Infinity (philosophy)

→ In philosophy, infinity can be attributed to infinite dimensions, as for instance in Kant's first antinomy. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes. In Greek philosophy, for example in Anaximander, 'the Boundless' is the origin of all that is. He took the beginning or first principle to be an endless, unlimited primordial mass (ἄπειρον, apeiron). In Judeo-Christian theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In ethics infinity plays an important role designating that which cannot be defined or reduced to knowledge or power.

The snake

Euronome the goddess of fertility danced on the waves of the ocean until Ophion , the serpent impregnated her. Euronome took the form of a dove and layed the Universal (in Greek this is phonetically translated as katholikossee ) egg (sometimes this is translated as oon ). The serpent encircled the egg seven times and on the seventh encircle the egg was split and from the egg all of that which exists was produced.

— Bradway & McCoard

See also : The Tree and The Snake

The origin of ∞ , is perhaps ergo the testiment to the testicles of Oceanus with the twisted scrotum.

Chaos

The orphic cosmogonies are thought the development across the boundary between the earlier reliance wholly on primitive myth to instead the aquisition of knowledge through speculation on metaphysical reality. The principle to which Orphism first departs is of Chaos that is meaning the infinite, and Ether by this was understood as the limited. The first word meaning of the word Chaos attributed to the work of Hesiod means at the word-root to gape, at a later time the word developed from to pour " from a false derivation (Guirand) " and after meaning a space in a state of randomness (" a confused and unorganised mass of elements scattered through space (Guirand)").

The River

In Homer's Odyssey

...the ship went down the stream of the river Oceanus...

— Odyssey Book XI

The Oceanus or Okeanus is :

...flowing-back-on-itself...

...encircled the whole earth...

...dwelt at the furthest limits of the earth...

— Homer's Iliad

OKEANOS the god and TETHYS his wife had for a time a celibate marriage,and in the end they seperated. While they were together they begot children. Okeanos did though hate his children, and had to bury them within the Earth. Tethys was angry at this and decided to punish her husband. Of all the children only Cronos assisted her in punishing Okeanos. One night while Okeanos was resting from his dayly activities in his bed, Cronos used the blade that Tethys had created to castrate Okeanos. the testicles first flung behind the son, ultimately sinking into the sea (a result of which Love was born in the shape of Aphrodite). Okeanos was then alone and had only to continue his endless circular flux within the world. At the time of the succession of the gods from Cronos, having ruled from his father, to Zeus, only Okeanos was kept within the pantheon, and sometimes Tethys (Kerényi 1957).

see also : ἀγάπη and the wound is a gape

Early Greek

Main : Ancient Philosophy

Ancient greek for not limited is ἀπέρας

In an attempt to understand an element of the λόγος ( logos ), in the context especially of the origin of all things, without recourse to experiment, Anaximander student of Thales stated:

ARCHE TON ONTON TO APEIRON

...Τό μπέιζον is the first principle of all things, that the universe, though variable in it's parts, as a whole is immutable ; and that all things are produced from ἀπέρας and terminate in it...(Enfield & Brucker 1792).

Hippasus of Metapontum is credited as having introduced to ancient thought the concept of incommensurability. Tradition states that the idea emerged in the mid-part of the fifth century B.C.

Philolaus of Croton introduced the concepts of limited (perainonta) and άπειρος (apeira) meaning unlimited in On Nature.

Plato states of God:

...and he made the universe moving in a circle, one and solitary, yet by reason of it's excellence able to converse with itself, and needing no other friendship or acquintance... the entire compound he divided lengthways into two parts, which he joined together at the centre like a letter χ , and bent them into a circular form connecting them with themselves at the point opposite to the original meeting point ; and , comprehending them in a uniform revolution upon the same axis, he made the one the outer and the other the inner circle. Now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the diverse...

