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

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.

Carroll’s ire is said to have been directed at Ireland’s greatest mathematician, William Rowan Hamilton , who coined the term “vector” and invented the four-dimensional numbers – quaternions – that contained them. He created something so new that it broke the rule of commutativity, which mathematicians had taken for granted for thousands of years. From his eureka moment in 1843 came the maths you need to manipulate a robot, a realistic computer image, or a spacecraft. But it was not until 1981 that Nasa first used quaternions to routinely guide its rockets. Following in Hamilton’s footsteps was Arthur Cayley , seen as the founder of matrix algebra, the branch of mathematics that drives Google’s search engine.

During these turbulent times for science, sexism was rife. In 1856, the American climate scientist Eunice Newton Foote published her landmark paper on the heating effect of atmospheric carbon dioxide – seven years before John Tyndall made the same claim. But it is Tyndall whose name is associated with discovering the greenhouse gas effect. Emmy Noether, dubbed the “mother of modern algebra”, was key to developing breakthroughs that helped Albert Einstein with the problem of gravitational energy. Yet she was denied a university post for four years, from 1915, because it was considered unthinkable that a woman be allowed to teach.

Scientific advances are often not the efforts of a single genius but more like a palimpsest in which layer upon layer of thought has been inscribed by brilliant heretics. Prof Arianrhod ends her book in the magical world of tensors , explaining how Einstein’s breakthroughs depended on them. The equations of general relativity have unlocked so many mysteries of the universe.

Einstein’s theories were inspired by James Clerk Maxwell. The Scottish physicist came up with the groundbreaking theory of electromagnetism. He died the same year that Einstein was born, which is symbolic, writes Prof Arianrhod, because Einstein did for tensors what Maxwell had done for vectors: he was the first major physicist to show their practical power. It took a quarter of a century to verify Maxwell’s prediction of radio waves, and it took a hundred years to detect Einstein’s gravitational waves. The lesson of history is that what might seem abstruse mathematical inquiry today is likely to determine the shape of tomorrow.