The other day, I was talking to a couple of friends of mine - a literary theorist and a historian/political scientist - about the (apparently) recently proved Poincaré Conjecture. The Poincaré Conjecture involves a statement about the 3-sphere - that is to say, something like the usual sphere, but one dimension bigger. In order to visualise a 3-sphere, you would need to picture it in four dimensions, just as the usual sphere (2-sphere) is usually considered to be a two-dimensional surface, embedded in three-dimensional space. The idea of four dimensions can provoke puzzlement among humanities students of a sort to startle a blasé mathematician for whom such things are second nature. My friend the student of history and politics asked, with a slightly concerned look on his face,
"Can mathematicians actually picture four dimensional space?"
"Roger Penrose says he did it - briefly - once," I said, grinning .
"No, but are there people out there who can actually..."
"Not that I know of."
My historian friend was relieved. My literary theorist friend was confused. "If you can't picture it," he asked, "how could you have any intuition about it? I mean, you could just say whatever you wanted about it and no-one would be able to refute you."
The first way that students of mathematics start dealing with four (or more) dimensions is fairly simple. If you're in two dimensions, you can describe any point with, say, an x co-ordinate and a y co-ordinate.
To go to three dimensions, you need a third co-ordinate in, say, the z-direction (use the previous picture but imagine the third axis coming out of the screen towards you.). To go to four dimensions, add a fourth number, and there you have a way of describing four dimensional (flat/Euclidean) space.
It gets more complicated when you start dealing with curved surfaces, but the idea is similar. If, earlier, you wondered why we consider the surface of a sphere to be two-dimensional, think of it this way. The Earth is approximately a sphere. We can describe positions on the Earth's surface with two co-ordinates: latitude and longitude. Two co-ordinates, two dimensions. Exact definitions might worry about the poles, where there is no real longitude to speak of - it could be anything - but there are fairly simple ways of getting around that which need not concern us here.
Besides the question of four dimensions, there is a broader principle raised by my friend's comment. The extent to which intuition plays any role at all in the basic system of mathematical proof is arguable. This is a fascinating topic in its own right that deserves a post of its own, so for now I shall simply promise to address it later.
 He says it in The Emperor's New Mind.