Some people seem to be very fascinated with the concept of dimensions. The ideas of “alternate dimensions” abound in science fiction, not to mention pseudoscience. For instance, I recently came across this gem of a video (and associated book) in which the author claims to explain the 10 dimensions of superstring theory. In reality, the 10 dimensions he explains have nothing to do with string theory, but are just a somewhat arbitrary 10-dimensional subspace of the configuration space of the universe. The fact that humans can’t directly visualize more than three dimensions of space means that the idea of higher dimensions is constantly misunderstood.
In mathematics, we have the concept of a space. A space in its barest form is simply a mathematical set (a collection of some type of elements), together with some kind of system for describing how different elements of the set—called the points of the space—are connected to one another. For instance, a space might possess a metric, which is a function that gives the distance between two points. Distance is one of the most fundamental ways in which points can be related; as a thought experiment, if we have a metric space (i.e. a set of points together with a distance function), we can use the metric to define straight lines and circles—the building blocks of geometry—and from there all the geometric concepts with which we’re familiar.
Dimension, then, is an attribute of a space that can be deduced from the way its points are connected. For instance, we’re all familiar from high school with Euclidean space and the ordinary Cartesian coordinate system, and we know the distinction between two-dimensional and three-dimensional Euclidean space from daily life. In the two-dimensional space, each point can be identified by two numbers, $x$ and $y$. In three-dimensional space, $x$ and $y$ aren’t enough—we need a third, $z$. This is an example of what mathematicians call the Hamel dimension of a space; it’s defined for vector spaces, and tells you how many numbers you need to pin down the location of a single point in the space.
Another way of thinking about it is that it describes how well connected a space is. The greater the dimension, the “more” directions there are and therefore the “more” points in the neighborhood of any given point. Some additional mathematical definitions of dimension that formalize this idea are the Lebesgue covering dimension and the Hausdorff dimension (which has the interesting property that it gives a non-integer dimension for fractal sets).
The Hamel dimension extends naturally to manifolds, which (roughly speaking) are curved versions of Euclidean space. This is the type of dimension referred to in superstring theory, where events takes place within a 10-dimensional manifold. Physically, the model identifies three of those dimensions with the three-dimensional space we experience, a fourth with time, and the remaining six are assumed to be inaccessible to us, either because they are very small, or because our universe is confined to a three-dimensional brane embedded in the 10-dimensional spacetime manifold.
No doubt owing to the fact that we live in a spatial environment that closely approximates three-dimensional Euclidean space, our brains have a large amount of specialized circuitry that enables us to comprehend three dimensions; unfortunately, we are totally incapable of visualizing spaces with four or more dimensions in the same intuitive way. Nevertheless, such spaces are mathematically well-defined and well-understood; there is nothing mystical about them. They are not other “places” to which one can contemplate “going”; it makes no sense (no matter how often they say it on Star Trek) to say that something is “from another dimension” or that one has “entered another dimension”.