Infinity

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.

Ancient view of infinity

The traditional view derives from Aristotle:

"... it is always possible to think of a larger number: for the number of
times a magnitude can be bisected is infinite. 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, 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:

"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 an
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 [reference].

Early modern views

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

     1, 2, 3, 4, ...
     2, 4, 6, 8, ...

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.

"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.

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.

"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 É 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 the light of the discovery by Evangelista Torricelli,
of a figure (Gabriels Horn) whose surface area is infinite, but whose volume
is finite.

Mathematical conception

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 views

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 impassioned atack upon Axiomatic set theory, and upon
the idea of the actual infinite, during his "middle period".

Infinity is now seperated into aleph-null, a countable series such as
natural numbers, and aleph-one, an uncountable series such as the number of
possible arcs in a circle or the points on a line.

"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 empricists, he thought that the infinite was in some
way given to sense experience

"...I can see in space the possibility of any finite experience ... we
recognise [the] essential infinity of space in its smallest part." "[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."

"... what is infinite about endlessness is only the endlessness itself"