Infinity or the infinite (from the Latin "finitus", meaning limited -- mathematical symbol: ∞) is that which is not "finite", that which has no limit in space or time. In mathematics, where this is known as the "transfinite"; it is that which is not merely finite, but which may have some limit beyond that.
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2 Early modern views 3 Mathematical conception 4 Modern views 5 The Absolute 6 Current Views |
The traditional view derives from Aristotle:
Ancient view of infinity
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 finite numbers without restriction. For example "For any integer n, there exists an integer m > n such that Phi(m)". The second view is found in a clearer form in medieval writers such as William of Ockham:
Galileo (during his long house arrest in Sienna after his condemnation by the Inquisition) was the first to notice that we can place a set of infinite numbers into one-to-one correspondence with one of its proper subsets (any part of the set, that is not equivalent to the whole). For example, we can match up the "set" of even numbers {2, 4, 6, 8 ...} with the natural numbers {1, 2, 3, 4 ...} as follows
Early modern views
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. He 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.
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 appearance or "impressions", and since all sense impression is inherently finite, so too for our thoughts and ideas. Our idea of infinity is merely negative or privative.
The modern mathematical conception of the infinite developed in the late nineteenth century from work by Georg Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of sets. Their approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its "proper " parts.
Thus Cantor showed that infinite sets can even have different sizes, distinguished between countably infinite and uncountable sets, and developed a theory of cardinal numbers around this. His view prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the surreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
Our intuition gained from finite sets breaks down when dealing with infinite sets. One example of this is Hilbert's paradox of the Grand Hotel.
An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no boundaries. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might operate on a similar principle; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point.
Modern discussion of the infinite is now regarded as part of set theory and mathematics, and 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".
Infinity is now separated into many kinds of infinite sets, such as aleph-null, a countable series such as natural numbers, and beth-one, an uncountable series such as the number of possible arcs in a circle or the points on a line, and an infinite number of others.
Another question is whether the mathematical conception of infinity has any relation to the religious concept of God. This question was addressed by both Cantor, with his concept of the Absolute Infinite which he equated with God, and Kurt Gödel with his "ontological proof" of the existence of an entity he related to God.
As of lately the most general idea of the infinite has been used to promote new philosophical and theological movements, which have the infinite as their central theme or device. A good reference source may be found through the Principia Cybernetica website, which is considered one of the most sophisticated philosophical exchanges available on the world-wide-web. Of particular note is the contemporary philosopher Jean-Pierre Ady Fenyo's views concerning the potential social psychological implication/application of the concept of infinity in general (he is listed in the Millennium Edition of MARQUIS' Who's Who In The World). In his 1994 book (copyright reg. U.S. Library of Congress) he argues that the dimensions of wisdom are open mindedness, long-term orientedness, and depth of thought combined, that given its very nature of being boundless and eternal thinking regularly about the infinite generates such dimensions of wise thought and that if the majority, or at least a significant number, of individuals in the world were instigated/inspired to do so (by all non-violent means possible) then the world would change significantly and become a much more decent place than it currently is.
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See also: InfinitesimalMathematical conception
Modern views
Unlike the traditional empricists, he thought that the infinite was in some way given to sense experienceThe Absolute
Current Views