— p.20

In Sophist stating that Forms are all divisible, except infima species of this nothing smaller is possible through dividing, and contemplation of anything smaller is indefinable (apeira) tending in number to a number that has no limit, in this therefore incomprehensable άπειρος (indefinite). Conceiving apeira as the foundation for the generation of numbers to all higher numbers,this taught in the Academy as the generation of number to the απέτρος Unlimited and (α)πέτρος Limited begins from prior numbers. Although earlier;

"[if] ἀπέρας, (they would be) altogether unknowable, since knowledge limits and sets bounds to what is known.” - Anaxagoras [of which was stated] "...by apeira he probably meant incomprehensible and unknowable to us." (Simplicius, Cael. 608.24; quoted in Guthrie, I, pp. 420-4). →[10]

Aristotle stated:

But Plato has two ἀπέρας, the ουσία (Great) and the ἀρχή (Small)

— Physics, book 3, chapter 4.

The first is concerning the receptacle, into this all matter is contained that exists everywhere (i.e. in the contemporary understanding is meant astronomical space), and the second are the things that are the smallest indivisible stable substances of Nature, this ἀρχή is produced in the mind in the thinking on theoretical reduction of matter through division (i.e.in the contemporary manifestation are atoms).

In respect to quantification of ἀπέρας he stated:

Continuum is the essential factor in consideration of magnitude of quantities some are discrete (diôrismenon), others continuous (suneches)… . Discrete are numbers and language; continuous are lines, surfaces, bodies, and also, besides these, time and place.

by this is interpreted that contemplation of ἀπέρας is a wholly physical consideration excluding numerical attribution, and a factor of the inorganic. Further the infinite continuum nature of physical bodies to the extent of becoming immeasurable, is identified as a difficulty in defining an imaginary concept with no physical reality by:

... It is always possible to think of a larger number: for the number of times a magnitude can be bisected is ἀπέρας. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number.

— Physics 207b8

This is often called potential infinity.

Leucippus and his student Democritus, and Epicurus accounted for how the things of the world came to be by saying all material things are made of small parts:

... ἀπέρας in multitude,and they believed them to be atoms and indivisible (Simplicius's Commentary on Aristotle's On the Heavens)

and furthermore Simplicius stated of them later:

..those who abandoned division to ἀπέρας on the ground that we are not able to divide to infinity and therefore cannot guarantee that the division be unending, declared that the bodies are composed of indivisible things and are divided into indivisibles (Commentary on Aristotle's Physics)

The school of Leucippus found the infinite as synonymous with the void, by this is meaning the vacuum of outer space.

..primary bodies are always moving in the void (that is the ἀπέρας)... ( Aristotle On the Heavens)

and also that the

...primary bodies, the atoms are moving in the ἀπέρας void, by compulsion (Simplicius's Commentary on Aristotle's On the Heavens)

See also : Infinity (Oriental thought)

Rome

The Latin for without boundary is given as either infinitas or infinitus:

īnfīnītum is a fathomless gulf, into which all things vanish. Marcus Aurelius

Lucretius provides the story of an understanding of the physicality of objects in the thought of physics by stating in DE RERUM NATURA :

... When the life of men lay foully grovelling before our eyes, crushed beneath the weight of a Religion, who displayed her head from the regions of the sky, lowering over mortals with terrible aspect, a man of Greece was the first to raise mortal eyes against her...to break the closed bars of nature's portals...proceeded far beyond the flaming battlements of this world , and in mind and thought traversed the whole immensity of space; hence triumphant he declares what can arise into being, and what can not, in fine, in what way the powers of all things are limited, and a deeply fixed boundary assigned to each...

— p.6

and the limits to the world,

... Moreover, in things before our eyes, object seems to bound object; the air sets-a-boundary to the hills, and the hills to the air ; the land limits the see, and the sea, on the other hand, limits the entire land; but as to the whole there is nothing beyond which is a boundary.on all sides vast abundance of space lies open for all things, all limit being set aside everywhere and in every direction...

