In the 1930s the logician Kurt Gödel showed that if you set out proper rules for mathematics, you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite contrived. But are they about to enter mainstream mathematics?

It requires only a little processing power, but it's a giant leap for robotkind: engineers at the University of Southampton have developed a way of equipping spacecraft and satellites with human-like reasoning capabilities, which will enable them to make important decisions for themselves.

Quantum mechanics is usually associated with weird and counterintuitive phenomena we can't observe in real life. But it turns out that quantum processes can occur in living organisms, too, and with very concrete consequences. Some species of birds, for example, use quantum mechanics to navigate. Last year we talked to physicists Simon Benjamin and Erik Gauger, and found out that studying these little creatures' quantum compass may help us achieve the holy grail of computer science: building a quantum computer.

It's been nearly 18 months since the Large Hadron Collider at CERN started up and scientists are eagerly awaiting their first glimpse into the cosmic mysteries it was designed to explore. But when can we realistically expect the first ground-breaking discoveries to come through? Last week, John Ellis, outgoing leader of the theory division at CERN, addressed an audience of physicists at the University of Cambridge to update them on the current state of play. Plus went along and also managed to catch Ellis for a quick interview.

Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. But it's not quite like that. You can actually build different versions of maths in which statements are true or false depending on your preference. So is maths just a game in which we choose the rules to suit our purpose? Or is there a "correct" set of rules to use? We find out with the mathematician Hugh Woodin.

The world we live in is strictly 3-dimensional: up/down, left/right, and forwards/backwards, these are the only ways to move. For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like? Or might it even be true that we already inhabit such a space, that our 3-dimensional home is no more than a slice through a higher dimensional realm, just as a slice through a 3-dimensional cube produces a 2-dimensional square?

That geometry should be relevant to physics is no surprise — after all, space is the arena in which physics happens. What is surprising, though, is the extent to which the geometry of space actually determines physics and just how exotic the geometric structure of our Universe appears to be. Plus met up with mathematician Shing-Tung Yau to find out more.

This book tells the fascinating story of strange geometric objects that have achieved some fame outside of maths and even inspired a Woody Allen joke: they're called Calabi-Yau manifolds. When Plus recently interviewed one of its authors he was adamant that maths should be brought to the masses without dumbing down or glossing over the tricky parts. And this is just what this books sets out to achieve.

Quantum mechanics is usually associated with weird and counterintuitve phenomena we can't observe in real life. But it turns out that quantum processes can occur in living organisms, too, and with very concrete consequences. Some species of birds use quantum mechanics to navigate. And as Plus found out at a recent conference, studying these little creatures' quantum compass may help us achieve the holy grail of computer science: building a quantum computer.

Quantum physics is a funny thing. With counterintuitive ideas such as superposition and entanglement, it doesn't seem to resemble reality as we know it, yet quantum physics is an incredibly successful theory of how the physical world operates. Plus attended the conference Quantum Physics and the Nature of Realtiy at the University of Oxford in September 2010. We spoke to Andrew Briggs, John Polkinghorne, Nicolas Gisin, David Wallace, Roger Penrose and Andrea Morello about how we can resolve the mysteries of quantum physics with our experience of reality. And we find out why quantum physics is just like riding a bike...

Benoît Mandelbrot, the father of fractal geometry, died last Thursday at the age of 85. Born in Poland in 1924, Mandelbrot had dual French and American citizenship and spent most of his working life in the US. He died of cancer in a hospice in Cambridge, Massachusetts.

Well, it goes to no-one because there isn't a Nobel Prize for maths. Some have speculated that Alfred Nobel neglected maths because his wife ran off with a mathematician, but the rumour seems to be unfounded. But whatever the reason for its non-appearance in the Nobel list, it's maths that makes the science-based Nobel subjects possible and it usually plays a fundamental role in the some of the laureates' work. Here we'll have a look at two of the prizes awarded this year, in physics and economics.