"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."

"And in that moment, I swear we were infinite."

"Anaximander gave up the idea that water or any other known substance might be the first principle, and held that this is of the nature of the infinite, that is, matter without any determinate property except that of being infinite. All things are developed out of this and return to it again, so that an infinite series of worlds have been generated and have in turn become again resolved into the abstract mass."

"Game theory is logically demanding, but on a practical level, it requires surprisingly few mathematical techniques. Algebra, calculus, and basic probability theory suffice. ...the stress placed on game-theoretic rigor in recent years is misplaced. Theorists could worry more about the empirical relevance of their models and take less solace in mathematical elegance. ...[I]f a proposition is proved for a model with a finite number of agents, it is... irrelevant whether it is true for an ifinite number... There are... only a finite number of people, or even bacteria. Similarly, if something is true in games in which payoffs are finitely divisible... it does not matter whether it is true when payoffs are infinitely divisible. There are no payoffs in the universe... infinitely divisible. Even time... continuous in principle, can be measured only by devices with a finite number of s. Of course, models based on the real and complex numbers can be hugely useful, but they are just approximations... There is... no intrinsic value of a theorem that is true for a continuum of agents on a , if it is also true for a finite number of agents of a finite choice space."

"The Greeks failed to comprehend the infinitely large, the infinitely small, and infinite processes. They "shrank before the silence of the infinite spaces.""

"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."

"Another source of the sublime is infinity... Infinity has a tendency to fill the mind with that sort of delightful horror, which is the most genuine effect and truest test of the sublime. There are scarce any things which can become the objects of our senses, that are really... infinite. But the eye not being able to perceive the bounds... they seem... infinite, and they produce the same effects... We are deceived in the like manner, if the parts of some large object are so continued to any indefinite number, that the imagination meets no check... Whenever we repeat an idea frequently, the mind... repeats it long after the first cause has ceased... multiplied without end. ...This is the reason of an appearance very frequent in madmen; that they remain... in the constant repetition of some remark... complaint, or song... every repetition reinforces it with new strength... unrestrained by the curb of reason, continues... to the end of their lives."

"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."

"[A]s the great extreme of dimension is sublime, so the last extreme of littleness is in the same measure sublime... when we attend to the infinite divisibility of matter, when we pursue animal life into these excessively small, and yet organized beings... when we push our discoveries yet downward... in tracing which the imagination is lost as well as the sense; we become amazed and confounded at the wonders of minuteness; nor can we distinguish in its effects this extreme of littleness from the vast itself. For division must be infinite as well as addition; because the idea of a perfect unity can no more be arrived at, than that of a complete whole, to which nothing can be added."

"Each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite."

"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."

"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."

"Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist."

"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."

"If a thing loves, it is infinite."

"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."

"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 sequences1, 2, 3, 4, 5, 6, ... , n, ... 12, 22, 32, 42, 52, 62, ..., 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 one-to-one 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. ...A class whose elements can be put in a one-to-one 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 non-denumerable set. A basic course in calculus starts from the theory of point sets. The distinction between denumerable and non-denumerable 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."

"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."

"Wallis rejected as absurd and inconceivable the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity."

"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."

"It is not possible, I think, to rise from the perusal of the arguments of Clark and Spinoza without a deep conviction of the futility of all endeavors to establish, entirely à priori, the existence of an Infinite Being, His attributes, and His relation to the universe. The fundamental principle of all such speculations, viz. that whatever we can clearly conceive, must exist, fails to accomplish its end, even when its truth is admitted. For how shall the finite comprehend the infinite? Yet must the possibility of such conception be granted, and in something more than the sense of a mere withdrawal of the limits of phænomal existence, before any solid ground can be established for the knowledge, à priori, of things infinite and eternal."

"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."

"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 matter--that is why it is inconsistent in them to make it what contains, and not what is contained."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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?"

"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 whole--that 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 such--the 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."

"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."

"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 it--on 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."

"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."

"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."

"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."

"Belief in the existence of the infinite comes mainly from five considerations: 1) From the nature of time--for it is infinite. 2) From the division of magnitudes-for 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 everybody--not 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."

"The physicists... always regard the infinite as an attribute of a substance which is different from it and belongs to the class of the so-called elements--water 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 contact-compounded of the homogeneous parts."

"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."

"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 point--it 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."

"Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self-subsistent 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."

"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."

"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?"

"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."

"All things were together, infinite both in number and in smallness; for the small too was infinite."

"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."

"Empedocles holds that the corporeal elements are four, while all the elements-including those which initiate movement-are six in number; whereas Anaxagoras agrees with Leucippus and Democritus that the elements are infinite."