Infinite Frustration. The History of Humanity s Conception of Infinity. Infinity in Religion and Philosophy

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1 Infinite Frustration The History of Humanity s Conception of Infinity Infinity in Religion and Philosophy

2 Infinity in Religion and Philosophy 1. Biblical Judaism: I Am that I Am 2. Ancient India 3. Ancient Greece and the Hellenistic World 4. Rome and Early Christianity 5. Islam 6. The Late Middle Ages 8. The Renaissance and Enlightenment 9. The 19 th Century 10. Modern Views

3 Biblical Judaism I Am that I Am God conceived as creating the finite world from outside, hence infinite God generally anthropomorphized, becoming more abstract conceptually in later writings God s existence presented as an infinite, self-referential loop Tradition recognizes power of names as finite symbols for infinite concepts

4 Biblical Judaism I Am that I Am And Moses said to God: Look, when I come to the children of Israel, and say to them: The God of your fathers has sent me to you; and they say to me: What is His name? What shall I say to them? And God said to Moses: I AM THAT I AM ; and He said, Thus shall you say to the children of Israel: I AM has sent me to you. - Exodus 3:13-14

5 I AM (I exist) THAT (because)

6 Ancient India - The Jain religion (c. 9 th century BCE) was the first to believe that the universe was infinite in space and time - The Surya Prajnapti (c. 400 BCE) classified all numbers into three categories: enumerable, innumerable and infinite - The Yajurveda (c. 300 BCE): If a part is added to or removed from infinity, what remains is still infinity That is full, this is full From the full, the full is subtracted When the full is taken from the full The full still will remain - Isha Upanishad of the Yajurveda

7 Ancient Greece and the Hellenistic World - Anaximander The concept of to apeiron (to apeiron) - Pythagoras Infinity as chaos - Parmenides and Zeno of Elea Paradoxes - Leucippus and Democritus Discreteness vs. continuity - Plato Idealism - Aristotle Synthesis: Potential infinity vs. actual infinity

8 Anaximander ( BCE) - Teacher of Pythagoras - Conducted the first recorded scientific experiment; believed that physical forces, rather than supernatural powers, create order in the universe - First to define the infinite, or to apeiron (to apeiron), as the source of all things; conceptualized the infinite as indefinite, formless, and boundless

9 The principle and origin of existing things is to apeiron. And into that from which existing things come to be they also pass away according to necessity; for they suffer punishment and make amends to one another for their injustice, in accordance with the ordinance of time. -- Anaximander

10 Anaximander s Universe to apeiron The Intelligible World

11 Anaximander s universe to apeiron (infinite) The intelligible world peras (finite) Tension of opposites hot.. cold light... dark good evil

12 Pythagoras (572?-490? BCE) - Believed that number was the source of all things - All numbers were rational (ratios of positive integers) - Pupil Hippasus discovered the irrationality of 2 - Discovery led Pythagoras to associate to apeiron with chaos and incomprehensibility Bust of Pythagoras in the Capitoline Museum, Rome

13 The Pythagoreans Table of Opposites Peras Limited One Rest Straight Good Purposeful To Apeiron Unlimited Many Motion Crooked Bad Purposeless

14 Parmenides ( BCE) - Founder of Eleatic School of philosophy; teacher of Zeno - Argued that reality is a timeless, static unity and that all variety and motion are only apparent - Influenced by Pythagoras - Pupil Melissus stated that the true reality, the One, is unlimited in magnitude (i.e. infinite)

15 Zeno of Elea (490? 430? BCE) - Pupil and lover of Parmenides (who adopted him) - Attempted to prove that motion is an illusion by developing several famous paradoxes based on the assumption of the infinite divisibility of time and space - Described in Plato s dialogue Parmenides and in Aristotle s Physics

16 Zeno s Paradoxes Achilles and the Tortoise In the time it takes Achilles to reach point 1, the tortoise moves forward to point 2, and so forth - If space and time are infinitely divisible, Achilles can never catch the tortoise, because you can not move through an infinite number of steps in a finite time

17 Zeno s Paradoxes The Arrow - At each instant of time the arrow occupies a space equal to itself - When an object occupies a space equal to itself, it is not moving - If space and time are infinitely divisible, then the arrow never moves

18 Leucippus (1 st half of 5 th century BCE) Democritus (460?-370? BCE) - Believed that reality consists of an infinite space filled with an infinite number of indivisible atoms (atomos: atomos) whose attributes and motions determine what we perceive - First to propose that reality is not continuous - Influenced Epicurus and his followers Bust of Democritus

19 Democritus Atoms If we chop the carrot into finer and finer pieces, eventually we will come to a smallest piece which can be cut no further (a tomos = cannot be cut).

20 Empedocles ( BCE) - Originated idea that there are four basic elements - Taught that elements are conserved neither created nor destroyed; only their patterns change - Believed the universe undergoes eternal recurrence its configuration repeats over infinite time - Influenced Zeno of Citium, Stoics

21 Plato ( BCE) - Accepted Parmenides distinction between the true world (the world of Ideas ) and the world of appearances - Each Idea (or Form ) abstracted the defining element(s) of an infinite variety of instances in the world of appearances - To apeiron was the indeterminate, which had a peras imposed upon it to create the instances reflecting the Ideas

22 Plato s concept of Ideas The Real World to apeiron Ideals or Forms (peras)

23 Plato s concept of Ideas The Real World to apeiron Ideal Chair (peras)

24 Plato s concept of Ideas The Real World to apeiron Ideal Chair (peras) The World of Appearances (infinite variety and number)

25 Plato s concept of Ideas The Real World to apeiron Ideal Chair (peras) The World of Appearances (infinite variety and number)

26 Plato s concept of Ideas The Real World to apeiron Ideal Chair (peras) The World of Appearances (infinite variety and number)

27 Plato s concept of Ideas The Real World to apeiron Ideal Chair (peras) The World of Appearances (infinite variety and number)

28 Plato s concept of Ideas The Real World to apeiron Ideal Chair (peras) The World of Appearances (infinite variety and number)

29 Plato s concept of Ideas The Real World to apeiron Ideal Chair (peras) The World of Appearances (infinite variety and number)

