This is an absolutely fascinating book for anyone interested in the way that science really works, bearing in mind the difficulties of having to base our models and theories on induction. Andrew Jaffe introduces the difficulties we face when trying to take a scientific view because largely we are dependent on induction: predicting the future from what has previously been observed. He explores what probability is, the two key ways of looking at it (frequentist and Bayesian) and how scientists use (or misuse it) to work out the implications of their experiments for hypotheses. This is then expanded into looking at the nature of scientific models and the philosophy of science before heading out to entropy, quantum randomness and attempting to achieve meaningful cosmology with its potential dearth of evidence.  The topic might sound a little dry, but in fact Jaffe does it with good humour and a very readable style. For example, he uses measuring his daughter's height by making marks on...

Ever since computers became useful tools, mathematicians have had mixed opinions about using the technology to solve mathematical problems. Obviously this is particularly topical when we look at what AI can (and can't) do - but there are plenty of opportunities for the brute force of computing to deal with a tricky mathematical road block, and that's what Paul Nahin sets out to cover here. I'll say straight up front that I think this book would have been better and would have had a wider audience if it had not made excessive assumptions of what a reader knows. In his introductory chapter, Nahin says 'There is nothing in this book that attentive high school students who have taken an AP-calculus or AP-statistics class will find beyond them. This assumption lets me, for example write (as I do in the first chapter) the symbol X without explanation with the expectation that a reader will instantly recognise it as denoting the binomial coefficient...' (I have written X h...

Ian Stewart is arguably the UK's best raconteur of mathematics - here he takes on some of the extremes of the mathematical world, and in doing so gives us some real insights into what makes mathematicians tick. There's a good mix here of the flashy fun aspects of maths - think, for instance of the wonders of infinity or the monster group - and the solid everyday that nonetheless can turn up surprises. The book is littered with little insights. For example, if we think it's easy to work out the area of a rectangle by dividing it up into unit squares, what do you do with one that measures square root of two by pi? You'll find yourself jumping around from what lies beneath calculus to game theory (rock, paper, scissors anyone? - I hadn't realised a version of this game dates back around 2,000 years). One minute you'll be considering colouring maps and the next finding the shortest distance between two points on a curved surface. Some of the mathematics here has eve...

Subtitled 'overlooked and omitted topics in mathematics', the obvious concern is that there is a good reason these topics are overlooked and omitted. Thankfully, this is not the case, but it's fair to say that despite attempts to dress it up that way, this isn't a recreational maths book. There's a fair description in the blurb: 'the topics which every undergraduate mathematics student "should" know, but has probably never encountered... magnificent secrets that are beautiful, useful and accessible.' As someone who many years ago did a degree with a fair amount of mathematics in it, I think it probably would have appealed back then - though to be honest a lot of it has disappeared from my memory, strongly reducing the entertainment value. Here's an example. The first real page contains the sentence:  'If you are handed a real number 𝓍 ∈  ⁠ ⁠,  one way to tell if 𝓍 is rational or irrational is to look at sequences of rational numbers q n ...

If we are to believe the media we are bombarded with misinformation and disinformation - there's certainly a lot of it out there and Alex Edmans sets out to give a guide to the many ways that information can be badly or misleadingly presented, and how we can defend ourselves from it. At the heart of his argument are two biases. I'm so glad he limits it to two - I get totally lost trying to keep on top of all the biases that psychologists introduce, so sticking to confirmation bias plus black and white thinking as the key errors to look out for, both in how we receive information ourselves and how others present it, is very helpful. At the heart of the book is a ladder of levels of something like quality of information. These are statement, fact, data, evidence and proof. Edmans goes into plenty of detail on each rung - how we get, for example, from statement to fact, or data to evidence. Most of all, he demonstrates brilliantly how both those undertaking studies and those inter...

Networks are of huge significance to life and technology, so it was refreshing to read a popular maths title on the subject. I was a little concerned when, in the introductory chapter, Anthony Bonato spent quite a while discussing network/graph theory jargon - we really don't need to know that a network is referred to as G with nodes represented by V and edges by E, let alone the meaning of 'heavy tailed'. There is absolutely no need to have such technical-speak in a popular title.  Unfortunately, he goes bombarding us with terminology in further chapters: it can be painful, especially when we are told a directed graph is known as a digraph, when I suspect most people outside of the graph theory community would consider a digraph a two letter phoneme. Despite the relentless terminology we do learn a lot about networks and their applications, from Google's PageRank to Bacon numbers and from COVID infections to optimising security camera placement. As a writer, I was inte...

One of the biggest developments in the history of maths was moving from describing relationships and functions with words to using symbols. This interesting little book traces the origins of a whole range of symbols from those familiar to all, to the more obscure squiggles used in logic and elsewhere. On the whole Raúl Rojas does a good job of filling in some historical detail, if in what is generally a fairly dry fashion. We get to trace what was often a bumpy path as different symbols were employed (particularly, for example, for division and multiplication, where several still remain in use), but usually, gradually, standards were adopted. This feels better as a reference, to dip into if you want to find out about a specific symbol, rather than an interesting end to end read. Rojas tells us the sections are designed to be read in any order, which means that there is some overlap of text - it feels more like a collection of short essays or blog posts that he couldn't be bothered ...

A great book for the right audience. When I was a teenager I loved Martin Gardner's Mathematical Puzzles and Diversions books. They combined mathematical oddities, such as there being more than one size of infinity, with numerical tricks and some physical maths-based objects, most notably flexagons. As someone who is physically inept, I never got much beyond a basic hexaflexagon, but these paper-based three dimensional structures that could change shape when manipulated, a bit like more sophisticated versions of paper fortune tellers, were fascinating. Now we have a chunky, large format title exploring flexagons in far more depth. The first part of the book takes us into different flexagon structures and the range of possibilities enabling flexing. Some of these require quite sophisticated manipulation of the paper structure (which I did struggle with somewhat, being clumsy). Each flexagon type comes with a flat template to reproduce. These are sometimes quite small and for me ben...

A classic curate's egg of a book. Some aspects of it are brilliant, but there is enough that isn't to make it frustrating. Wisely, Oliver Johnson decided to do a book on very practical aspects of maths - applications that are a wonderful counter to the old moan at school of 'but what use is it to me?' This is great, but two aspects are less positive. One is that this would be a sensible argument if we taught school students this stuff. Just as I think we should teach interesting physics, this is genuinely interesting maths that doesn't necessarily involve more work to learn the basics. But we don't. The second issue is that Johnson decided to do maths without formulae and equations. This is a common enough practice in popular science, where you can often get away without the mathematics, but in popular maths it is a real stumbling block. When, for example, Johnson is telling us about Bayesian methods - really useful stuff - rather than presenting us a with a ver...

This seemed to be a book that had a lot going for it. The topic of 'the science of certainty' appealed to a reader like me who is fascinated by probability and statistics, and I enjoyed the way the introduction made use of the uncertainty of the impact of the Eyjafjallakökull volcano on flight safety, then the delight that is the Monty Hall problem. But although the rest of the book had some highlights, I couldn't get on with much of it. In a way, the title is highly misleading, because the book isn't really about 'proof' - after all, very little science involves proof. Certainly most of the studies we see misreported in the press don't. We can only prove something with perfect knowledge. This is fine when applying basic logic. We can make deductions, for example, if we are able to make a statement like 'no square is circular'. But such statements are rarely applicable in the real world. Instead we have to rely on induction or abduction, which is usu...