*Jordan Ellenberg is an algebraic geometer at the University of Wisconsin and a blogger at Slate. His book *How Not To Be Wrong* was new when he sent The Aperiodical a copy to review ages ago.*

*How Not To Be Wrong *sets out its stall in the introductory chapter, *‘**When Am I Going To Use This?’.* Mathematics doesn’t consist of the repetitive exercises that take up so much of school maths, but is “the extension of common sense by other means”, a skill you can bring to bear to enhance your existing reasoning power.

This assertion is backed up by the story of Abraham Wald, a Hungarian mathematician who, having emigrated to the US following the Nazi invasion of Austria, worked at a classified statistics programme during the Second World War. He was asked to study the pattern of bullet holes on returning aircraft to determine how best to place the limited amount of armour that the planes could handle. His insight was that the intuitive answer – to protect the most bullet-ridden parts of the returning planes – was exactly wrong: those are the parts of the planes that can take damage and still have the plane return. The parts that need protection are the ones that down the plane on their first hit, so the skewed sample of the returning planes show no damage to those areas. Whether you think this is sounds like a genuine mathematical insight or a flash of inspiration that a trained social scientist could equally have arrived at, we see plenty of evidence that Wald’s maths training played a huge role in properly utilising this realisation, including a page of his actual mathematics, typewritten in the terrifying pre-LaTeX days. Wald’s story recurs throughout the book as the mathematical concepts underlying it are fleshed out.

The bulk of *How Not To Be Wrong* is split into five sections: *Linearity*, *Inference*, *Expectation*, *Regression*, *Existence*. The first four-and-a-half of these are collections of clearly and entertainingly described discussions of “maths in real life”-type topics (plus a decent amount of pure maths to keep people like me happy). If you’ve followed Ben Goldacre’s crusade to popularise evidence-based medicine or spent 2011 reading angry blog posts decrying the quality of debate surrounding the AV referendum, you may find yourself covering old ground during the lengthy discussions of *p*-values and electoral paradoxes. But there is plenty otherwise to keep you occupied, including a brilliant explanation of why everyone will (not) be obese by 2048; the maths of the Bible Code; why you should miss more trains; and a hearteningly excellent discussion of topics related to the 1+2+3… furore (the recently-controversial series isn’t directly mentioned but a brief diversion into the Grandi series and a well-chosen G. H. Hardy quote should settle the nerves of anyone still on edge about the whole affair – very probably just me). A few of these vignettes have been published on the author’s Slate blog as a sort of taster for the book, which you can read and thus render reviews such as this one essentially pointless.

The second half of *Existence, *along with the final standalone chapter *How To Be Right*, seems to comprise the author’s thesis on the nature of maths. From the notion of the axiomatic basis of a mathematical system (cleverly explained through examples of counterintuitive geometries) and its similarity or otherwise to systems of law and democracy, Ellenberg discusses the foundations of maths, formalism, Russell’s paradox and the concept of mathematical genius. The book concludes, as books should, on its soapbox, with a rousing defence of the necessary uncertainty that goes with a mathematical approach to life.

For me as a mathematician, the book, especially in the opening and closing parts, displays exactly the philosophy and tone (and quality of diagram) I associate with the subject. Hopefully readers without a maths background will by the end of *How Not To Be Wrong* understand that I mean that as a strong compliment.

How Not To Be Wrong* is on sale now, published by Penguin Press. Amazon are selling it for \$16.77 in the US and £16.00 in the UK.*