30 Aristotle ( BCE) - Rejected Eleatic/Platonic distinction between the real and the apparent - Believed that nothing was actually infinite in the sense of the entire infinity being present at once - Developed the idea of potential infinity - Framed all subsequent discussion of the infinite until late 1800s

31 It is incumbent upon the person who treats of nature to discuss the infinite and to enquire whether there is such a thing or not, And, if there is, what it is [and] all who have touched on this kind of science in a way worth considering have formulated views about the infinite. - Aristotle

32 Aristotle s arguments against the infinite - Infinity can not be an actual thing in itself, because then it would have to have parts that are themselves infinite, which is absurd - No set of actual objects can be infinite in number, because counting them would require traversing an infinite sequence of numbers all at once, which cannot be done - Infinity can not be a property of any actual single body, because actual bodies are bounded by surfaces, and hence finite

33 Aristotle s arguments against the infinite - No actual body can be infinitely divisible, because of Zeno s paradoxes - No actual body can be infinitely large, because if it were, there would be no room for any other bodies, and because it could not move there would be no place for it to go other than where it was, since it would take up all available space

34 Aristotle s arguments in favor of the infinite - Time appears infinite, both by addition and by division - Matter appears infinitely divisible - The continual generation and destruction of new things appears impossible without an infinite supply of matter - Whatever is finite appears limited by something else beyond, so that there may not be any ultimate limits - In mathematics, both the sequence of natural numbers and space are infinitely extendible

35 Aristotle s Compromise Potential Infinity [The infinite] is not what has no part outside it but what always has some part outside it One thing is taken after another, what is taken always being finite, but ever other and other. - Aristotle

36 Aristotle s Compromise Potential Infinity Infinity is not a permanent actuality, but consists in a process of coming to be, like time and the number of time. The infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number. - Aristotle

37 Examples of Potential Infinity 1. Number: Infinite by addition 1, 2, 3, 4,, n, n+1, No matter how big a number you pick, there is always a bigger one.

38 Examples of Potential Infinity 1. Number: Infinite by addition 1, 2, 3, 4,, n, n+1, No matter how big a number you pick, there is always a bigger one 2. Time: Infinite by division t, t/2, t/4, t/8,, t/n, t/2n, No matter how short a time span you pick, there is always a shorter one

39 Examples of Potential Infinity 1. Number: Infinite by addition 1, 2, 3, 4,, n, n+1, No matter how big a number you pick, there is always a bigger one. 2. Time: Infinite by division t, t/2, t/4, t/8,, t/n, t/2n, No matter how short a time span you pick, there is always a shorter one 3. Space: Infinite by addition and division Division: Between any two points there are more points Addition: No matter how far you go, you can go farther

40 Time and Aristotelian potential infinity - The act of constructing an infinite set item by item presupposes that it happens over an extended period of time n, n+1, n+2, t, t+1, t+2, - Therefore a potential infinity can not exist all at once - Aristotle believed that past time was infinite (there was no creation moment)

41 Aristotle on the Continuum - The continuum can not be composed of indivisible points or atoms, because a point can not be in direct succession to another point, since there is always a line between any two points

42 Rome and Christianity through the Early Middle Ages - Roman philosophy: Reactions to Aristotle, Plato, etc. - Neoplatonism: Introduction of mysticism into apprehension of the metaphysical infinite - Early Christianity: Attempts to reconcile philosophical concepts of infinity with religious concepts of God

43 Lucretius (99?-55? BCE) - Roman aristocrat, poet and Epicurean philosopher - Wrote De Rerum Natura (On the Nature of Things) - Believed that space must be infinite because all matter would fall to the bottom if it were not, and because the assumption that it has a boundary leads to a paradox

44 Lucretius Paradox? The edge of the universe - If a spear is thrown at the edge of the universe, either it will hit the edge and pierce it, or it will not. If it pierces the edge, then the edge was not the edge. If it does not, then the universe has no edge

45 Plotinus ( ) - Originator of Neoplatonism; believed in the Platonic distinction between the apparent and the real - Saw underlying reality as a mystical, transcendent unity, the One, or the All, which was metaphysically infinite and beyond all categories or attributes; identified with God, or the Good - Influenced many other philosophers including St. Augustine

46 We must therefore take Unity as infinite not in measureless extension or numerable quantity but in fathomless depths of power And if this [the unity or One] is all things, that must be above and outside of all, so, must transcend real being. And again, if that secondary [intellectualprinciple] is all things, and if above its multiplicity there is a unity not ranking among those things, once more this unity transcends real being The One has never known measure and stands outside of number, and so is under no limit either in regard to anything external or internal - Plotinus

47 St. Augustine ( ) - Deeply influenced by Neo- Platonism, which he attempted to integrate with Christianity - Believed God to be both actually infinite and transcendent - Held that time in the apparent world was potentially infinite, but that Eternity in the mind of God was actually infinite, and distinct from the time we perceive - Explored the problem of how the finite self finds the actually infinite God through the potentially infinite mind St. Augustine, by Botticelli

48 Every number is known to Him whose understanding cannot be numbered. Although the infinite series of numbers cannot be numbered, this infinity is not outside His comprehension. It must follow that every infinity is, in a way we cannot express, made finite to God. - St. Augustine

49 Great is the power of memory. It is a true marvel, O my God, a profound and infinite multiplicity! And this is the mind, and this I myself am. What, then, am I, O my God? Of what nature am I? A life various, and manifold, and exceedingly vast. Behold in the numberless halls and caves, in the innumerable fields and dens and caverns of my memory, full without measure of numberless kinds of things -- present there either through images as all bodies are; or present in the things themselves as are our thoughts; or by some notion or observation as our emotions are, which the memory retains even though the mind feels them no longer, as long as whatever is in the memory is also in the mind -- through all these I run and fly to and fro. I penetrate into them on this side and that as far as I can and yet there is nowhere any end. - St. Augustine

50 Problems integrating Aristotle and faith - If God is omnipotent, why are there no actual infinities? Couldn t God create something actually infinite? - If the universe is eternal (if past time is potentially infinite by addition), then it is possible that there are an infinity of souls - If the world was created by God, then past time is not potentially infinite St. Augustine s retort

