The charm of infinity

Evelina Petitto


The concept of infinity is extremely complex and complicated, referring to something endless, unlimited, without bounds. This gave the headache to philosophers and mathematicians, that for a long time tried to define its properties, characteristics and even questioned its existence. To further complicate the situation we can distinguish three types of infinities: mathematical, physical and cosmological.

The symbol of infinity, called the lemniscate, was first used in a seventeenth century treatise on conic sections. It caught on quickly and was soon used to symbolize infinity or eternity in a variety of contexts; the appropriateness of the symbol ∞ for infinity lies in the fact that one can travel endlessly around such a curve.

The first to introduce this symbol with its mathematical meaning has been John Wallis in 1655. It was conjectured that he used a variant form of the Latin numerical for 1000 (CIƆ or CƆ), that was also used to indicate “many”.

Before entering the world of mathematics formulations of infinity, let’s talk simple: what is infinity? It isn’t something big, or extremely huge… it is something endless. This is a concept that actually can give you the headache: in the word we experience, in fact, we are not used to deal with things that have no end and no beginning.

Infinity is something that does not grow: it’s already fully formed, and it has always been like it.

Infinity is not a number, it is an idea, and as such it cannot be measured.


Mathematical infinities occur, for instance, as the number of points on a continuous line or as the size of the endless sequence of counting numbers: 1, 2, 3…

The ancient Greeks expressed infinity by the word apeiron, which had connotations of being unbounded, indefinite and formless.

In mathematics, we encounter the concept of infinity for the first time in regard with the ratio between the diagonal and the side of a square. Pythagoras and his followers initially believed that any aspect of the world could be expressed using whole numbers only (0, 1, 2, 3,…); however, they soon realised that the diagonal and the side of a square are incommensurable, that is, their lengths cannot both be expressed as whole number multiples of any shared unit. ). In modern mathematics this discovery is expressed by saying that the ratio is irrational and that it is the limit of an endless, nonrepeating decimal series.

One thinker that disliked the idea of infinity was Aristotele: he in fact rejected the concept of “actual” infinity (spatial, temporal, or numerical), which he distinguished from the “potential” infinity of being able to count without end.

Following this path Eudoxus of Cnidus and Archimedes tried to avoid the use of actual infinity by using a technique later named as the method of exhaustion: an area was calculated by halving the measuring unit at successive stages until the remaining area was below some fixed value (the remaining region having been “exhausted”).

The issue of infinitely small numbers led to the discovery of calculus by the English mathematician Isaac Newton and the German mathematician Gottfried Wilhelm Leibniz. In particular Newton introduced his own theory of infinitely small numbers, or infinitesimals, to justify the calculation of derivatives, or slopes.

A more direct use of infinity in mathematics arises with efforts to compare the sizes of infinite sets, such as the set of points on a line (real numbers) or the set of counting numbers. When talking about infinite sizes, ordinary intuitions about numbers are misleading; for example, thinkers of the medieval period were already aware of the paradoxical fact that line segments of varying lengths seemed to have the same number of points. Let’s try to draw two concentric circles, with the one on the outside being twice as big as the inner one:


Surprisingly, each point P on the outer circle can be paired with a unique point P′ on the inner circle by drawing a line from their common centre O to P and labelling its intersection with the inner circle P′.  However, intuitively the outside circle, being twice as big, should also have twice as many points as the inner circle, but infinity in this case is the same as twice infinity.

In the early 1600s, the Italian scientist Galileo Galilei addressed a similar non intuitive result now known as Galileo’s paradox. He demonstrated that the set of counting numbers could be put in a one-to-one correspondence with the apparently much smaller set of their squares. Galileo concluded that “we cannot speak of infinite quantities as being the one greater or less than or equal to another.”

The German mathematician Georg Cantor helped to resolve the confusion about infinite numbers. First, he demonstrated that the set of rational numbers (fractions) is the same size as the counting numbers; hence, they are called countable. Second, he proved the surprising result that not all infinities are equal. Using a so-called “diagonal argument,” Cantor showed that the size of the counting numbers is strictly less than the size of the real numbers. This result is known as Cantor’s theorem.


Because the status of physical infinities is still quite undecided, the science of physical infinities is much less developed than the science of mathematical ones.

Let’s talk for example about the universe: a lot of us have the idea that the universe is infinite in time and space and although some speculations have been done confirming this way of thinking , cosmologists generally believe that the universe is curved in such a way as to make it finite but unbounded, resembling the surface of a sphere. Other cosmological theories conceptualise the universe as being embedded in a higher-dimensional “superspace”, which could be infinite in extent.


In the light of the big-bang model of the origin of the universe, cosmologists generally believe that the universe has a finitely long past; whether it might have an endless future is an open question.

Under the “infinite future” view, space may continue much as it is now, with the galaxies drifting farther and farther apart, the stars burning to dust, and the remaining particles possibly decaying into radiation. Alternatively, in the “finite future” view, a cosmic catastrophe at some definite time in the future may destroy the universe. Some believe that the end of the universe we live in may be followed by the born of a new universe… in which case we can almost say that the future if the universe is still infinite.


Probably the most familiar context for discussing infinity is in metaphysics and theology, with the discussions about the concept of Absolute.

Although Plato thought of the Absolute as finite, all theologians and metaphysicians from Plotinus on have supposed the Absolute to be infinite. What is meant by “the Absolute” depends, of course, upon the philosopher in question; it might be taken to mean God, an overarching universal mind, or simply the class of all possible thoughts.


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