Infinity

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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

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.

[edit] Quotes

  • 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)
  • Mind is infinite and self-ruled, and is mixed with nothing, but is alone itself by itself.
    • Anaxagoras, Frag. B 12 from Early Greek Philosophy, Chapter 6, John Burnet (1920)
  • 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.
  • 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.
  • 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.
    • Aristotle, Physics Bk III.4, Hardie and Gaye
  • 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.
    • 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 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.
    • 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 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.
    • 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 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.
    • 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 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.
    • 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 matter--that 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
Each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite. ~ Georg Cantor
The least particle ought to be considered as a world full of an infinity of different creatures. ~ Georg Cantor
  • 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.
  • 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.
  • 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.
  • 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 non-existent. 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.

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