51 How, then, shall I respond to him who asks, What was God doing before He made heaven and earth? I do not answer, as a certain one is reported to have done facetiously (shrugging off the force of the question). He was preparing hell, he said, for those who ask such questions. - St. Augustine

52 Islam - Islamic philosophers initially tried to reconcile Aristotle s ideas on infinity with Islamic concepts of God - Adopted Hindu mathematics, but not Indian concepts of infinity, which were tied in with Hindu and Jain religion - Crusades: Reaction against Greek philosophy, return to Islamic fundamentalism in Middle East - Philosophical tradition continued in Islamic Spain, where it influenced Western thought during the late Middle Ages

53 Al-Kindi (Alkindus) ( ) - Early Islamic philosopher and polymath - Instrumental in introducing Greek philosophy and Indian mathematics to the Islamic world - Argued that if, as Aristotle said, no actual infinity existed in the universe, then the universe could not be eternal, and had to have been created by God

54 Ibn Sina (Avicenna) ( ) - Persian scholar and the foremost Islamic philosopher of his time - Made contributions to numerous sciences, especially medicine - Explicated and critiqued Aristotle from an Islamic perspective - Agreed with Aristotle that the age of the universe is infinite, but believed that no causal sequence or system could extend to infinity - Argued that there was a contradiction between the impossibility of an actual infinity and the immortality of the soul; resolved it on the grounds that souls do not take up any space

55 Al-Ghazali ( ) - Sufi philosopher, mystic and theologian - Argued against Avicenna and the Greek philosophers - Invented a method of philosophical skepticism that would not be equaled until Descartes 500 years later - Questioned the eternity of the world on the grounds that there would be an infinite number of souls and that the number of revolutions of the planetary spheres would be paradoxical

56 Al-Ghazali s Paradox (updated to reflect modern knowledge) - If the universe is infinitely old, then both Jupiter and the Earth have orbited the Sun an infinite number of times, which is equal - But the Earth orbits the Sun 13 times for each orbit of Jupiter around the Sun, which is not equal

57 Ibn Rushd (Averroes) ( ) - Andalusian philosopher and scholar, described as the founder of secular thought in Western Europe - Attempted to reconcile Aristotle with Islam; attacked al-ghazali s refutation of Avicenna - Believed in the eternity of the universe, its infinity in time - Demonstrated the existence of God from the impossibility of an infinite chain of causation and the necessity of a First Cause - Disagreed with Avicenna that there could be an actually infinite number of souls because it implied different sizes of infinities

58 The Late Middle Ages - Reawakening in the West of interest in science, mathematics and philosophy Fascination with dynamics the problems of motion leading to questions of infinitesimals, infinite series and the continuum - Cross pollination of ideas on infinity from Islamic civilization - Renewed interest in harmonizing Aristotelian ideas of infinity with Christian theology; Divine infinity studied through analogy with mathematical infinity and infinitely divisible continua : Fourth Lateran Council declares belief in the temporal beginning of the world to be an Article of Faith - Discovery of new paradoxes

59 Certain theologians allow that God can increase the volume of heaven, that He can, for example, make heaven be twice as large, three times as large, and so forth, indefinitely, such that, given any magnitude whatever, God can create a magnitude twice it in any case, the proposition, given any magnitude, God can make a magnitude twice it does not formally entail the proposition, God can make an actual infinite magnitude. One might state that any magnitude that can be conceived as potential can also exist in actuality that it would be formed by the simultaneous addition of the parts which have been created. I assert that this proposition is false if a magnitude can be conceived as potential by the simple addition of preexisting parts without the creation of new parts, an equal magnitude can exist in actuality. This allows one to respond easily to any difficulty that one can oppose to the growth of forms to infinity. - Walter Burley (1275?-1345), Professor of theology, Sorbonne

60 Peter of Spain (c ) - Identity uncertain; possibly a monk born in Lisbon who later became Pope John XXI - Distinguished between the categorematic and syncategorematic uses of words and applied this distinction to discussions of infinity - Example: This body can move infinitely fast Categorematically: This body is capable of achieving an infinite speed Syncategorematically: There is no limit to the finite speeds which this body is capable of achieving - Related to Aristotle s actual vs. potential infinity distinction

61 St. Thomas Aquinas (1225?-1274) - Catholic theologian and Dominican priest, student of Albertus Magnus, influenced by Aristotle and Averroes - Taught that God s perfection and unity implied God s metaphysical infinity, but that this infinity was different than mathematical infinity - Believed nothing in Creation was mathematically infinite, but that reason allows the contemplation of mathematical infinity

62 Aquinas view of mathematical infinity - Our ability to conceptualize categories allows us to imagine an infinite number of members of categories - For example, because we understand the concept of greenness, we can imagine and recognize a potentially ( relatively ) infinite number of green things

63 Things other than God can be relatively infinite, but not absolutely infinite. For with regard to infinite as applied to matter, it is manifest that everything actually existing possesses a form; and thus its matter is determined by form. But because matter, considered as existing under some substantial form, remains in potentiality to many accidental forms which are absolutely finite, it can be relatively infinite; as, for example, wood is finite according to its own form, but still it is relatively infinite, inasmuch as it is in potentiality to an infinite number of shapes. - St. Thomas Aquinas

64 The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited. it is impossible for an actually infinite multitude to exist, even accidentally. But a potentially infinite multitude is possible; because the increase of multitude follows upon the division of magnitude; since the more a thing is divided, the greater number of things result. Hence the infinite is to be found potentially in the division of the continuous - St. Thomas Aquinas

65 John Duns Scotus (1266?-1308) - Scottish philosopher and theologian - Argued that continuous lines can not be made up of infinitely many indivisible points

66 - Clearly the inner circle is smaller than the outer circle Duns Scotus Argument

67 - Clearly the inner circle is smaller than the outer circle Duns Scotus Argument

68 Duns Scotus Argument - Clearly the inner circle is smaller than the outer circle - But we can pair all points on the inner circle with corresponding points on the outer circle

69 Duns Scotus Argument - Clearly the inner circle is smaller than the outer circle - So the two circles must have the same number of points, which is a contradiction - But we can pair all points on the inner circle with corresponding points on the outer circle

70 Thomas Bradwardine (1295?-1349) - Noted mathematician at Oxford University and Archbishop of Canterbury - Wrote De Continuo, which discusses the properties of all mathematical and physical continuua and attempts to refute atomism No continuum is made up of atoms, since every continuum is composed of an infinite number of continua of the same species.

