
 From the mathematical point of view there are infinitely many... numbers... Thus the first task of "scientific" arithmetic—as contrasted with... "practical" knowledge...— consists in finding such arrangements and orders of the assemblages of monads as will completely comprehend their variety under welldefined properties, so that their unlimited multiplicity may at last be brought within bounds (cf. Nichomachus I, 2). ...When we recall how Plato (Theaetetus 147 C ff.) makes Theaetetus, speaking from a very advanced stage of scientific geometry and arithmetic, describe his procedure... What... appears to Plato so exemplary for Socrates' present inquiry concerning "knowledge", and indeed for every Socratic inquiry of this kind[?]. Theaetetus... divides "the whole realm of number"... into two domains: to one of these belong all those numbers which may arise from a number when it is multiplied by itself... to the other, all those which may arise from the multiplication of one number with another. The first number domain he calls "square," the second "promecic" or "heteromecic" (oblong), designations which recur in all later arithmetical presentations (cf. Diogenes Laertius III, 24). Thus two eide [kinds, forms, or species]... allow us to articulate and delimit a realm of numbers previously incomprehensible because unlimited, especially if we substitute the various eide of polygonal numbers for the one eidos of oblong numbers.
 Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (19341936)
 Greek mathematical thought does, indeed deal first and last with different kinds of numbers. Were it otherwise, how would it be possible to come to terms with the limitlessness of the material with which arithmetic is confronted? Therefore... theoretical arithmetic... attempts to comprehend all possible groupings of monads in general under arrangements which are determinate, i.e., which possess unambiguous characteristics and which may, in turn, be reduced to their own ultimate elements...
Now the most comprehensive eide, those which come closest to the rank of arche and are therefore termed "the very first"... are the odd and the even. ...each of these halves nevertheless comprising an unlimited multitude of numbers. But each of these... is now in turn gathered "into one"... by means of certain unambiguous characteristics... Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (19341936)
 In the field of nonEuclidean geometry, Riemann... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length.
...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent nonEuclidean geometry. Morris Kline, Mathematics and the Physical World (1959) Ch. 26: NonEuclidean Geometries, p. 454
 The Greeks failed to comprehend the infinitely large, the infinitely small, and infinite processes. They "shrank before the silence of the infinite spaces."
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 The Pythagoreans associated good and evil with the limited and unlimited, respectively.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 Aristotle says the infinite is imperfect, unfinished, and therefore unthinkable; it is formless and confused. Only as objects are delimited and distinct do they have a nature.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 To avoid any assertion about the infinitude of the straight line, Euclid says a line segment (he uses the word "line" in this sense) can be extended as far as necessary. Unwillingness to involve the infinitely large is seen also in Euclid's statement of the parallel axiom. Instead of considering two lines that extend to infinity and giving a direct condition or assumption under which parallel lines might exist, his parallel axiom gives a condition under which two lines will meet at some finite point.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 The concept of the infinitely small is involved in the relation of points to a line or the relation of the discrete to the continuous, and Zeno's paradoxes may have caused the Greeks to shy away from this subject.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 The relationship of point to line bothered the Greeks and led Aristotle to separate the two. Though he admits points are on lines, he says that a line is not made up of points and that the continuous cannot be made up of the discrete. This distinction contributed also to the presumed need for separating number from geometry, since to the Greeks numbers were discrete and geometry dealt with continuous magnitudes.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 Because they [the ancient Greeks] feared infinite processes they missed the limit process. In approximating a circle by a polygon they were content to make the difference smaller than any given quantity, but something positive was always left over. Thus the process remained clear to the intuition; the limit process, on the other hand, would have involved the infinitely small.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the selfsufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine. ...Simplicius cites others who worked on the problem and says further that people "in ancient times" objected to the use of the parallel postulate.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes in Postulate 2 that a straightline segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length—for example in Book I, Propositions 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 Aristotle had considered the question of whether space is infinite and gave six nonmathematical arguments to prove that it is finite; he foresaw that this question would be troublesome.
 Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
 In an infinite, eternal universe, the point is that anything is possible, and it's unlikely that we can even begin to scratch the surface of the full range of possibilities.

