Infinity is the strangest idea humans have had. Philosophers feared it, mystics loved it, mathematicians tamed it, and it still breaks our intuitions. Here is its story, from ancient Greece and India to the discovery that there is more than one size of infinity.
Around 450 BCE Zeno of Elea argued that motion is impossible. To cross a room you must first cross half of it, then half of what remains, and so on forever: infinitely many steps. And swift Achilles can never catch a tortoise with a head start, because each time he reaches where it was, it has moved on. Aristotle answered that such infinities exist only potentially, never all at once (Physics, Books III and VI). Mathematicians finally resolved the puzzles only with the theory of limits, two thousand years later.
Indian thinkers were unafraid of the infinite. A famous verse of the Upanishads describes the infinite as something that can be divided without being diminished:
“That (the invisible Brahman) is full, this (the visible Brahman) is full. This full (visible Brahman) proceeds from that full (invisible Brahman). On grasping the fulness of this full (visible Brahman) there is left that full (invisible Brahman).”Brihadaranyaka Upanishad 5.1.1 (tr. F. Max Müller)
Indian mathematicians also gave the world zero as a true number. In 628 CE Brahmagupta wrote rules for calculating with it, and in the 12th century Bhaskara II described a quantity divided by zero as infinite, unchanged however much is added to it.
For the 17th-century mathematician and believer Blaise Pascal, the new astronomy’s endless universe was a source of dread as much as awe:
“The eternal silence of these infinite spaces frightens me.”Pascal, Pensées 206 (tr. W. F. Trotter)
Illustration: infinite spaces
In the 1870s the German mathematician Georg Cantor proved something that shocked his colleagues: infinities come in different sizes. The whole numbers 1, 2, 3… are infinite, and so are the even numbers, and the two can be paired off exactly, so they are the same size. But in 1891 Cantor showed, with his famous “diagonal argument”, that the points on a line can never be listed one by one: there are more of them than there are whole numbers. Above that lies an endless ladder of ever larger infinities.
David Hilbert dramatised the strangeness with an imaginary hotel with infinitely many rooms, all full, which can still always take another guest: everyone simply moves one room up. Cantor was attacked as a corrupter of mathematics in his lifetime; Hilbert answered that no one would drive us from the paradise Cantor had created.
Continue: What Is Time?, Our Place in the Cosmos, or Kabbalah’s Ein Sof, the Infinite.
The mathematician David Hilbert liked to describe a hotel with infinitely many rooms, every one occupied. A new guest arrives. No problem: move the guest in room 1 to room 2, room 2 to room 3, and so on forever, and room 1 is free. Infinitely many new guests arrive? Move everyone to the room with double their number, and all the odd-numbered rooms are free. The hotel shows why infinity breaks our intuition: a part can be as large as the whole.
A new guest takes room 1. A part can be as large as the whole.
Cantor wondered whether there is an infinity bigger than the counting numbers but smaller than the points on a line. He could not prove either answer, and the strain of the work, along with fierce opposition from some colleagues, darkened his later life. The answer came in two halves: Kurt Gödel showed in 1940 that “no” could not be disproved from the standard rules of mathematics, and Paul Cohen showed in 1963 that “yes” could not be disproved either. On this question, mathematics itself is silent.
Infinity on Wikipedia · Zeno’s paradoxes · Stanford Encyclopedia: Infinity · Pascal’s Pensées (Project Gutenberg)