71 Gregory of Rimini (1300?-1358) - General of the Augustinian Monks - Developed a comprehensive theory of categorematic (actual) infinity as a magnitude greater than any finite quantity, no matter how large One says, infinity is something such that when one takes any part of it whatever, there always remains another part to be taken But once equal parts of the infinity are not taken in equal times, but in times whose durations decrease in geometric progression there is no longer any inconsistency in the infinity being taken in its totality - Argued that one infinite could be considered larger than another if it contains all the units of the latter plus some additional units

72 Jean Buridan ( ) - French priest, teacher at the University of Paris - Developer of the theory of inertia (impetus); most famous for the parable of Buridan s Ass - Attacked Gregory of Rimini s theory of categorematic infinity on the grounds that in constructing an infinite set of items there is never a last item It is not possible for God to create a stone of one cubic foot in each of the proportional parts of an hour, for He would have to do it by a process that would distinguish all the proportional parts from each other, and would enumerate them in such a way that after these parts there are no others and that is impossible, for in this sense, there are no parts that are all the parts.

73 Nicholas Oresme (1323?-1382) - Bishop of Lisieux, counselor of King Charles V of France - Philosopher, mathematician, physicist, theologian; one of the most original thinkers of the 14 th Century; invented rectilinear coordinate systems 300 years before Descartes and believed the earth rotated around its axis - Contended that space was infinite and that all reality was continuous and infinitely divisible, creating a paradox when things were measured using numbers, which are discrete

74 Every measurable thing except numbers is imagined in the manner of continuous quantity. Therefore, for the mensuration of such a thing, it is necessary that points, lines, and surfaces, or their properties, be imagined. For in them [the geometrical entities] as the Philosopher [Aristotle] has it, measure or ratio is initially found Although indivisible points or lines are non-existent, still it is necessary to feign them mathematically for the measures of things and for the understanding of their ratios. - Nicholas Oresme

75 Nicholas of Cusa ( ) - German Cardinal of the Roman Catholic Church - Believed that the universe was actually infinite, without external limits ( privatively infinite ) as seen from the human point of view - Held that the earth was not the center of the universe, because an infinite space can have no center - Identified the metaphysically infinite with God and Truth; accepted that the infinite encompassed contradictions when considered from the finite human perspective

76 A finite intellect cannot by means of comparison reach the absolute truth of things. Being by nature indivisible, truth excludes more or less, so that nothing but truth itself can be the exact measure of truth In consequence, our intellect, which is not the truth, never grasps the truth with such precision that it could not be comprehended with infinitely greater precision. - Nicholas of Cusa

77 It is self-evident that there is no comparative relation of the infinite to the finite.... Therefore, it is not the case that by means of likeness a finite intellect can precisely attain the truth about things.... For truth is not something more or something less but is something indivisible. Whatever is not truth cannot measure truth precisely.... For the intellect is to truth as an inscribed polygon is to the inscribing circle. - Nicholas of Cusa

78 Giordano Bruno ( ) - Italian priest and philosopher - Deeply influenced by Nicholas of Cusa s ideas on infinity and indeterminacy - Developed a pantheistic philosophy of a transcendent, ineffable God Who contained contradictions because of His metaphysical infinity - Held that the universe was actually infinite in space and time, that the Earth was not its center (following Copernicus) and that there were an infinite number of worlds besides Earth - Charged with blasphemy and heresy and burned at the stake

79 The Renaissance and Enlightenment - Rationalists: Gave logic priority over perception in the search for truth; believed in the logical necessity of the infinite Descartes Pascal Spinoza Leibniz - Empiricists: Believed in the primacy of perception; argued the impossibility of perception of the infinite by finite beings, and therefore against its actual existence Locke Hume Berkeley - Kant: Synthesized the two positions

80 Rene Descartes ( ) - Founder of rationalist philosophy, which gave deduction primacy over perception - Believed that the infinite enjoyed a logical and epistemological priority over the finite - Argued that only God is actually infinite (referring to the potentially infinite as the indefinite ) and that God implanted the idea of the infinite in the finite human mind

81 I never use the word infinite to mean only what has no end, something which is negative and to which I have applied the word indefinite, but to mean something real, which is incomparably bigger than whatever has an end. - Descartes

82 Blaise Pascal ( ) - French mathematician and philosopher - Believed that nature is actually infinite and that human reason is capable of a nobility and dignity that transcend human finitude - Expressed with an eloquence that has never been surpassed the wonder, awe and dread that the human mind experiences when confronted with the infinite

83 From Pascal s Pensees (1670) Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed. Nature is an infinite sphere of which the center is everywhere and the circumference nowhere. Reason s last step is the recognition that there are an infinite number of things which are beyond it. Through space the universe grasps me and swallows me up like a speck; through thought I grasp it. The eternal silence of these infinite spaces terrifies me.

84 Baruch (Benedict) Spinoza ( ) - Rationalist philosopher, influenced by Descartes - Argued that God is actually and metaphysically infinite, with an infinity of attributes, and that therefore God is everything that exists - Excommunicated from the Jewish community of Amsterdam

85 Since God is Being absolutely infinite, of whom no attribute can be denied which expresses the essence of substance and since He necessarily exists it follows that if there were any substance besides God, it would have to be explained by some attribute of God, and thus two substances would exist possessing the same attribute, which is absurd; and therefore there cannot be any substance except God, and consequently none other can be conceived Hence it follows with the greatest clearness, firstly, that God is one, that is to say in nature there is but one substance, and it is absolutely infinite - Spinoza

86 Gottfried Wilhelm Leibniz ( ) - German mathematician and philosopher; with Newton, with whom he fought for priority, inventor of the calculus - Regarded space and time as ideal continua, infinite both by addition and by division - Believed that perceptual reality (matter and energy) consisted of an infinite number of primitive metaphysical entities called monads which gave extension and consciousness to matter

87 But it is also true that the one who made [the Universe], and governs it, is of a yet more infinite perfection, since he encompasses an infinity of possible worlds, from which he selected the one that pleased him. - Leibniz

