Mathematics

"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."

"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."

"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 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?"

"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."

"If 'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible or sensible."

"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."

"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?"

"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."

"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."

"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."

"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."

"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."

"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 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."

"Motion is supposed to belong to the class of things which are continuous; and the infinite presents itself first in the continuous--that 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."

"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."

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

"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."

"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."

"Should not a true understanding of life promote care for the future along with the present? This is the immediate duty of every scientist. Until now scientists have dealt with life as finite — is it not now their mission to see life as extending into Infinity? 553."

"Ford, there’s an infinite number of monkeys outside who want to talk to us about this script for Hamlet they’ve worked out."

"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."

"As there are an infinity of possible worlds, there are also an infinity of laws, certain ones appropriate to one; others, to another, and each possible individual of any world involves in its concept the laws of its world."

"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."

"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."

"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 straight-line 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."

"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 self-sufficiency 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."

"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."

"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."

"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."

"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."

"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."

"The Pythagoreans associated good and evil with the limited and unlimited, respectively."

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

"In the field of non-Euclidean 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 non-Euclidean geometry."

"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 ' 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..."

"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 well-defined 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."

"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 line-up would never recur."

"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."

"All this arguing of infinities is but the ambition of school boys."

"Heisuke Hironaka, as quoted by Allyn Jackson: (quote from p. 1019)"

"At the very base of all these different kinds of mathematics, there is an eternal question or eternal mission of mathematics, and that is infinity. Consciously or unconsciously, what mathematicians do is a finitization of infinity. It is impossible to put infinite things into a computer. It doesn’t matter how good a computer may be, how fast it becomes; it cannot compute infinite things. But there the mathematician has a job to do: to formulate a . The model may not match exactly the original phenomenon, but it helps you to understand it. And the model is finite: you can put it in a computer, and the computer computes the exact answer, at least for the model. So mathematicians give infinity a finite shape or a finitely computable and understandable form. This is quite an interesting feature of human nature. To my way of thinking, humans are different from other animals in that humans have a notion of infinity. They never see infinity, they never live infinitely, and even the universe may not last infinitely long. But humans cannot live without the idea of infinity."

"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."

"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 energy-quanta. 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."

"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."

"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."

"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."

"Marvin Minsky was a long term friend of Feynman ... He recalls Feynman's suspicions of continuous functions and how he liked the idea that space-time might in fact be discrete. Feynman was fascinated by the question 'How could there possibly be an infinite amount of information in any finite volume?'"

"[E]ven in the most precise part of sience, in mathematics, we cannot avoid using concepts that involve contradictions. ...[I]t is well known that the concept of infinity leads to contradictions that have been analyzed, but it would be practically impossible to construct the main parts of mathematics without this concept."