If you like mathematics because things are either true or false, then you'll be worried to hear that in some quarters this basic concept is hotly disputed. In this article Phil Wilson looks at constructivist mathematics, which holds that some things are neither true, nor false, nor anything in between.
When it comes to describing natural phenomena, mathematics is amazingly — even unreasonably — effective. In this article Mario Livio looks at an example of strings and knots, taking us from the mysteries of physical matter to the most esoteric outpost of pure mathematics, and back again.
What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? In this article Richard Elwes explores how these questions brought triumph to one man and ruin to another, ventures to the limits of mathematics and finds that, with infinity, you're spoilt for choice.
Runner up in the general public category. Great minds spark controversy. This is something you'd expect to hear about a great philosopher or artist, but not about a mathematician. Get ready to bin your stereotypes as Rebecca Morris describes some controversial ideas of the great mathematician David Hilbert.
When Kurt Gödel published his incompleteness theorem in 1931, the mathematical community was stunned: using maths he had proved that there are limits to what maths can prove. This put an end to the hope that all of maths could one day be unified in one elegant theory and had very real implications for computer science. John W Dawson describes Gödel's brilliant work and troubled
Kurt Gödel, who would have celebrated his 100th birthday next year, showed in 1931 that the power of maths to explain the world is limited: his famous incompleteness theorem proves mathematically that maths cannot prove everything. Gregory Chaitin explains why he thinks that Gödel's incompleteness theorem is only the tip of the iceberg, and why mathematics is far too complex ever to be
described by a single theory.
Suppose you walk past a barber's shop one day, and see a sign that says
"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself?