A Very Short Introduction

*(2017) Ian Stewart, Oxford University Press, £7.99 / US$11.95, trdpbk, xvi + 143pp, ISBN 978-0-198-7553-4*

The border between science and science fiction is a long and rich one with each influencing the other: science underpins much SF while SF enthuses many into studying, and indeed working in, science. SF is also known for its big concepts that impinge upon reality and even the very fabric of space time. One idea that straddles this border is the concept of infinity and almost by definition it is a proverbial 'big' topic that engenders much ' 'sense-of-wonder' (sensawunda). Who better to give a potted guide to this mind-boggling topic than a mathematician *cum* SF author and non-fiction SF as well as popular science writer Ian Stewart. (Not to mention SF fan and Fellow of the Royal Society.)

Infinity is a real mathematical concept but is not a real number. It is real in the sense that a simple equation will generate infinity the reciprocal of zero (1 divided by 0) is infinity. But it is not a real number in that you cannot do easy sums with it or define it as an actual number. Sums such as infinity plus 1 are meaningless because infinity by definition is the greatest/highest number of all and so you cannot add anything to it to make a greater number.

Ian explains all this in a concise way in this remarkable short book (booklet really). Along the way we come across some fascinating phenomena arising out of the concept of infinity. For example, the geometric construct Gabriel's horn has an infinite surface area yet counter-intuitively holds a finite volume!

My one (minor) grumble is that this is such an interesting topic I would have liked more. Alas my desire for more does not chime with Oxford U. Press' (OUP) 'A Very Short Introduction' series and nor is it fair to have a go at what is left out of these remarkable books. Having said that I was less interested in historical views of infinity and within that chapter I had zero interest in religious views of infinity: what I would have liked instead would have been to have had a few pages on the real-life mathematical applications of infinity-related mathematics to which Ian Stewart alludes but does not have the space to give examples.

The value of the OUP 'A Very Short Introduction' series to my mind is two-fold. There are some topics which one has some idle, but genuine, curiosity about and in which one would like to dip, but not to spend a lot of time on. There are also topics with which one feels one should have a vague working knowledge. SF fans, especially (non-mathematical) scientists – if that is not oxymoronic as scientists use maths as maths is one of the (theoretical) constructs that underpins all science – are likely to have a fascination with the concept of infinity. Somewhat analogous to Gabriel's horn, this finite book provides an insight into the infinite.

Jonathan Cowie

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