 Stanley Kubrick Playboy Interview (1968) [2]
L[edit]
 For the present, such a state of instantaneous transition from inequality to equality, from motion to rest, from convergence to parallelism, or anything of the sort, can be sustained in a rigorous or metaphysical sense, or whether infinite extensions successively greater and greater, or infinitely small ones successively less and less, are legitimate considerations, is a matter that I own to be possibly open to question; but for him who would discuss these matters, it is not necessary to fall back upon metaphysical controversies, such as the composition of the continuum, or to make geometrical matters depend thereon. Of course, there is no doubt that a line may be considered to be unlimited in any manner, and that, if it is unlimited on one side only, there can be added to it something that is limited on both sides. But whether a straight line of this kind is to be considered as one whole that can be referred to computation, or whether it can be allocated among quantities which may be used in reckoning, is quite another question that need not be discussed at this point.
 Gottfried Wilhelm Leibniz, "Reply to Nieuwentijt Undated," (~1695) The Early Mathematical Manuscripts of Leibniz, 1920, p. 149.
 It will be sufficient if, when we speak of infinitely great (or more strictly unlimited), or of infinitely small quantities (i.e., the very least of those within our knowledge) it is understood that we mean quantities that are indefinitely great or indefinitely small, i.e., as great as you please, or as small as you please, so that the error that any one may assign may be less than a certain assigned quantity. Also, since in general it will appear that, when any small error is assigned, it can be shown that it should be less, it follows that the error is absolutely nothing; an almost exactly similar kind of argument is used in different places by Euclid, Theodosius and others; and this seemed to them to be a wonderful thing, although it could not be denied that it was perfectly true that, from the very thing that was assumed as an error, it could be inferred that the error was nonexistent. Thus by infinitely great and infinitely small, we understand something indefinitely great, or something indefinitely small, so that each conducts itself as a sort of class, and not merely as the last thing of a class. If any one wishes to understand these as the ultimate things, or as truly infinite, it can be done, and that too without falling back upon a controversy about the reality of extensions, or of infinite continuums in general, or of the infinitely small, ay, even though he think that such things are utterly impossible; it will be sufficient simply to make use of them as a tool that has advantages for the purpose of the calculation, just as the algebraists retain imaginary roots with great profit. For they contain a handy means of reckoning, as can manifestly be verified in every case in a rigorous manner by the method already stated. But it seems right to show this a little more clearly, in order that it may be confirmed that the algorithm, as it is called, of our differential calculus, set forth by me in the year 1684, is quite reasonable.
 Gottfried Wilhelm Leibniz, "Reply to Nieuwentijt Undated," (~1695) The Early Mathematical Manuscripts of Leibniz, 1920, p. 150.
 Outside the ordered universe [is] that amorphous blight of nethermost confusion which blasphemes and bubbles at the center of all infinity—the boundless daemon sultan Azathoth, whose name no lips dare speak aloud, and who gnaws hungrily in inconceivable, unlighted chambers beyond time and space amidst the muffled, maddening beating of vile drums and the thin monotonous whine of accursed flutes.
 H.P. Lovecraft The DreamQuest of Unknown Kadath, in At The Mountains of Madness, p. 308.
M[edit]
 The concept of infinity came in relatively late, even in Egypt, and... its first fathers were more likely metaphysicians than theologians. In looking backward, as in looking forward, early man was quite unable to imagine endless time. Always he concluded that the animal creation, including his own kind, must have a beginning, and the earth he walked on, with it. Sometimes he ascribed the act of creation to the gods, or to one of them, and sometimes he laid it to a potent being of lesser dignity, usually to a totem animal.
 H.L. Mencken, Treatise on the Gods (1930).
 Swamp Thing: You thought ...that it could not...get worse...you imagined...that things...had reached their limits. Do not...delude yourselves...there are...no limits.
