Mathematics. It's a word that either transports you back to a dark classroom with impossible-looking long division on the blackboard, or it fills you with the unending, enlightening joy of the mysterious worlds that the logical human mind can conjure up.
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If you're in the former camp, it might be difficult to understand how anyone could be interested or impassioned by something so seemingly dry and difficult. Numbers, shapes, equations – how can any of that create beauty or pleasure? Strings of data are dull, you might think, percentages and sums best left to calculators (or, these days, Google).
Anyone in the other group, and Alex Bellos is one of these people, would disagree. For these folk, mathematics is a proud human endeavour more profound than science and more creative than art. For people like Bellos, mathematics is beautiful.
Bellos starts his tour of the mathematical world with some anthropology, asking whether numbers are something natural to humans, or whether they are learned and constructed. Our counting numbers (1, 2, 3, etc) are probably less than 10,000 years old, an offshoot of language, and there were probably no more than a handful of these discrete units for most of that time.
Studies of Mundukuru tribespeople, who live in a remote part of the Amazon, show that they can count to four or five and then their next quantity is "many". The system works for them because it allows them to quickly assess quantities. They have no need to count lots of things and, indeed, see counting endlessly as a ludicrous activity.
Counting proper (and associated manipulations of numbers and quantities) took off when civilisations started to grow and rulers needed to know how much land their citizens had and, therefore, how much tax they should pay. The Babylonians, Sumerians, Greeks, Chinese, Indians and Persians all had slightly different ways of quantifying objects and areas and slightly different ways of writing down their numbers.
The modern, base-10 number system came from elements of all these ancient cultures but it was the Greek and Arabic mathematicians who did most to begin what we would nowadays call mathematics: taking patterns and numbers measured in the physical world and abstracting them into universal proofs. In this sense, maths is a more ancient and fixed base for knowledge than science, which is continually improved and changed in light of new evidence. The maths of Pythagoras is the maths we use today, whereas the scientific thinking of Aristotle has largely been consigned to history.
Mathematicians have explored ever more abstract worlds and geometries, floating in dimensions that may or may not exist and finding symmetries and patterns in hard-to-imagine shapes. They have studied the properties and patterns in numbers, straight lines, curves, surfaces, cubes and hypercubes, all in a bid to understand how these things fit together and what those details might reveal about the deeper logic of mathematics.
Some of the expeditions have led to scientific insights about the world around us (the universe might turn out to have a hyperbolic shape first imagined by mathematicians hundreds of years ago) and the physical laws that govern it. But most were just attempts to tackle puzzles and problems that the mathematically minded felt compelled to solve.
From the thoughts of generations of mathematicians, our eyes have been opened to special numbers such as pi – the ratio of the circumference of a circle to its diameter that pops up everywhere in scientific theories – and the golden ratio, a number that exists in nature in the way leaves are arranged around stems and the shape of a nautilus shell, and is now used by designers of beautiful objects (such as iPods).
We found logarithms (which were the only way to do complex engineering sums before calculators came along) and realised that the collective behaviour of people or molecules was predictable, even if it seemed random. Statistics therefore became important for states, for economists, and to discover and understand climate change. Gamblers wanted to know how to beat the house and, by examining the mathematical patterns and probabilities in a game, were rewarded with intricate ways of gaining a tiny edge.
There have been books about the history of mathematics before and, I hope, there will be many more in the future. There are scores of textbooks about the history of mathematics too, many of which could tell you roughly the same things that Bellos's book covers. But you would be hard-pressed to find a book on this subject with the same humour, wonder, and with the comfort of knowing that the author is resolutely on your side on this (sometimes difficult) adventure through the land of numbers and shapes.
Alex's Adventures in Numberland is worth celebrating because Bellos does not accept the tiresome cliche that maths is boring and therefore needs to be made interesting to uninterested people. Instead, he effortlessly reveals the truth of just how fascinating, how human, how intensely interesting this subject (and its history) really is.
Royal Society Winton Prize for Science Books: The shortlist
Alex's Adventures in Numberland by Alex Bellos
Through the Language Glass: How Words Colour Your World by Guy Deutscher
The Disappearing Spoon by Sam Kean
The Wavewatcher's Companion by Gavin Pretor-Pinney
Massive: The Missing Particle That Sparked the Greatest Hunt in Science by Ian Sample
The Rough Guide to the Future by Jon Turney