What are ‘Dimensions’

January 12, 2018

Relativity says we live in four dimensions. String theory says it’s 10. What are ‘dimensions’ and how do they affect reality?

Writing away at my desk, I reach my hand up to turn on a lamp, and down to open a drawer to take out a pen. Extending my arm forward, I brush my fingers against a small, strange figurine given to me by my sister as a good-luck charm, while reaching behind I can pat the black cat snuggling into my back. Right leads to the research notes for my article, left to my pile of ‘must-do’ items (bills and correspondence). Up, down, forward, back, right, left: I pilot myself in a personal cosmos of three-dimensional space, the axes of this world invisibly pressed upon me by the rectilinear structure of my office, defined, like most Western architecture, by three conjoining right angles.

Our architecture, our education and our dictionaries tell us that space is three-dimensional. The OED defines it as ‘a continuous area or expanse which is free, available or unoccupied … The dimensions of height, depth and width, within which all things exist and move.’ In the 18th century, Immanuel Kant argued that three-dimensional Euclidean space is an a priori necessity and, saturated as we are now in computer-generated imagery and video games, we are constantly subjected to representations of a seemingly axiomatic Cartesian grid. From the perspective of the 21st century, this seems almost self-evident.

Yet the notion that we inhabit a space with any mathematical structure is a radical innovation of Western culture, necessitating an overthrow of long-held beliefs about the nature of reality. Although the birth of modern science is often discussed as a transition to a mechanistic account of nature, arguably more important – and certainly more enduring – is the transformation it entrained in our conception of space as a geometrical construct.

Over the past century, the quest to describe the geometry of space has become a major project in theoretical physics, with experts from Albert Einstein onwards attempting to explain all the fundamental forces of nature as byproducts of the shape of space itself. While on the local level we are trained to think of space as having three dimensions, general relativity paints a picture of a four-dimensional universe, and string theory says it has 10 dimensions – or 11 if you take an extended version known as M-Theory. There are variations of the theory in 26 dimensions, and recently pure mathematicians have been electrified by a version describing spaces of 24 dimensions. But what are these ‘dimensions’? And what does it mean to talk about a 10-dimensional space of being?

In order to come to the modern mathematical mode of thinking about space, one first has to conceive of it as some kind of arena that matter might occupy. At the very least, ‘space’ has to be thought of as something extended. Obvious though this might seem to us, such an idea was anathema to Aristotle, whose concepts about the physical world dominated Western thinking in late antiquity and the Middle Ages.

Strictly speaking, Aristotelian physics didn’t include a theory of space, only a concept of place. Think of a cup sitting on a table. For Aristotle, the cup is surrounded by air, itself a substance. In his world picture, there is no such thing as empty space, there are only boundaries between one kind of substance, the cup, and another, the air. Or the table. For Aristotle, ‘space’ (if you want to call it that), was merely the infinitesimally thin boundary between the cup and what surrounds it. Without extension, space wasn’t something anything else could be in.

Centuries before Aristotle, Leucippus and Democritus had posited a theory of reality that invoked an inherently spatialised way of seeing – an ‘atomistic’ vision, whereby the material world is composed of minuscule particles (or atoms) moving through a void. But Aristotle rejected atomism, claiming that the very concept of a void was logically incoherent.

By definition, he said, ‘nothing’ cannot be. Overcoming Aristotle’s objection to the void, and thus to the concept of extended space, would be a project of centuries. Not until Galileo and Descartes made extended space one of the cornerstones of modern physics in the early 17th century does this innovative vision come into its own. For both thinkers, as the American philosopher Edwin Burtt put it in 1924, ‘physical space was assumed to be identical with the realm of geometry’ – that is, the three-dimensional Euclidean geometry we are now taught in school.

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