88 I am so much in favor of the actual infinite, that I hold that nature affects it everywhere, in order the better to mark the perfection of its author. So I believe that every part of matter is, I do not say divisible, but actually divided. - Leibniz

89 it is the confusion of the ideal and the actual which has embroiled everything and produced the labyrinth concerning the composition of the continuum. Those who compose a line of points have sought first elements in ideal things or relations, otherwise than was proper; and those who have found that relations such as number, and space cannot be formed of an assemblage of points, have been mistaken in denying, for the most part, the first elements of substantial realities, as if they had no primitive units, or as if there were no simple substances. In actuals there is nothing but discrete quantity, namely the multitude of monads or simple substances. - Leibniz

90 John Locke ( ) - English empiricist philosopher; inspired American revolutionaries - Opposed to the rationalist prioritization of logic over perception - Believed that it was possible to conceptualize the potential infinity of space and time by addition and division, but that no other infinities existed or could be imagined because there were no models in experience for them

91 Every one that has any idea of any stated lengths of space, as a foot, finds that he can repeat that idea; and joining it to the former, make the idea of two feet; and by the addition of a third, three feet; and so on, without ever coming to an end of his additions, whether of the same idea of a foot, or, if he pleases, of doubling it, or any other idea he has of any length, as a mile, or diameter of the earth, or of the orbis magnus: for whichever of these he takes, and how often soever he doubles, or any otherwise multiplies it, he finds, that, after he has continued his doubling in his thoughts, and enlarged his idea as much as he pleases, he has no more reason to stop, nor is one jot nearer the end of such addition, than he was at first setting out: the power of enlarging his idea of space by further additions remaining still the same, he hence takes the idea of infinite space This, I think, is the way whereby the mind gets the idea of infinite space. - John Locke

92 Whatever positive ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression. - Locke

93 David Hume ( ) - Scottish empiricist philosopher - Maintained that we cannot experience, and therefore we cannot have an adequate idea of, either the infinite or the infinitesimal - Argued that there is a minimum perceptible size, below which smaller division is impossible

94 Tis universally allowed, that the capacity of the mind is limited, and can never attain a full and adequate conception of infinity: And tho it were not allow d, twould be sufficiently evident from the plainest observation and evidence. - Hume

95 Put a spot of ink upon paper, fix your eye upon the spot, and retire to such a distance that at last you lose sight of it; 'tis plain that the moment before it vanished the image or impression was perfectly indivisible. - David Hume

96 Immanuel Kant ( ) - Critiqued both the rationalist and the empiricist perspective and achieved a synthesis of the two - Believed that the external world does impinge upon consciousness through experience, but only because consciousness has certain epistemic faculties and an a priori framework that makes it able to receive, interpret and structure experience - Maintained that knowledge of the infinite is part of the a priori framework, as are causality, space, time, number, etc.

97 Kant s System Things in Themselves (flowers) (woman at beach) (house on a lake)

98 Kant s System Things in Themselves (flowers) (woman at beach) A priori knowledge - Space - Time - Number - Infinity - Causality - (house on a lake)

99 Things in Themselves (flowers) (woman at beach) (house on a lake) Kant s System A priori knowledge - Space - Time - Number - Infinity - Causality - The World We Perceive (a posteriori)

100 Kant and infinity - Kant did not believe that we can know, or even imagine, the universe as either finite or infinite in space or time For Kant, space and time are only forms of a priori knowledge The universe as a whole, as the place of things in themselves, does not exist in space or time

101 ... in order to conceive the world, which fills all space, as a whole, the successive synthesis of the parts of an infinite world would have to be looked upon as completed; that is, an infinite time would have to be looked upon as elapsed, during the enumeration of all coexisting things. - Immanuel Kant

102 Kant and infinity - However, Kant believed that consciousness was also self-conscious, or introspective, and thus infinite in the sense of self-referentiality or infinite regress One can perceive One can know that one is perceiving One can know that one knows etc. - This receptivity of consciousness to infinity is the root of a priori knowledge of mathematical infinity of space and time by addition and division

103 Kant and infinity the Antinomies The First Antinomy - Thesis: The world has a beginning in time, and is also limited as regards space - Antithesis: The world has no beginning, and no limits in space; it is infinite as regards both time and space

104 Kant and infinity the Antinomies The First Antinomy - Thesis: The world has a beginning in time, and is also limited as regards space - Antithesis: The world has no beginning, and no limits in space; it is infinite as regards both time and space Kant s position: - Both thesis and antithesis can be justified with equal force; therefore, neither can be proven or disproven

105 Kant and infinity the Antinomies The First Antinomy - Thesis: The world has a beginning in time, and is also limited as regards space - Antithesis: The world has no beginning, and no limits in space; it is infinite as regards both time and space Kant s position: - Both thesis and antithesis can be justified with equal force; therefore, neither can be proven or disproven - Time and space are infinitely extendible in the imagination, but this applies only to the framework in which we receive and organize our perceptions, and is not true of things in themselves

106 Kant and infinity the Antinomies The Second Antinomy - Thesis: Every composite substance in the world is made up of simple parts, and nothing anywhere exists except the simple or what is composed of the simple - Antithesis: No composite thing in the world is made up of simple parts, and there nowhere exists in the world anything simple; i.e., the world is made up of infinitely divisible continuua

107 Kant and infinity the Antinomies The Second Antinomy - Thesis: Every composite substance in the world is made up of simple parts, and nothing anywhere exists but the simple or what is composed of the simple - Antithesis: No composite thing in the world is made up of simple parts, and there nowhere exists in the world anything simple; i.e., the world is made up of infinitely divisible continuua Kant s position: - Both thesis and antithesis can be justified with equal force; therefore, neither can be proven or disproven

108 Kant and infinity the Antinomies The Second Antinomy - Thesis: Every composite substance in the world is made up of simple parts, and nothing anywhere exists but the simple or what is composed of the simple - Antithesis: No composite thing in the world is made up of simple parts, and there nowhere exists in the world anything simple; i.e., the world is made up of infinitely divisible continuua Kant s position: - Both thesis and antithesis can be justified with equal force; therefore, neither can be proven or disproven - Space and matter are infinitely divisible in the imagination, but this applies only to the framework in which we receive and organize our perceptions, and is not true of things in themselves

