New Scientist The Collection

LOVE it or loathe it, there is no escape: mathematics is out there. We live in a world that is in some sense mathematical – although in precisely what sense is still hotly debated by mathematicians, physicists, philosophers and others. We are, as a result, innately mathematical beings. Crossing the road, catching a ball, stacking the dishwasher: in our everyday lives we are constantly, unconsciously, manipulating numbers, assessing shapes, calculating position, geometry and motion. We are doing maths. But most of us see the conscious pursuit of the subject as something best left to the experts. The abstractions, conjectures and proofs of formal mathematics belong to a higher, rarefied plane few of us can access. This latest issue of New Scientist: The Collection aims to bridge the divide between the…

TO THE Iranian mathematician Maryam Mirzakhani, the first woman to win the Fields medal, mathematics often felt like “being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks”. “With some luck,” she added, “you might find a way out.” Mirzakhani, who died in July 2017 at the age of 40, ventured deeper into the mathematical jungle than most. Nonetheless, most of us have spent enough time on its periphery to have a sense of what the terrain looks like. Increasingly, it seems as if humans are the only animals with the cognitive ability to hack their way through the undergrowth. But where does this ability come from? Why did we develop it? And what is it for?…

For most of us, maths means numbers, and that’s not wrong. The ability to understand and manipulate numbers in the abstract (think addition, subtraction, multiplication and division) is the foundation on which a formidable edifice has been built (see main story). Broadly speaking, this edifice consists of three pillars: geometry, analysis and algebra. Geometry is probably the most familiar to us. It begins with a sense of space, codified into principles that describe how static things in space relate to each other, like a triangle’s sides. When you have to consider things that move and change with time, you come to analysis, a field that includes calculus, whether it’s integral or differential calculus, or its many variations. Algebra is what allows us to process knowledge in terms of numbers, symbols…

The debate over whether our sense of exact numbers is innate has often turned to animals for support. If our distant cousins can be shown to share certain mathematical abilities, then that implies our own must predate the development of culture. Certainly, some individual animals have been shown to display remarkable talent. Alex, an African grey parrot trained by Irene Pepperberg, could correctly identify sets of between two and six objects 80 per cent of the time. Ai, a chimpanzee trained by Japanese primatologist Tetsuro Matsuzawa, could do much the same. But too much emphasis is placed on research involving animals, says Rafael Núñez at the University of California, San Diego, at the expense of data from human cultures that have sophisticated languages and yet don’t show exact numerosity (see…

“It is familiar to anyone writing about (or teaching) mathematics: no one very much likes the subject,” writes mathematician David Berlinski in his book One, Two, Three. This distaste, even fear, of mathematics is common – most of us know the feeling. Berlinski says this can be attributed to its use of arcane symbols. Symbols are strange, plus using them in the forms of theorems and proofs demands great attention, and the pay-off is never obvious (have you ever asked “how is learning algebra going to help me in real life?”). “In mathematics, something must be invested before anything is gained, and what is gained is never quite so palpable as what has been invested,” writes Berlinski. Maths anxiety, a tendency to panic when asked to perform mathematical tasks, is…

” Gödel revealed the awkward possibility that arithmetic sometimes could not supply any answers at all” IF YOU were forced to learn long division at school, you might have had cause to curse whoever invented arithmetic. A wearisome whirl of divisors and dividends, of bringing the next digit down and multiplying by the number you first thought of, it almost always went wrong somewhere. And all the while you were plagued by that subversive thought – provided you were at school when such things existed – that any sensible person would just use a calculator. Well, here’s an even more subversive thought: are the rules of arithmetic, the basic logical premises underlying things like long division, unsound? Implausible, you might think. After all, human error aside, our number system delivers…