*(2013) Alfred S. Postmentier & Ingmar Lehmann, Prometheus Books, £15.99 / Can$25 / US$24, hrdbk, 296pp, ISBN 978-1-616-14747-1*

This is actually quite an important book even though many shy away from mathematics. However if you have got, or are young enough to currently being taught, maths A-level (a qualification taken in Britain around the age of 18 or 19 which I think is called a high school qualification in the USA) then do bear with me as this is one title you really should get.

Now if you are happening upon this review and are wondering what on Earth it is doing on a science fact and science fiction website, let me point out that many SF fans are science geeks and that maths is a STEM subject. (STEM stands for science, technology, engineering, maths and medicine for those operating in the world of education or who are policy wonks.) Indeed maths is fundamental to all of science and engineering even if a lot of us in science and engineering shy away from much of maths (physicists and computer scientists [as opposed to computer technologists] being the notable exceptions). This is why nearly all large research institutes have a maths or statistics unit whose good folk check our methodologies and sums. So this is why maths, if not of interest, is important to us.

* Magnificent Mistakes in Mathematics* is a grand tour of the mathematical howlers that have taken place throughout history as well as misconceptions. It covers the almost mundane, such as the nineteenth century attempt to calculate pi to a record number of decimal places. This staggering attempt took 15 years to complete. It would have been a bit of a bummer for the mathematician William Shanks to learn in 1946 that there was an error at the 528th decimal place but for Shanks having already died in 1882. Them's the breaks.

It also contains wonderful 'proof's' such as 1 = 0 at a level of maths that even I (with just O-level maths) can follow. And then there are the illusions. These are not just geometric illusions of the eye – though these are included too – but logical illusions. One of the most famous (but still surprisingly little known) is Monty Hall's illusion which I will give you as a taster. Consider someone showing you three boxes who says that in one of them is a prize which you will get if you pick the correct box. You choose one, but (without opening that box) the person decides to give you a second chance making the choice easier. The person opens one of the boxes that does not have the prize and then asks whether you want to stick with your original choice or to change it? What should you do if anything that will give you the best chance of winning?

Now the answer, believe it or not, is not to stick with your original choice. As to the answer why this is you will have to get the book. (Or Google Monty Hall paradox.)

The reason why this book is so important is, as said, because maths underpins all science. But this is me being glib. The thing is that most of us in science tend to specialise early, and most science career paths tend to lead us down further specialisation. This means that society ends up with a cadre of senior but largely highly specialised scientists, and the danger here is that their lack of breadth of knowledge means that their ability to function outside their specialist area is impeded. And so, for example to go beyond the book, we get a few non-climate scientists who are anthropogenic climate change deniers, and genetically modified organism (GMO) technology haters (when actually some GMO is beneficial with negligible ecological risk) or conversely unwavering GMO supporters (whereas some GMO technology is ecologically harmful and can result in unwanted side effects such as promoting pesticide resistance). To get by, you need to know that, while science may be logical, we are human and that human perceptions – even our perceptions of logic – can be illusionary. (This is a thesis visited before and notably by the biologist Louis Wolpert *FIBiol* in *The Unnatural Nature of Science* (1992).) What the *Magnificent Mistakes in Mathematics* does is give you the tools to see through such logical illusions. And if you can do it with maths, it is only a short hop, skip and a jump beyond the book for yourself to see how it can be done with words. And if in turn you have cracked that the you then can begin to see through a lot of the balderdash and bumpkin that bombards us in the media, by politicians, and, yes, even by our peers and – dare I say it – ourselves when we get into slovenly thinking. (We are all human.)

OK, this is maths and some of it is hard. But to be honest you can skip over some of the really hard stuff and there is still plenty there that the average person with good mid-school level (O-level) maths can cope. Indeed it is worth having a crack at following some of the challenging sections as brain teasers. Here you will find the achievement very rewarding indeed. OK, so this book will sort out the men from the boys and the ladies from the girls, but challenges are a necessary part of life and this book certainly gives far, far more than it takes in effort. Do give it a go. And if you are studying science or are a scientist, consider it essential reading.

Jonathan Cowie

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