109 we are not entitled to say of a whole which is divisible to infinity that it is made up of infinitely many parts. For although all parts are contained in the intuition of the whole, the whole division is not so contained, but consists only in the continuous decomposition, that is, the regress itself, whereby the series first becomes actual. Since this regress is infinite, all the members or parts at which it arrives are contained in the whole, viewed as an aggregate. But the whole series of the division is not so contained, for it is a successive infinite and never whole and cannot, therefore, exhibit an infinite multiplicity, or any combination of an infinite multiplicity in a whole. - Kant

110 The 19 th and 20 th Centuries Modern Views - Reactions to Kant - Reactions to discoveries in mathematics - Wittgenstein and the role of language - The Existentialists and the anti-postmodernists; infinity as a symbol of self-actualization, fidelity and truth

111 Georg Wilhelm Friedrich Hegel ( ) - Creator of German idealism, along with Fichte and Schelling - Attempted to unify contradictions in the creation of synthesis through a process of dialectic, or confrontation of opposites - Believed that what is rational is physically real - Distinguished between false infinity (endlessness) and true infinity - Saw continuity and discreteness as a dialectic of quantity

112 Hegel s False and True Infinities - False infinity, or endlessness You can always increase it; it simultaneously exists as itself and as its increment. At every stage it is finite, but there is no last limit Similar to a concept of mathematical infinity - True infinity Standing beyond all relation to anything finite Absolute, self-complete, incorporating all Similar to a concept of metaphysical infinity

113 The image of the progress to [false] infinity is the straight line, at the two limits of which alone the infinite is and always is only where it [the line] and it is determinate being is not, and which goes out beyond to this its non-determinate being, i.e. into the indeterminate; the image of genuine infinity, bent back into itself, becomes the circle, the line which has reached itself, which is closed and entirely present, without beginning and end. - Hegel

114 Looked at from the point of view of mere discreteness, substance, matter, space and time, and so on, are absolutely divided, and the One is their principle. From the point of view of continuity, this One is merely suspended; division remains divisibility, the possibility of dividing remains possibility, without ever actually reaching the atom still continuity contains the moment of the atom, since continuity exists simply as the possibility of division Each of the two opposed sides contains the other in itself, and neither can be thought of without the other; and thus it follows that taken alone, neither determination has truth, but only their unity. This is the true dialectic consideration of them, and the true result. - Hegel

115 Edmund Husserl ( ) - Father of phenomenology, the theory of how subjective experience relates to phenomena in the world - Influenced Carnap, Heidegger, Sartre, Derrida and other major 20 th century philosophers - Believed that the idea of infinity was itself non-infinite, and could thus be intuited, even though infinities were beyond human experience - Emphasized the importance of the formulation of ideas of infinity (the plenum ) on the structure of culture and the mind

116 Scientific culture under the guidance of ideas of Infinity means, then, a revolutionization of the whole culture, a revolutionization of the whole manner in which mankind creates culture. It also means a revolutionization of historicity, which is now the history of the cutting-off of finite mankind s development as it becomes mankind with infinite tasks. - Husserl

117 through a universal causal regulation, all that is together in the world has a universal immediate or mediate way of belonging together; through this the world is not merely a totality (Allheit) but an allencompassing unity (Alleinheit), a whole (even though it is infinite). - Husserl

118 Henri Bergson ( ) - French philosopher of the foundations of consciousness and the perception of time and duration - Held that continuity is basic and not divisible into points, but that consciousness perceives continuous phenomena as if they are divisible

119 If I raise my hand from A to B, this movement appears to me under two aspects at once. Felt from within, it is a simple, indivisible act. Perceived from without, it is the course of a certain curve, AB. In this curve I can distinguish as many positions as I please, and the line itself might be defined as a certain mutual co-ordination of these positions. But the positions, infinite in number, and the order in which they are connected, have sprung automatically from the indivisible act by which my hand has gone from A to B the movement is more than the positions and their order; for it is sufficient to make it in its indivisible simplicity to secure that the infinity of the successive positions as also their order be given at once with something else which is neither order nor position but which is essential, the mobility. - Henri Bergson

120 Ludwig Wittgenstein ( ) - Austrian-British philosopher of language - Said that language (including mathematics) is a game played with symbols and signs according to rules that are socialized - Argued against misuse of the language of infinity; called Cantor s transfinite set theory a cancerous growth on the body of mathematics; maintained that mathematical infinity is a procedural system for generating cases or signs rather than a sign in itself

121 But what makes a sign an expression of infinity? What gives the peculiar character that belongs to what we call infinite? The dots in are just the four dots: a sign, for which it must be possible to give certain rules. (The same rules, in fact, as for the sign and so on ad inf. ) This sign does in a manner ape enumeration, but it isn t an enumeration. And that means that the rules governing it don t totally agree with those which govern an enumeration; they agree only up to a point The difficulty here is to fight off the thought that possibility is a kind of shadowy reality. - Wittgenstein

122 Jean-Paul Sartre ( ) - French Existentialist - Held that the universe is indifferent to human fate or human existence, and that human freedom is given meaning only through the willed choices of the individual - Believed that consciousness is metaphysically infinite, but that it requires interaction with an Other to prove or to display its own existence

123 Just as the geometrician is free to create a particular figure which pleases him but cannot conceive of one which does not immediately enter into an infinity of relations with the infinity of other possible figures, so our free choice of ourselves, by causing the upsurge of a certain evaluative order of our past, causes the appearance in the world of an infinity of relations of this past to the world and to the Other. And this infinity of relations is presented to us as an infinity of conducts to be adopted since it is in the future that we evaluate our past. - Jean-Paul Sartre

124 Alain Badiou (1937-) - French-Moroccan antipostmodernist philosopher - Appropriates the axioms and language of Cantorian set theory to attempt to recover the concepts of being, truth and the subject, which are denied by the postmodern ethos - Connects the infinite (in set-theoretic terms) with truth and fidelity