 Alan Moore, Saga of the Swamp Thing #53, pg. 25
N[edit]
 It seems to me, that if the matter of our sun and planets and all the matter of the universe, were evenly scattered throughout all the heavens, and every particle had an innate gravity towards all the rest, and the whole of space throughout which this matter was scattered was but finite, the matter on [toward] the outside of this space would, by its gravity, tend towards all the matter on the inside, and, by consequence, fall down into the middle of the whole space, and there compose one great spherical mass. But if the matter was evenly disposed throughout an infinite space it could never convene into one mass; but some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered at great distances from one another throughout all that infinite space.
 Isaac Newton, Four Letters to Bentley (1692) first letter
 I fear what I have said of Infinities, will seem obscure to you; but it is enough if you understand, that Infinities when considered absolutely without any Restriction or Limitation, are neither equal nor unequal, nor have any Proportion one to another, and therefore the Principle that all Infinities are equal, is a precarious one.
OP[edit]
 Let man then contemplate the whole of nature in her full and grand majesty, and turn his vision from the low objects which surround him. Let him gaze on that brilliant light, set like an eternal lamp to illumine the universe; let the earth appear to him a point in comparison with the vast circle described by the sun; and let him wonder at the fact that this vast circle is itself but a very fine point in comparison with that described by the stars in their revolution round the firmament. But if our view be arrested there, let our imagination pass beyond; it will sooner exhaust the power of conception than nature that of supplying material for conception. The whole visible world is only an imperceptible atom in the ample bosom of nature. No idea approaches it. We may enlarge our conceptions beyond all imaginable space; we only produce atoms in comparison with the reality of things. It is an infinite sphere, the center of which is everywhere, the circumference nowhere. In short it is the greatest sensible mark of the almighty power of God, that imagination loses itself in that thought.
 Note: Havet traces the statement about nature's infinite sphere to Empedocles
 Blaise Pascal, Pensées, 72 (1669).
 For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either. The ends of things and their beginnings are impregnably concealed from him in an impenetrable secret. He is equally incapable of seeing the nothingness out of which he was drawn and the infinite in which he is engulfed.
 Blaise Pascal, Pensées, 72 (1669).
 ...as nature has graven her image and that of her Author on all things, they almost all partake of her double infinity. Thus we see that all the sciences are infinite in the extent of their researches. For who doubts that geometry, for instance, has an infinite infinity of problems to solve? They are also infinite in the multitude and fineness of their premises; for it is clear that those which are put forward as ultimate are not selfsupporting, but are based on others which, again having others for their support, do not permit of finality. ...Of these two Infinites of science, that of greatness is the most palpable, and hence a few persons have pretended to know all things. ...the infinitely little is the least obvious. Philosophers have much oftener claimed to have reached it, and it is here they have all stumbled. ...We need no less capacity for attaining the Nothing than the All. Infinite capacity is required for both, and it seems to me that whoever shall have understood the ultimate principles of being might also attain to the knowledge of the Infinite. The one depends on the other, and one leads to the other. These extremes meet and reunite by force of distance, and find each other in God, and in God alone.
 Blaise Pascal, Pensées, 72 (1669).
 Excessive qualities are prejudicial to us and not perceptible by the senses; we do not feel but suffer them. Extreme youth and extreme age hinder the mind, as also too much and too little education. In short, extremes are for us as though they were not, and we are not within their notice. They escape us, or we them. This is our true state; this is what makes us incapable of certain knowledge and of absolute ignorance.
 Blaise Pascal, Pensées, 72 (1669).
 We sail within a vast sphere, ever drifting in uncertainty, driven from end to end. When we think to attach ourselves to any point and to fasten to it, it wavers and leaves us; and if we follow it, it eludes our grasp, slips past us, and vanishes for ever. Nothing stays for us. This is our natural condition, and yet most contrary to our inclination; we burn with desire to find solid ground and an ultimate sure foundation whereon to build a tower reaching to the Infinite. But our whole groundwork cracks, and the earth opens to abysses.
 Blaise Pascal, Pensées, 72 (1669).
 I hold it equally impossible to know the parts without knowing the whole, and to know the whole without knowing the parts in detail. The eternity of things in itself or in God must also astonish our brief duration.
 Blaise Pascal, Pensées, 72 (1669).
 Unity joined to infinity adds nothing to it, no more than one foot to an infinite measure. The finite is annihilated in the presence of the infinite, and becomes a pure nothing. So our spirit before God, so our justice before divine justice.
