13 December 2014 | NewScientist | 39

Mathematics is like a language – but one that, thanks to its inbuilt logic, writes itself. That’s how mathematician Ian

Stewart sees it, anyway. “You can start writing things down without knowing exactly what they are, and the language makes suggestions to you.” Master enough of the basics, and you rapidly enter what sports players call “the zone”. “Suddenly it gets much easier,” Stewart says.

“You’re propelled along.”But what if you don’t have such

a maths drive? It’s wrong to think it’s all down to talent, says mathematician and writer Alex Bellos: even the best exponents can take decades to master their craft. “One of the reasons people don’t understand maths is they don’t have enough time,” he says. “It’s not supposed to be easy.”

Sketching a picture of the problem helps. Take negative numbers. Five sheep are easy enough to envisage, but what

about minus five? “We can’t see the minus five sheep, so you can’t get your head around it,” says Bellos. It was only when someone had the bright idea of arranging all the existing numbers 0,1,2,3… on a line that it became obvious where the negative numbers fitted in. Similarly, complex numbers – 2D numbers that underpin the mathematics of quantum theory, among other things – only really took off with the advent of a “complex plane” in which to depict them.

Analogies also help. If thinking about ellipses oppresses you, think about a circle that’s been squashed and work from there, says Stewart. Overall, contrary to the impression of mathematics as a discipline of iron logic, the best way to attack a problem of any sort is often to get a brief overview of it, skip over anything you can’t work out and then go back and fill in the details. “A lot of mathematicians say it’s important to be able to think vaguely,” says Stewart. Catherine de Lange

Frank Close has a question. “If you step off the top of a cliff, how does the Earth down there ‘know’ you are up there for it to attract you?” It’s a question that has taxed many

illustrious minds before him. Newton’s law of gravitation first allowed such apparently instantaneous “action at a distance”, but he himself was not a fan, describing it in a letter as “so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it”.

Today we ascribe such absurdities to fields. “The idea of some physical mediation – a field of influence – is more satisfying,” says Close, a physicist at the University of Oxford. Earth’s gravitational field, for example, extends out into space in all directions, tugging at smaller objects like the moon and us on top of a cliff; the Earth itself is under the spell of the sun’s gravitational field.

But hang on: what exactly is a field?On one level, it is just a map.

“Ultimately, a field is something that depends on position,” says Frank

Wilczek, a theoretical physicist at the Massachusetts Institute of Technology. A gravitational field tells us the strength of gravity at different points in space. Temperatures or isobars on a weather chart are a field. A field is a mathematical abstraction – numbers spread over space.

But there is more to it than that. Witness what physicist Michael Faraday saw in the 19th century, and many a schoolkid has since: iron filings neatly ordering themselves along the lines of a magnetic field, reaching out into space from the magnet itself and influencing nearby objects (though at the speed of light, not instantaneously). “It made a huge impression on Faraday, that this strange thing had a physical reality,” says Wilczek.

Arguably the modern world is built on the principle of electromagnetic induction that Faraday developed out of his new understanding of fields: magnetic fields and electric fields power the motors of our civilisation. A mere abstraction?

The modern era has shed some further light on fields, but also added

confusion. Quantum fields – ultimately, the electromagnetic field is one – have tangible products in the form of particles, which pop up as disturbances within them. For the electromagnetic field, this entity is the photon. The Higgs field, long postulated to pervade empty space and to give elementary particles their mass, was discovered in 2012 by squeezing out its particles in high-energy collisions.

But quantum fields are complicated beasts, formed of “superpositions” of many classical fields. That’s far away from anything we can envisage as a map, or delineate as neat lines. “At that point I have to rely on equations,” says Wilczek, who won a Nobel prize for his work on the quantum fields of the strong nuclear force.

One thing’s for sure: fields are everywhere. Quantum theory teaches us that even seemingly empty space is a roiling broth of fields and their associated particles. “The idea that nothing’s there is extremely naive,” says Wilczek. Aside from anything else, fields are the proof that nature does indeed abhor a vacuum. Richard Webb

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