Any point in a mathematical space can be defined by a set of coordinates, and the number of coordinates required to define that point is the number of dimensions the space possesses. In a 3-dimensional space like you’re used to, for example, every point can be uniquely defined by precisely three coordinates — three pieces of information (length, width, and height). Each dimension represents a degree of freedom within the space.
Though physicists talk about dimensions in terms of space (and time), the concept of dimensions extends far beyond that. For example, the matchmaking Web site eHarmony.com provides a personality profile that claims to assess you on 29 dimensions of personality. In other words, it uses 29 pieces of information as parameters for its dating matches.
Say you wanted to find a potential romantic partner. You’re trying to target a specific type of person by entering different pieces of information: gender, age range, location, annual income, education level, number of kids, and so on. Each of these pieces of information narrows down the “space” that you’re searching on the dating site.
If you have a complete space consisting of every single person who has a profile on the dating site, when your search is over you’re narrowed to searching only among those who are within the ranges that you’ve specified.
Say Jennifer is a female, age 30, in Dallas, with a college degree and one child. Those coordinates “define” Jennifer (at least to the dating site), and searches that sample those coordinates will include Jennifer as one of the “points” (if you think of each person as a point) in that section of the space.
The problem with this analogy is that you end up with a large number of points within the dating site space that have the same coordinates. There may be another girl, Andrea, who enters essentially identical information as Jennifer. Any search of the sample space that brings up Jennifer also brings up Andrea. In the physical space that we live in, each point is unique.
Each dimension — in both mathematics and in the dating site example — represents a degree of freedom within the space. By changing one of the coordinates, you move through the space along one of the dimensions. For example, you can exercise a degree of freedom to search for someone with a different educational background or a different age range or both.
When scientists talk about the number of dimensions in string theory, they mean the degrees of freedom required for these theories to work without going haywire. The bosonic string theory required 25 space dimensions to be consistent. Later, superstring theory required 9 space dimensions. M-theory seems to require 10 space dimensions, and the later F-theory includes 12 total dimensions.