Infinity: A Very Short Introduction
This is one of the small booklets (literally pocket books: 174 x 111 mm) appearing in the Oxford series Very Short Introductions that treat diverse subjects from accounting to zionism. Infinity, is a concept mainly of importance and practically useful in mathematics, but it has also philosophical and even religious aspects. Stewart is as broad as "a very short introduction" allows and adds a lot of history to his discussion. So much is to be told on only 143 small pages. Although there is obviously a lot of overlap, Stewart's treatment is wider than Marcus Du Sautoy's How to Count to Infinity and Eugenia Chen's Beyond Infinity who stay more on a mathematical playground.
Infinity, and certainly the infinitely large, has long been something fuzzy that was discussed on a philosophical basis. The Greek were arguing over a distinction between an actual (existing) infinity and a potential version, i.e. that "something" that is beyond all natural numbers, which is never reached by enumeration. They got away with the infinitely small by their concept of commensurability in what was mainly a geometric approach to mathematics. The infinitely small was beyond any possible subdivision of a finite length. Their fundamental common measure was thus finite and that led Zeno to his paradoxes. The infinitely small was somehow tackled when calculus was developed by Newton and Leibniz in the eighteenth century introducing infinitesimals. They represented something almost zero but not quite. When used in calculations one could divide by them, since they were not zero, but at some point, when suitable for the result, they were assumed to be zero. Not very rigorous mathematics that was. It was not until towards the end of the nineteenth century that Georg Cantor brought more insight into the nature of the infinitely large. Stewart guides us through this history and illustrates how the concept of infinity has played a role in several disciplines that all have somehow contributed to how we think of the concept today.
With a first chapter, Stewart puts forward some puzzles or paradoxes that involve infinity to illustrate that it is not sufficient to say that infinity is that "something" that is beyond all numbers. More precise definitions are needed for the infinitely large as well as for the infinitely small. Examples are the processes that hide irrational numbers like a staircase approximating the diagonal of a square converging to a straight line when its steps become finer and finer, and the regular polygon converging to the circle as it gets more edges. These demonstrate the problem of evaluating $0\times\infty$ in a sensible way. Hilbert's hotel is illustrating that a more precise definition of the infinitely large is required, and Stewart gives some other examples. These puzzles and paradoxes are first raised as questions for the reader to think about. Stewart's explanations of all these confusing statements are given afterwards.
The second chapter illustrates that infinity is not hidden away in higher mathematics but that it is also embedded into elementary calculus. Gabriel's horn is obtained by revolving $1/x$ for $x>1$ around the $x$-axis. This has the surprising property that its volume is finite even though the surface is infinite. Of course infinity is also hidden in 0.9999... being equal to 1, a fact that astonishes many an undergraduate student, and of course infinity resonates in the decimal representation of irrational numbers. Distinguishing discrete from continuous would not be possible without infinity. Here as in the other chapters Stewart gives quite some attention to history: Dedekind defining the real numbers as sections which are essentially infinite objects, Lambert who proved the irrationality of $\pi$. In the Jain religion of India (600 BCE), people distinguished infinity from enormously large numbers, etc.
Chapter three is further exploring the historical views of infinity. Space and time were traditionally assumed to be infinite, but when looking at the infinitely small, the situation is different. People had difficulty in dealing properly with infinitely small things. Zeno's paradoxes are examples that illustrate that a sum of infinitely many nonzero numbers can be finite. Since the ancient Greek there has been a distinction between an actual infinity and a potential infinity, a discussion that has continued throughout the centuries among philosophers. Even some theologians claimed that God was the only existing impersonation of something infinite. Some proofs for the existence of God were based on this belief. For mathematics, this distinction is not essential. Mathematical existence is abstract and does not coincide with physical or actual existence.
The next chapter is a discussion of the infinitely small and how this has triggered the development of calculus. The original historical concept of infinitesimals is now replaced by the concept of a limit. The infinitesimals where revived when in the 1960's Abraham Robinson developed non-standard analysis.
In geometry, infinity is where the horizon is. It led to the development of perspective in the Renaissance. This is extensively discussed in chapter six, explaining why a ship seems to become smaller as it approaches the horizon, and how this has led to the concept of a point or a line at infinity. The Euclidean plane can be modelled as a disk where infinity is represented by its boundary. More concretely, the line at infinity makes it easy to produce perspective drawings. Eventually this discussion ends in ideas of projective geometry and the mapping of the plane to a sphere and vice versa by stereographic projection, the point at infinity corresponding to the North Pole on the sphere.
Infinity is a useful concept in mathematics, but how does it appear in a physical world? That is what the next chapter is about. In physical sciences, infinity often leads to a nasty singularity. Stewart discusses three examples. The analysis of the rainbow phenomenon is an optical example. If light is incident at a certain angle, then the intensity of the rainbow would be infinite according to ray optics. This singularity entailed that light had to be reconsidered as a wave. In Newton's gravitation theory a singularity occurs when the distance between particles becomes zero and the potential becomes infinite. For example Zhihong Xia proved in 1988 that by solving equations in a five-body problem, dramatically non-physical solutions are obtained after a singularity. Black holes are singularities in general relativity theory and in cosmology the Big Bang is obviously a singularity. Stewart also explains here why cosmologists are wrong when they use curvature as a parameter that determines whether our universe is finite or not.
Te last chapter is the discussion of how Cantor came to his proof that the real number are not countable and how this has led to set theory and his transfinite numbers, and how this resulted in a revision of the foundations of mathematics. This story is best known by mathematicians or anyone who is a bit familiar with this kind of mathematical background literature. But again here Stewart follows the historical evolution of who did what and why in brewing up the eventual result.
This is a lot of information and because of the compact presentation, it will not always be casual reading for a general reader. There are a few references provided per chapter, which might be of interest if the reader wants to look up more details. Some aspects are elaborated more than what is needed for explaining the impact of infinity (e.g. the computation of the angle of the rainbow, the geometry of perspective) but these topics are of course interesting in their on right, and they are usually not found in other treatments of infinity. If you are interested in only the strict mathematical concept of infinity, then Du Sautoy's or Chen's treatises that were mentioned above might be simpler alternatives. But in this booklet, even the experienced reader may have more occasion to learn something new. Some of these non-essential but nevertheless flashes of a that's-interesting-I-didn't-know-that experience will make it worthwhile reading.