125 For the process of a truth to begin, something must happen I call it an event An event is linked to the notion of the undecidable... The undecidability of the event induces the appearance of a subject of the event. Such a subject is constituted by an utterance in the form of a wager. This utterance is as follows: This event has taken place, it is something which I can neither evaluate, nor demonstrate, but to which I shall be faithful. To begin with, a subject is what fixes an undecidable event, because he or she takes the chance of deciding upon it. This decision opens up the infinite procedure of verification of the true. This procedure is the examination, within the situation, of the consequences of the axiom that decided upon the event. Such a procedure is an exercise of fidelity. - Alain Badiou

126 For example, the work of Sophocles is a subject for the artistic truth or procedure of Greek tragedy, a truth begun by the event of Aeschylus. This work is creation; that is, a pure choice in what, before it, was indiscernible. And it is a finite work. However, Tragedy itself, as an artistic truth, continues to infinity. The work of Sophocles is a finite subject of this infinite truth. In the same way, the scientific truth decided by Galileo is pursued to infinity. But the laws of physics which have been successively invented are finite subjects of this truth. - Alain Badiou

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128 Introduction and Basic Concepts Infinity as a thing vs. infinity as an attribute of things ( x is infinite )

129 Introduction and Basic Concepts Infinity as a thing vs. infinity as an attribute of things ( x is infinite ) Infinity of the small ( infinitesimal ) vs. infinity of the large

130 Introduction and Basic Concepts Infinity as a thing vs. infinity as an attribute of things ( x is infinite ) Infinity of the small ( infinitesimal ) vs. infinity of the large Continuity as infinity (relationship between geometry and numbers)

131 Continuity as Infinity Discrete or Quantized

132 Continuity as Infinity: Truly Continuous

133 Introduction and Basic Concepts Infinity as a thing vs. infinity as an attribute of things ( x is infinite ) Infinity of the small ( infinitesimal ) vs. infinity of the large Continuity as infinity (relationship between geometry and numbers) The metaphysically infinite vs. the mathematically infinite

134 Mathematical vs. Metaphysical Infinity Mathematical Boundlessness Endlessness Unlimitedness Immeasurability Eternity Greater than any given quantity Always capable of being added to Always capable of being divided further Metaphysical Completeness Wholeness Unity Universality Absoluteness Perfection Self-Sufficiency Autonomy Transcendence

135 Introduction and Basic Concepts Infinity as a thing vs. infinity as an attribute of things ( x is infinite ) Infinity of the small ( infinitesimal ) vs. infinity of the large Continuity as infinity (relationship between geometry and numbers) The metaphysically infinite vs. the mathematically infinite Self-reference as infinity ( infinite looping )

136 Paradoxes of self-reference - The Cretan Paradox (Epimenides of Knossos, 7 th Century BCE) A Cretan (Epimenides) says, All Cretans are liars. - The Liar Paradox (Eubulides of Miletus, 4 th Century BCE) A man says, What I am saying now is lies. Is he telling the truth? - This sentence is false. Is the sentence true?

137 Paradoxes of self-reference - Godel s sentence (1931): This theorem can not be proved. Is the theorem true or false? If it can not be proved, then it is true, yet you cannot prove it If it can be proved, then it is true that it can not be proved, which is a contradiction Shows that mathematics with finite axiom systems is either incomplete (there are truths that can not be proven) or inconsistent (you can prove contradictions)

138 Aristotle on the Continuum Things are called continuous when the touching limits of each become one and the same and are contained in each other. Continuity is impossible if these extremities are two Continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one. - Aristotle

139 The belief in eternity [infinite past time] the way Aristotle sees it that is, the belief according to which the world exists in virtue of necessity, that no nature changes at all, and that the customary course of events can not be modified with regard to anything destroys the Law [Scripture] in its principle, necessarily gives the lie to every miracle, and reduces to inanity all the hopes and threats that the Law has held out, unless by God! one interprets the miracles figuratively also this, however, would result in some sort of crazy imaginings. - Maimonides

140 We have intended, then, to say, that all our intuition is nothing but the representation of appearances; that the things which we intuit are not in themselves the same as our representations of them in intuition, nor are their relations in themselves so constituted as they appear to us; and that if we take away the subject, or even only the subjective constitution of our senses in general, then not only the nature and relations of objects in space and time, but even space and time themselves disappear; and that these, as appearances, cannot exist in themselves, but only in us. - Immanuel Kant

141 What may be the nature of objects considered as things in themselves and without reference to the receptivity of our sensibility is quite unknown to us. We know nothing more than our own mode of perceiving them Space and time are the pure forms thereof; sensation the matter. The former alone can we know a priori, that is, antecedent to all actual perception; and for this reason such knowledge is called pure intuition. The latter is that in our knowledge which is called knowledge a posteriori, that is, empirical intuition Supposing that we should carry our empirical intuition even to the very highest degree of clearness, we should not thereby advance one step nearer to the appearances of the constitution of objects as things in themselves. - Immanuel Kant

142 Again unless there shall be a least, the very smallest bodies will consist of infinite parts, inasmuch as half of a half will always have a half and nothing will set bounds to the division. Therefore between the sum of things and the least of things what difference will there be? There will be no distinction at all; for how absolutely infinite soever the whole sum is, yet the things which are smallest will equally consist of infinite parts. - Lucretius

143 Anaxagoras (500?-428 BCE) - Friend of Pericles and Euripides - Believed that all matter existed since the beginning, but in the form of infinitesimal pieces that were formed into the objects we perceive by the operation of mind (nous) - Stated that however small anything was, there was something else still smaller; and however big anything was, there was something else still bigger

144 Marcus Aurelius ( ) - Roman Emperor, Stoic philosopher - Believed in the infinity of time and space and disbelieved in the possibility of an afterlife; held that life s meaning lay in devotion to duty and rational virtue in the face of the indifference of the universe to human existence

145 These are the properties of the rational soul It traverses the whole universe and the surrounding void, and surveys its form, and it extends itself into the infinity of time, and embraces and comprehends the periodic renovation of all things, and it comprehends that those who come after us will see nothing new, nor have those before us seen anything more, but in a manner he who is forty years old, if he has any understanding at all, has seen, by virtue of the uniformity that prevails, all things that have been and all that will be. - Marcus Aurelius

