3. Historical views of infinity

‘Historical views of infinity’ focuses on historical attitudes to infinity in philosophy, religion, and mathematics, including Zeno’s famous paradoxes. Infinity is not a thing, but a concept, related to the default workings of the human mind. Zeno’s paradoxes appear to be about physical reality, but they mainly address how we think about space, time, and motion. A central (but possibly dated) contribution was Aristotle’s distinction between actual and potential infinity. Theologians, from Origen to Aquinas, sharpened the debate, and philosophers such as Immanuel Kant took up the challenge. Mathematicians made radical advances, often against resistance from philosophers.

6. Physical infinity

‘Physical infinity’ moves from mathematics to the real world and tackles questions such as ‘is space infinite?’ In many areas of physics, the presence of an infinite quantity (often called a singularity) is construed as a warning that the theory is losing touch with reality. For instance, according to classical ray optics, the intensity of light at the focus of a lens is infinite. The physical resolution of this difficulty involves replacing light rays by waves. Singularities are discussed in three physical contexts: optics, Newtonian gravity, and Albert Einstein’s relativity. However, there’s one area of physics in which an actual infinity—physical, not conceptual—is presented as a possible truth: cosmology.

4. The flipside of infinity

‘The flipside of infinity’ examines a logical counterpart of the infinite: infinitesimals. These are quantities that are infinitely small, instead of infinitely large. Historically, such quantities formed the basis of calculus, one of the most useful branches of mathematics ever invented. However, they caused considerable head-scratching, starting an argument that took about two centuries to resolve. This was achieved using a version of Aristotle’s potential infinity—namely, potential infinitesimality. Exhaustion is also explained, along with the modern concept of a limit, which abolished infinitesimals. Then it considers how infinitesimals were reinstated and outlines non-standard analysis, which provides a logical framework for infinitesimals.

1. Puzzles, proofs, and paradoxes

‘Puzzles, proofs, and paradoxes’ introduces nine typical examples of reasoning about the infinite—puzzles, paradoxes, even a few proofs—including Galileo’s squares and numbers problem; David Hilbert’s hotel scenario that illustrated Cantor’s theory of infinite numbers; Guido Grandi’s proof of the Creation from 1703; and concepts concerning the area of a circle, the largest number, and the diagonal of a square. Each of these nine examples is discussed briefly, before the chapter analyses whether the methods or the answers are logically acceptable. Some of the concepts deserve further discussion, and are returned to in later chapters.

2. Encounters with the infinite

‘Encounters with the infinite’ raises some common misconceptions about infinity, and shows how infinity naturally appears in elementary arithmetic. It aims to show how deeply embedded infinity is, even in basic areas of mathematics, and to clarify possible confusion about topics that we think we understand. It begins with two issues: firstly, infinity is not just a synonym for a very big number; it is not a very large limit, but the absence of any limit. Secondly, infinity is not just some esoteric invention in advanced mathematics—we run into it quite early on at school level when we start learning about the decimal system.

Introduction

Why do we need to think about the infinite—a concept we never encounter directly? There are many reasons. Even in elementary mathematics, we encounter aspects of infinity, but more generally, our minds seem to require the idea that things might ‘go on forever’—in space and in time, in the future and the past. The Introduction explains how infinity is a fascinating concept, full of subtleties, logical pitfalls, puzzles, and paradoxes, and then considers the subtle distinctions which philosophers, theologians, and mathematicians have been forced to make when contemplating infinity. One of the greatest paradoxes of the infinite is that it’s turned out to be extremely useful.

7. Counting infinity

‘Counting infinity’ returns to the mathematics of infinity, discussing Cantor’s remarkable theory of how to count infinite sets, and the discovery that there are different sizes of infinity. For example, the set of all integers is infinite, and the set of all real numbers (infinite decimals) is infinite, but these infinities are fundamentally different, and there are more real numbers than integers. The ‘numbers’ here are called transfinite cardinals. For comparison, another way to assign numbers to infinite sets is mentioned, by placing them in order, leading to transfinite ordinals. It ends by asking whether the old philosophical distinction between actual and potential infinity is still relevant to modern mathematics, and examining the meaning of mathematical existence.

5. Geometric infinity

Much of the philosophical and mathematical fun comes from trying to tease the different meanings apart, and deciding which make sense, and why. A clear example occurs in ‘Geometric infinity’, where the discussion takes a sharp turn into a different realm of the infinite: projective geometry. As Euclid insisted in one of his axioms, parallel lines never meet. But the painters of the Italian Renaissance, analysing perspective, stumbled across a rich vein of geometry in which it makes sense to insist that parallels do meet—at infinity. If you’ve ever stood at a railway station watching the tracks converge as they disappear into the distance, you’ve caught a glimpse of geometric infinity.