 Blaise Pascal, Pensées, 233 (1669).
 We know that there is an infinite, and are ignorant of its nature. As we know it to be false that numbers are finite, it is therefore true that there is an infinity in number. But we do not know what it is. It is false that it is even, it is false that it is odd; for the addition of a unit can make no change in its nature. Yet it is a number, and every number is odd or even (this is certainly true of every finite number). So we may well know that there is a God without knowing what He is. Is there not one substantial truth, seeing there are so many things which are not the truth itself?
 Blaise Pascal, Pensées, 233 (1669).
 We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite, and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because He has neither extension nor limits.
 Blaise Pascal, Pensées, 233 (1669).
 When we would pursue virtues to their extremes on either side, vices present themselves insensibly there, in their insensible journeys towards the infinitely little; and vices present themselves in a crowd towards the infinitely great, so that we lose ourselves in them, and no longer see virtues.
 Blaise Pascal, Pensées, 357 (1669).
 We must relax our minds a little; but this opens the door to debauchery. Let us mark the limits. There are no limits in things. Laws would put them there, and the mind cannot suffer it.
 Blaise Pascal, Pensées, 380 (1669).
 All that is not thought is pure nothingness...And yet—strange contradiction for those who believe in time—geologic history shows us that life is only a short episode between two eternities of death, and that, even in this episode, conscious thought has lasted and will last only a moment. Thought is only a gleam in the midst of a long night. But it is this gleam which is everything.
 Henri Poincaré, The Value of Science (1905) Ch. 11: Science and Reality, Tr. (1907) George Bruce Halsted p.142
 This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n  1, it is true of n, and thence conclude that it is true for all the whole numbers. ..Here then we have the mathematical reasoning par excellence, and we must examine it more closely.
...The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.
...to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem...
In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general. Henri Poincaré, Science and Hypothesis (1901) Ch. I. On the Nature of Mathematical Reasoning, Tr. (1905) George Bruce Halstead
 Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite...
This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive. George Pólya, Induction and Analogy in Mathematics (1954) Vol. 1 Of Mathematics and Plausible Reasoning
QR[edit]
 This I have tested too frequently to be mistaken by offering to indifferent spectators forms of equal abstract beauty in half tint, relieved, the one against dark sky, the other against a bright distance. The preference is invariably given to the latter... the same preference is unhesitatingly accorded to the same effect in Nature herself.
Whatever beauty there may result from effects of light on foreground objects... there is yet a light which the eye invariably seeks with a deeper feeling of the beautiful, the light of the declining or breaking day, and the flakes of scarlet cloud burning like watchfires in the green sky of the horizon; a deeper feeling... having more of spiritual hope and longing, less of animal and present life...
I am willing to let it rest on the determination of every reader, whether the pleasure which he has received from these effects of calm and luminous distance be not the most singular and memorable of which he has been conscious...
It is not then by nobler form, it is not by positiveness of hue, it is not by intensity of light... that this strange distant space possesses its attractive power. But there is one thing that it has, or suggests, which no other object of sight suggests in equal degree, and that is—Infinity. It is of all visible things the least material, the least finite, the farthest withdrawn from the earth prisonhouse, the most typical of the nature of God, the most suggestive of the glory of his dwellingplace. For the sky of night, though we may know it boundless, is dark; it is a studded vault, a roof that seems to shut us in and down; but the bright distance has no limit, we feel its infinity, as we rejoice in its purity of light. ...this expression of infinity in distance... is of that value that no other forms will altogether recompense us for its loss; and... no work of any art, in which this expression of infinity is possible, can be perfect or supremely elevated, without it, and that, in proportion to its presence, it will exalt and render impressive even the most tame and trivial themes. And I think if there be any one grand division, by which it is at all possible to set the productions of painting, so far as their mere plan or system is concerned, on our right and left hands, it is this of light and dark background, of heaven light or of object light. John Ruskin, Modern Painters (1860) Vol. 2, Ch. V
S[edit]
 We know by actual observation only a comparatively small part of the whole universe. I will call this "our neighborhood." Even within the confines of this province our knowledge decreases very rapidly as we get away from our own particular position in space and time. It is only within the solar system that our empirical knowledge extends to the second order of small quantities (and that only for g_{44} and not for the other g_{αβ}), the first order corresponding to about 10^{8}. How the g_{αβ} outside our neighborhood are, we do not know, and how they are at infinity of space or time we shall never know. Infinity is not a physical but a mathematical concept, introduced to make our equations more symmetrical and elegant. From the physical point of view everything that is outside our neighborhood is pure extrapolation, and we are entirely free to make this extrapolation as we please to suit our philosophical or aesthetical predilections—or prejudices. It is true that some of these prejudices are so deeply rooted that we can hardly avoid believing them to be above any possible suspicion of doubt, but this belief is not founded on any physical basis. One of these convictions, on which extrapolation is naturally based, is that the particular part of the universe where we happen to be, is in no way exceptional or privileged; in other words, that the universe, when considered on a large enough scale, is isotropic and homogeneous.
