Infinity (philosophy)

In philosophy, infinity can be attributed to space and time, 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 (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.

History

Early Indian views of infinity

The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".

:Unicode|Pūrṇam adaḥ pūrṇam idam (That is full, this is full):Unicode|pūrṇāt pūrṇam udacyate (From the full, the full is subtracted):Unicode|pūrṇasya pūrṇam ādāya (When the full is taken from the full):Unicode|pūrṇam evāvasiṣyate (The full still will remain.) - Isha Upanishad

The essence of this verse is that the Infinite cannot be measured arithmetically - God is Infinite. The Infinite can be represented in Infinite ways and does manifest in infinite ways.

The Indian mathematical text "Surya Prajnapti" (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

* Enumerable: lowest, intermediate and highest
* Innumerable: nearly innumerable, truly innumerable and innumerably innumerable
* Infinite: nearly infinite, truly infinite, infinitely infinite

The Jains were the first to discard the idea that all infinites were the same or equal. They recognized different types of infinities: infinite in length (one dimension), infinite in area (two dimensions), infinite in volume (three dimensions), and infinite perpetually (infinite number of dimensions).

According to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number "N" of the Jains corresponds to the modern concept of aleph-null $aleph\_0$ (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number "N" is the smallest.

In the Jaina work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between IAST|"asaṃkhyāta" ("countless, innumerable") and "ananta" ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.

Early European views of infinity

In Europe the traditional view derives from Aristotle:

This is often called potential infinity; however there are two ideas mixed up with this. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. The other is that we may quantify over infinite sets without restriction. For example, $forall\; n\; in\; mathbb\{Z\}\; (exists\; m\; in\; mathbb\{Z\}\; [m\; >\; n\; wedge\; P(m)]\; )$, which reads, "for any integer n, there exists an integer m > n such that P(m)". The second view is found in a clearer form by medieval writers such as William of Ockham:

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.

Views from the Renaissance to modern times

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
…

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.

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.

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

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". [See also cite web | title= Logic of antinomies | url=http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an=0724.03003&format=complete | accessdate=November 14 | accessyear=2005 ]

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

Ethics

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,

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 mind" (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.

Use of infinity in common speech

In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, and we had to wait forever to get tickets."

In video games, for example, infinite lives and infinite ammo refer to unlimited respawn capability and ammunition supply. An infinite loop in computer programming is a loop that never terminates. (See halting problem.) These terms describe things that are only potential infinities; it is impossible to play a video game for an infinite period of time or keep a computer running for an infinite period of time.

The expression Infinity plus 1 is also used sometimes in common speech.

Almost infinite is often used to refer to a large, but unknown, number (eg. the number of grains of sand on a beach), even by those who are well aware that any finite number is in fact infinitely distant from infinity.

References