What Is Infinity?

The Infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.

- David Hilbert, On the Infinite

Podcast of the Day

Melvyn Bragg and guests discuss the nature and existence of mathematical infinity. Jonathan Swift encapsulated the counter-intuitive character of infinity with insouciant style:

“So, naturalists observe, a flea
Hath smaller fleas on him that prey
And these hath smaller fleas to bite ‘em
And so proceed ad infinitum.”

Alas, the developing utility mathematicians put to the idea of infinity did not find the English philosopher Thomas Hobbes quite so relaxed. When confronted with a diagram depicting an infinite solid whose volume was finite, he wrote, “To understand this for sense, it is not required that a man should be a geometrician or logician, but that he should be mad”. Yet philosophers and mathematicians have continued to grapple with the unending, and it is a core concept in modern maths. So, what is mathematical infinity? Are some infinities bigger than others? And does infinity exist in nature?

Listen to the In Our Time episode on Infinity

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Short Article of the Day

In 1883, the brilliant German mathematician Georg Cantor produced the first rigorous, systematic, mathematical theory of the infinite. It was a work of genius, quite unlike anything that had gone before. And it had some remarkable consequences. Cantor showed that some infinities are bigger than others; that we can devise precise mathematical tools for measuring these different infinite sizes; and that we can perform calculations with them. This was seen an assault not only on intuition, but also on received mathematical wisdom. In due course, I shall sketch some of the main features of Cantor’s work, including his most important result, commonly known as ‘Cantor’s theorem’. But first I want to give a brief historical glimpse of why this work was perceived as being so iconoclastic. Ultimately, my aim is to show that this perception was in fact wrong. My contention will be that Cantor’s work, far from being an assault on received mathematical wisdom, actually served to corroborate it.

Continue reading A W Moore's article: Why some infinities are bigger than others

Further Reading

Working with the infinite is tricky business. Zeno’s paradoxes first alerted philosophers to this in 450 B.C.E. when he argued that a fast runner such as Achilles has an infinite number of places to reach during the pursuit of a slower runner. Since then, there has been a struggle to understand how to use the notion of infinity in a coherent manner. This article concerns the significant and controversial role that the concepts of infinity and the infinite play in the disciplines of philosophy, physical science, and mathematics.

Philosophers want to know whether there is more than one coherent concept of infinity; which entities and properties are infinitely large, infinitely small, infinitely divisible, and infinitely numerous; and what arguments can justify answers one way or the other.

Here are four suggested examples of these different ways to be infinite. The density of matter at the center of a black hole is infinitely large. An electron is infinitely small. An hour is infinitely divisible. The integers are infinitely numerous. These four claims are ordered from most to least controversial, although all four have been challenged in the philosophical literature...

Continue reading the Internet Encyclopedia of Philosophy article on The Infinite by Bradley Dowden

Bonus Webcomic

Related Topics

If you’re interested in infinity, check out some of the following related topics for more resources:

LogicMathematics | Thought Experiments

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