The Guardian view on the beauty of numbers: new ways of seeing reality | Editorial

Maths is a language that has helped humanity lay bare the mysteries of the universe

In Lewis Carroll's Alice's Adventures in Wonderland, the Mad Hatter observes that say what you mean" is not the same as mean what you say", for you might just as well say that I see what I eat' is the same thing as I eat what I see'!" Readers might smile in recognition at the author's tongue-twisting, logic-spinning nonsense. But the Victorian literature expert Melanie Bayley has suggested something much more interesting at play. Carroll was a pseudonym for Charles Dodgson, an Oxford maths don who satirised radical new ideas in algebra. In the passage above, he targeted the emerging form of multiplication known as noncommutativity: when a times b" does not equal b times a".

Yet such controversial concepts became widely accepted as the new ideas proved their worth. How these laid the foundation of the modern world is part of the story told in a remarkable new book, Vector, by Robyn Arianrhod, a historian of science. Understanding her text fully requires an undergraduate-level grasp of maths. But her broader theme is easier to recognise: how social and technological change are intertwined with the progress of mathematical thought.