— p.47

and the definition of infinity :

...All that exists therefore, I affirm, is bounded in no direction for if it were bounded, it must have some extremity, but it appears that there cannot be an extremity, unless there be something beyond, which may limit it;...

— p.45

Medieval

Main : Medieval Philosophy

This Aristotlian view is found in a clearer form by medieval writers such as William of Ockham:

Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes.

But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent.

The parts are actually there, in some sense. However, on this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more." Aquinas also argued against the idea that infinity could be in any sense complete, or a totality.

In the late Middle-Ages a crisis of the conceptualization of the self was occassioned amongst the intelligentia as a development of thinking about the relation of the infinite Divine and the creation of a finite human world.

Views from the Renaissance

Main : Renaissance Philosophy

Evangelista Toricelli is credited with having made the first vacuum, by this bringing the emptiness of space onto the planet.

Galileo was the first to notice that we can place an infinite set into one-to-one correspondence with one of its proper subsets (any part of the set, that is not the whole). For example, we can match up the set of square numbers {1, 4, 9, 16, ...} with the natural numbers {1, 2, 3, 4, ...} as follows:

1 → 1
2 → → 4
3 → → → 9
4 → → → →16
5 → → → → → 25
6 → → → → → → 36
7 → → → → → → → 49
8 → → → → → → → → 64

It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same "size". Galileo thought this was one of the difficulties which arise when we try, "with our finite minds," to comprehend the infinite.

So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.

— On two New Sciences, 1638

The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets. (Mathematicians from the time of Georg Cantor do apply the principle to infinite sets, and do have a notion of some infinite quantities being greater than others.)

Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. They believed all our ideas were derived from sense data or "impressions," and since all sensory impressions are inherently finite, so too are our thoughts and ideas. Our idea of infinity is merely negative or privative.

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.

— Essay, II. xvii. 7., author's emphasis

Famously, the ultra-empiricist Hobbes tried to defend the idea of a potential infinity in light of the discovery, by Evangelista Torricelli, of a figure (Gabriel's Horn) whose surface area is infinite, but whose volume is finite. Not reported, this motivation of Hobbes came too late as curves having infinite length yet bounding finite areas were known much before. Such seeming paradoxes are resolved by taking any finite figure and stretching its content infinitely in one direction; the magnitude of its content is unchanged as its divisions drop off geometrically but the magnitude of its bounds increases to infinity by necessity. Potentiality lies in the definitions of this operation, as well-defined and interconsistent mathematical axioms. A potential infinity is allowed by letting an infinitely-large quantity be cancelled out by an infinitely-small quantity.

Modern philosophical views

Main : Modern Philosophy

Zwei Dinge sind unendlich: Das Universum und die menschliche Dummheit. Beim Universum bin ich mir aber noch nicht ganz sicher

Modern discussion of the infinite is now regarded as part of set theory and mathematics. This discussion is generally avoided by philosophers. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period".

Does the relation $m=2n$ correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes... In the superstition that $m=2n$ correlates a class with its subclass, we merely have yet another case of ambiguous grammar.

— Philosophical Remarks § 141, cf Philosophical Grammar p. 465

Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience.

[Time] is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room.

The philosopher Emmanuel Levinas uses infinity to designate that which cannot be defined or reduced to knowledge or power. In Levinas' magnum opus Totality and Infinity he says,

The idea of infinity is not an incidental notion forged by a subjectivity to reflect the case of an entity encountering on the outside nothing that limits it, overflowing every limit, and thereby infinite....All knowing qua intentionality already presupposes the idea of infinity, which is preeminently non-adequation.

— p. 26-27

The idea of incommensurability is reassessed as more likely to have emerged during the most latter part of the fifth century or the early fourth century.

Aristotle's potential infinity is expressed, $\forall n\in \mathbb {Z} (\exists m\in \mathbb {Z} [m>n\wedge P(m)])$ , which reads, "for any integer n, there exists an integer m > n such that P(m)".

if

∀ is the universal quantification

∈ = set membership

ℤ = integers

P = projective space

in that we may quantify over infinite sets without restriction. Also understood as, it is always possible to find a number of things that surpasses any given number, even if there are not actually such things.