146 Is then the number not simply infinite, but so that we can always take more? No, the generation of number is not in the power of the numberer, but it is already limited and stands fast. Or, in the intelligible, just as the real beings are limited so is the number limited to as many as the real beings. But we, just as we make man many by many times applying beauty and the rest to him, so along with each image we generate an image of number, and, as we multiply the town though it does not really exist as multiple, in the same way we also multiply the numbers; and if we should be numbering times, we apply numbers to them from those which we have, and those numbers remain within us. - Plotinus

147 That which resumes all under a unity is a Principle in which all things exist together and the single thing is All. From this Principle, which remains internally unmoved, particular things push forth as from a single root which never itself emerges. They are a branching into part, into multiplicity, each single outgrowth bearing its trace of the common source. Thus, phase by phase, there is finally the production into this world; some things close still to the root, others widely separate in the continuous progression until we have, in our metaphor, bough and crest, foliage and fruit. At the one side all is one point of unbroken rest, on the other is the ceaseless process, leaf and fruit, all the things of process carrying ever within themselves the Reason- Principles of the Upper Sphere, and striving to become trees in their own minor order and producing, if at all, only what is in strict gradation from themselves. - Plotinus

148 The multiplicity [in Intellect] is unified and not allowed to be altogether multiplicity, being a one-multiple. And because of this it is less than the One, because it has multiplicity, and insofar as it is compared with the One, it is worse; and since it does not have the nature of that One, but has gone out from it, it has been diminished, but it keeps its majesty by the one in it, and it turned back its multiplicity to one and there it stayed. - Plotinus

149 in infinity contradiction is without contradiction because it is infinity. Infinity is simplicity itself of all that are spoken; contradiction does not exist without otherness. Yet in simplicity otherness exists without otherness because it is simplicity itself. For all that can be said of absolute simplicity coincides with it, because in absolute simplicity having is being. The opposition of opposites is an opposition without opposition, just as the end of finite things is an end without an end. You are, therefore, O God, the opposition of opposites, because You are infinite, and because You are infinite, You are infinity itself. In infinity the opposition of opposites is without opposition. - Nicholas of Cusa

150 Every natural body has a greater or smaller determinate quantity. Hence it is impossible for a natural body to be infinite. The same appears from movement; because every natural body has some natural movement; whereas an infinite body could not have any natural movement because nothing moves naturally by a direct movement unless it is out of its place; and this could not happen to an infinite body, for it would occupy every place, and thus every place would be indifferently its own place. - St. Thomas Aquinas

151 Rabbinical Judaism and Kabbalah - Rabbi Akiba (1 st Century) Ma aseh Merkavah (The Creation of the Chariot) Mystical contemplation of the infinity and unity of God Power of names as finite symbols of the infinite Story of pardes: Akiba, ben Zoma, ben Azzai, ben Abuya

152 Rabbinical Judaism and Kabbalah - Rabbi Akiba (1 st Century) Ma aseh Merkavah (The Creation of the Chariot) Mystical contemplation of the infinity and unity of God Power of names as finite symbols of the infinite Story of pardes: Akiba, ben Zoma, ben Azzai, ben Abuya - Rabbi Shimon bar Yochai (late 1 st Century 2 nd Century) Student and contemporary of Akiba Collected mystical traditions into an orally transmitted body

153 Rabbinical Judaism and Kabbalah - Rabbi Akiba (1 st Century) Ma aseh Merkavah (The Creation of the Chariot) Mystical contemplation of the infinity and unity of God Power of names as finite symbols of the infinite Story of pardes: Akiba, ben Zoma, ben Azzai, ben Abuya - Rabbi Shimon bar Yochai (late 1 st Century 2 nd Century) Student and contemporary of Akiba Collected mystical traditions into an orally transmitted body - Bahir (The Brilliant) and Sefer Yetzirah (Book of Formation) - 1 st Century 5 th Century

154 Rabbinical Judaism and Kabbalah - Rabbi Akiba (1 st Century) Ma aseh Merkavah (The Creation of the Chariot) Mystical contemplation of the infinity and unity of God Power of names as finite symbols of the infinite Story of pardes: Akiba, ben Zoma, ben Azzai, ben Abuya - Rabbi Shimon bar Yochai (late 1 st Century 2 nd Century) Student and contemporary of Akiba Collected mystical traditions into an orally transmitted body - Bahir (The Brilliant) and Sefer Yetzirah (Book of Formation) - 1 st Century 5 th Century - Moses de Leon (1250?-1305) Zohar Compendium of bar Yochai s traditions and others in the form of a commentary on the Five Books of Moses

155 The Kabbalistic View of Infinity - Ein Sof (Without End): The primeval, actually infinite Divine, beyond comprehension, outside time and space - Sefirot (spheres, emanations): 10 dimensions or attributes of the Divine creative impulse, each of which is infinite, but each of which is accessible to comprehension in varying degrees according to its proximity to Ein Sof - The Ein Sof contracts (tzimtzum) in order to allow room for the creation of time, space, matter and energy, injecting its creative impulse into the Universe through the Sefirot - Shechinah (Indwelling): The infinite presence of God in the created Universe (conceptualized as female) - Malchut (Kingdom): The dimension of actual existence

156 The Kabbalistic Tree of Life Ein Sof, the Unknowable Infinite Keter, the Divine Crown Chochmah, Wisdom Binah, Understanding Chesed (Gedulah), Mercy Din (Gevurah), Justice Tif'eret, Beauty Netzach, Eternity Hod, Glory Yesod, Foundation Malchut (Shechinah), God's Presence in the World

157 Attributes of God Considered As Names One must know that He is called Wise, with all kinds of wisdom, Understanding, with all kinds of understanding, Saintly, with all kinds of saintliness, Heroic, with all kinds of heroism, Counsel, with all kinds of counsel, Righteous, with all kinds of righteousness, and King, with all kinds of royalty, until infinity. At the various levels Of His existence, He might be called compassionate, while elsewhere He will be called Judge, and so forth on a number of levels until infinity Hence, all of God s names are His symbols because of His actions. - Zohar, III, 257b

158 Moses ben Maimon (Maimonides) ( ) - Greatest rationalist Jewish philosopher - Attempted to reconcile Aristotle and, to some extent, Plato with the Hebrew Scriptures - Opposed to Kabbalistic, mystical tradition believed God is unknowable and can only be described by means of negations - Held that God is infinite and eternal, outside time and space, and that all Creation (time, space and matter) is finite