(Image: Richard Drury/Stone/Getty) By Ian Stewart Read more: “The nature of nothingness” THE mathematicians’ version of nothing is the empty set. This is a collection that doesn’t actually contain anything, such as my own collection of vintage Rolls-Royces. The empty set may seem a bit feeble, but appearances deceive; it provides a vital building block for the whole of mathematics. “The empty set is the set with nothing in it. The number 1 is the set with only the empty set in it. And so on” It all started in the late 1800s. While most mathematicians were busy adding a nice piece of furniture, a new room, even an entire storey to the growing mathematical edifice, a group of worrywarts started to fret about the cellar. Innovations like non-Euclidean geometry and Fourier analysis were all very well – but were the underpinnings sound? To prove they were, a basic idea needed sorting out that no one really understood. Numbers. Sure, everyone knew how to do sums. Using numbers wasn’t the problem. The big question was what they were. You can show someone two sheep, two coins, two albatrosses, two galaxies. But can you show them two? The symbol “2”? That’s a notation, not the number itself. Many cultures use a different symbol. The word “two”? No, for the same reason: in other languages it might be deux or zwei or futatsu. For thousands of years humans had been using numbers to great effect; suddenly a few deep thinkers realised no one had a clue what they were. An answer emerged from two different lines of thought: mathematical logic,