Infinity

That which has no limit - and the long argument over whether anything actual can be like that.

Theme: Being and reality
Tradition: Greek
Term: ἄπειρον · apeiron
Term fixed: c. 570 BC

In plain terms

Infinity is not a very large number. Adding one to a large number gives a larger one; adding one to infinity gives infinity, which is a sign that it is a different kind of thing altogether.

The old and useful distinction is between a process that could always be continued and a completed totality. Counting has no last step, and that is uncontroversial. Whether there exists somewhere an actually complete infinite collection is a different and much harder claim.

Most paradoxes come from treating the second as though it behaved like the first. A whole can be the same size as one of its parts. Two collections can both be endless and one still be larger. These are not tricks; they are what the subject is like once you take it seriously.

As a philosopher would put it

Aristotle's distinction is between the potential infinite, which is always incomplete and unproblematic, and the actual infinite, which he denies. Zeno's paradoxes are the pressure that made the distinction necessary: if a distance contains infinitely many sub-distances, crossing it seems to require completing an infinite series.

Cantor overturns the prohibition in the 1870s. Completed infinite sets can be compared by pairing their members, some infinities are strictly larger than others, and the diagonal argument shows the real numbers cannot be paired with the natural numbers. Infinity becomes a mathematical object with an arithmetic rather than a limit on thought.

Philosophically the older questions survive. The kalām tradition argued that an actually infinite past is impossible and therefore that the universe began - an argument al-Ghazālī presses hard against the philosophers. Aquinas notably refuses it, holding that an eternal created universe is not demonstrably impossible and that creation is settled by revelation rather than by argument.

Where you meet it

How long is the coast of Britain? Lewis Fry Richardson noticed while studying border lengths that the answer depends on the ruler. Measure with a 200-kilometre stride and you get one figure; with a 50-kilometre stride you get a substantially larger one, because the shorter ruler enters bays the longer one stepped over.

The increase does not settle. Every reduction in the ruler finds more detail, down through inlets, rocks and grains of sand, and the measured length grows without limit. Mandelbrot took this up in 1967 and made it the founding case of fractal geometry.

So an ordinary question about an ordinary object has no finite answer, and no error has been made by anyone. Official figures for coastline length differ by factors of two between agencies for exactly this reason. The infinity here is not in some remote domain; it is in the measurement of a thing you can walk along.

What is actually disputed

Whether actual infinities exist outside mathematics.

Finitists hold that the potential infinite is all we are entitled to, and that Cantor's hierarchy, whatever its mathematical value, describes no reality. The kalām argument is the strongest version applied to the world: an actually infinite series of past events could never have been completed, so the past is finite.

Against this, standard set theory treats infinite sets as objects and mathematics has been done on that basis for a century and a half without incoherence, which is evidence of a kind. Physics is unhelpfully divided: infinities in a theory are usually read as a sign the theory has broken down, and yet spacetime is standardly modelled as a continuum, which is an actual infinity of points in the most ordinary object there is.

Commonly confused with

a very large number Infinity does not behave like a number. Adding to it changes nothing, and it has no place on the number line.
the universe being big Unbounded and infinite are different. A surface can be finite with no edge, which is why cosmology is careful with the word.

Defined against

Finitude The limit that human existence has and the infinite does not, which is why the pair carries more than mathematics

Philosophers

Anaximander of Miletus The apeiron as the first principle
Zeno of Elea The paradoxes that made the distinction necessary
Aristotle Potential infinity yes, actual infinity no
Al-Ghazali The argument that an infinite past cannot have been completed
Thomas Aquinas Declines that argument, and says so

Works

Metaphysics Where the distinction is drawn
The Incoherence of the Philosophers The case against an eternal world
The Incoherence of the Incoherence Ibn Rushd's reply
Summa Theologiae Why an eternal creation is not demonstrably impossible
Critique of Pure Reason The antinomies, where both answers are proved

Schools

Milesian School Where the unlimited is first a principle
Eleatic School Where the paradoxes come from
Peripatetic school (Aristotelianism) The potential-actual distinction
Ash’arism The kalam argument for a finite past
Falsafa The eternalist position it was aimed at

Related concepts

Nothingness c. 475 BC · Necessity c. 330 BC · Time c. 397 · Space 1716 · Possibility c. 330 BC

Anaximander's apeiron, around 570 BC, is the first use of the unlimited as a principle; Aristotle's distinction between actual and potential infinity is what makes it manageable.