Beyond Infinity: An Expedition to the Outer Limits of Mathematics

Eugenia Cheng is a professor of mathematics whose research field is higher dimensional category theory. She has made it one of her missions to counter mathphobia. Her credo is that mathematics is not the difficult part to deal with in life, but that on the contrary it is life that is difficult and mathematics helps us to make it simpler and manageable. She has tried to illustrate this by combining her love for cooking and her passion for mathematics in her previous book Cakes, Custard + Category Theory (reviewed here). In that book she gave attention to mathematics alright, but there were also proper recipes for cooking. The latter are interesting if you love cooking yourself and they are a springboard to make a link towards mathematics, but they do not really help to understand category theory.

In the present book however she is explaining a really important mathematical concept: infinity, and it is far from being the simplest one to explain for non-mathematicians. The approach here is that she does just that. Not like in her previous book where she placed cookery next to the mathematics. Here of course Chen is still Chen and she still can't hide her love for cooking and category theory. However cooking is now only used as an anecdote or as and introduction to a chapter, just like perhaps a hiking experience, of a concert she attended, can be.

So what is this book about? The first part is intended to explain what infinity really is, and it soon becomes clear that it is not as simple as saying it is larger than any number one can imagine. It cannot be a number since the usual arithmetic rules do not work as with finite numbers. And then there are the paradoxes like the well known Hilbert hotel with infinitely many rooms that can always accommodate infinitely many more guests, even when it is fully booked. So Chen uses a more systematic approach introducing the simplest number systems: natural numbers, integers, and rationals. She goes even further and defines the natural numbers in the set theoretic style of Frege, only she does not use the abstract concept of a set, but she uses 'bags' instead. So 0 corresponds to the empty bag, 1 to the bag containing only the empty bag, 2 to the bag containing the two previous bags, etc. Also concepts like injection, surjection, and countable are introduced here.

Then a stumble stone is blocking the development. It turns out that there are more than countably many real numbers. The reals are not properly defined yet, but using Cantor's diagonal argument, and using a binary representation, Chen shows that there are more irrational numbers than natural numbers. Thus there are gradations of infinity, at which point $\aleph_0$ is introduced. The 'smallest' infinity of a countable set, but there exist higher forms like $\aleph_1=2^{\aleph_0}$ the number of reals, and this can be iterated $\aleph_k=2^{\aleph_{k-1}}$. The continuum hypothesis is briefly touched upon, and it is noted that it can't be proved (Cohen) or disproved (Gödel). The distinction between ordinal and cardinal numbers clarifies the difficulty that infinity gives with the usual arithmetic operations.

All this work in the first part of the book, leading to an understanding of what infinity actually is, is like a journey uphill. In a second part Chen points to the sights that are possible from the top of the hill. With the recursive definition of the natural numbers, a proof by induction is within reach and one can solve all sorts of counting problems and even evaluate infinite sums. Although the latter needs more careful consideration. She also introduces higher dimensions, i.e., larger than 2 or 3. It may even grow to infinity for a continuum. When a relation or a property is associated with a dimension, this brings her to her beloved research subject: higher dimensional category theory. Perhaps, this doesn't fit so well in the otherwise rather elementary exposition, but it is a nice, be it a somewhat unusual example, of a higher dimensional mathematical object.

The move is then from the infinitely large to the infinitely small, leading back to infinite sums of diminishing terms and Zeno paradoxes. What is needed here is the concept of a limit. She however explains it essentially avoiding to use that name. Instead she illustrates the idea with hitting a target that becomes smaller and smaller. This way it can be explained what infinitesimals are and how they are applied. It can now be proved that the harmonic series diverges, and eventually also that irrational numbers do exist, which is done by approaching Dedekind's definition of the reals.

I find this a very pleasing way of introducing some elementary, but also some less elementary, mathematical concepts to the layperson. Taking infinity as the carrot to lead the reader uphill is an interesting choice. This is the most essential concept needed when moving from algebra to analysis. Chen is an excellent guide to show the reader the way uphill. With many analogies and illustrations and reformulations it seems like the reader is carried to the top, no toiling required. The story is told fluently. Side remarks, historical notes, or a slightly more advanced remark are inserted as a framed boxes in the text. I guess it will be too elementary for mathematicians of mathematics students, but it is warmly recommended for secondary school pupils. In fact anyone who has the slightest interest in what infinity actually means should read it. The word is used lightly in common language, but you will learn what it means in a more exact sense and thus what it means to a mathematician. It turns out that it triggered the development of calculus and it has shaken the foundations of mathematics as recently as in the early 20th century.

Adhemar Bultheel
Book details

Eugenia Cheng continues her crusade against mathphobia. In this book she explores the meaning of infinity. To properly define infinity she has to define the cardinality of the natural numbers, and thus also the definition of the latter. That includes solving the inconsistencies with arithmetic operations and the paradoxes that result. However it turns out that the real numbers are not countable, so that there are gradations of infinity and hence also the definition of the reals is needed. That requires to consider the infinitely small, which leads to infinitesimals that form the onset of calculus. All is brought to the reader avoiding the usually boring technical approach of mathematics, but using many analogies and elementary everyday language.

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978-1781252857 (pbk)
GBP 12.99 (pbk)