 Willem de Sitter, The Astronomical Aspect of the Theory of Relativity (1933)
 It is only by intuition that the infinite can be apprehended. But why is this? Why cannot the infinite be apprehended by concepts? To see this we must understand that the word "infinite," in the religious sense, has nothing at all to do with that sense of the word in which it is applied to space, time, and the number series. We may call this latter the mathematical infinite to distinguish it from the religious infinite. And it is the confusion between these two which misled us into the false trail of supposing that the infinity of God's mind refers to the amount of His knowledge and that the finitude of man's mind refers to his ignorance. The religious infinite, or in other words the infinity of God, means that than 'which there is no other'. In this sense neither space nor time could be infinite, since space is an "other" to time, and time is an "other" to space.
 Walter Terence Stace, Time and Eternity (1952), p. 4647
 What does infinity mean to you? Are you not infinity and yourself?
 Dejan Stojanovic in Circling, “Infinity” (Sequence: “Recircling”).
 Infinity is the end. End without infinity is but a new beginning.
 Dejan Stojanovic in The Sun Watches the Sun, “Infinity and End” (Sequence: “Skywalking”).
 By God, I mean a being absolutely infinite — that is, a substance consisting in infinite attributes, of which each expresses eternal and infinite essentiality.
Explanation — I say absolutely infinite, not infinite after its kind: for, of a thing infinite only after its kind, infinite attributes may be denied; but that which is absolutely infinite, contains in its essence whatever expresses reality, and involves no negation. Baruch Spinoza, in Ethics Geometrically Demonstrated (1677), Definition 6
TZ[edit]
 Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means.
 Hermann Weyl, "The Open World: Three Lectures on the Metaphysical Implications of Science," (1932) as quoted in Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics (2009) ed. Peter Pesic.
 In the Timaeus Plato had expounded a theory that outside the universe, which he regarded as bounded and spherical, there was an infinite empty space.
 Gerald James Whitrow, The Structure of the Universe: An Introduction to Cosmology (1949)
 During the Middle Ages the universe was regarded as finite, with the earth at its centre. The idea was abandoned during the Scientific Renaissance, and the universe came to be pictured as an indefinitely large number of stars scattered throughout infinite Euclidean space. This conception appeared to be a necessary consequence of the theory of gravitation; for, as Newton pointed out, a finite material universe in infinite space would tend to concentrate in one massive lump.
 Gerald James Whitrow, The Structure of the Universe: An Introduction to Cosmology (1949)
 A child might be overawed by a great city, but a civil engineer knows that he might demolish it and rebuild it himself. Husserl's philosophy has the same aim: to show us that, although we may have been thrust into this world without a 'by your leave,' we are mistaken to assume that it exists independently of us. It is true that reality exists apart from us; but what we mistake for the world is actually a world constituted by us, selected from an infinitely complex reality.
 Colin Wilson in Introduction to the New Existentialism, p. 63 (1966)
 When you reach the top of the mountain, keep climbing.
 Quoted as a Zen Koan in Kevin Grange, "Beneath Blossom Rain: Discovering Bhutan on the Toughest Trek in the World" (2011), p. 284.