Experiential response

Looking outward to the blackness of space, sprinkled with the glory of a universe of lights, I saw majesty - but no welcome. → Loren Acton, USA

in this context the concept of infinity is an outward looking dependent macrocosmic concept, compared to the indivisible of Plato as an inward looking dependent microcosmic concept. As the ancients fought to overcome the boundaries of their own understanding of Nature, so the modern equivalent has given form to the immeasurable distance of the Leucippucian void by Einstein and others with the light year. Still as observation instead of the light in the Universe promotes further inquiries into infinity the concept remains within the definition of incommensurable. For as each further advancement in skill and capabilities of contact with sources of energy at increasing distance from earth so the foundation of the concept has remained unmoved. This is going to prove to be a challenge and difficult to solve for humanity. As the universe is posited to be in expansion, and some are worried of everything flying outward too far perhaps into the gulf of Aurelius, find all things hence forward becoming too weak in energy until everything ceases. The only hope seems to be of weighing the entire universe and making the necessary calculation from the speed of change, to know then how long everything is going to take for all life to be ended on planet Earth.

Difficulties occuring in contemplation of the infinite are no doubt something due to do with the hierarchical nature of scintific inquiry, such as the principle according to Aristotle that

Any science deals with matters available to scientific exploration. The natural order is to start from the things which are more knowable and obvious to us and proceed towards those which are clearer and inherently knowable by nature

— Physics Book - 1

which by it's very nature would limit the capabilities of understanding until such time as all other prior available exploration by way of scientific method have been explored.

or else no such thing that is infinite exists physically on earth so no man is able to think or imagine of such a thing in Heaven nor anywhere else.

...And he said, Thou canst not see my face: for there shall no man see me, and {1} live. (1) For Moses did not see his face in full majesty, but as man's weakness could bear...

—  Geneva Study - The Christian and Judaic Holy Book

See also: The Operations of Houston

Invisibility of God

Trying to place God anywhere in the Universe, and in the desperate need for some feeling that immortality and the relief from the fear or terror or tension of death, from knowing of the existence of God or instead the possibility of God in the physical world, the location of the Divinity is naturally tending toward the place in the Universe where there are no more stars or astronomical structures and this place is the outside of the furthest physical thing in space from Earth (ουσία). So in thinking of this place there is perhaps only emptiness and the lack of light. This means no possibility for any thing on Earth to perceive anything within the emptiness on the other side of a last object, or at least at a certain hypothetical distance in the emptiness of total nothingness ( if that is the physical reality) in that there is no physical thing past nothing. In the unknowable there is mystery and in the impossible mystery is the origin of religious conceptualization. The existence of infinity is accepted naturally from this and from thought of the space that there was before all the Universe existed, they are the only two places infinity does in reality exist, in theory,although again from lack of accaptance of the annuling of consciousness and end to life that is death without recourse to an eternal afterlife. Another possible strategy of overcoming the fear of the appreciation of the least practical part of the Universe, is in the Ancient Greek division, of this sub-atomic particles are produced that are still measurable by science therefore having some boundary, whether of negligible mass or the range of energy produced from the particle in interaction, this as a type of boundary.

Three types of infinities

Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity. There are scientists who hold that all three really exist and there are scientists who hold that none of the three exists. And in between there are the various possibilities. Rudy Rucker, in his book Infinity and the Mind — the Science and Philosophy of the Infinite (1982), has worked out a model list of representatives of each of the eight possible standpoints. The footnote on p. 335 of his book suggests the consideration of the following names: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. Brouwer, David Hilbert, Bertrand Russell, Kurt Gödel and Georg Cantor.

The Serpent: Formal comparison of (eg)g & infinity symbol

$Egg$

and also compare with the letters Zeta and Delta.

See also

• Eternity
• Circularity
• Recursion: See semantic satiation