Science fiction authors and pop culture often describe “dimensions” as if they are alternate realities or completely different planes of existence. This is a misappropriation of the term, although respected scientists appear to have given up fighting the change in the language. Particle physicists use the term differently, to map out the spatial (and temporal) structure of the universe, and tracking these dimensions is fundamental for both Einstein’s Theory of Gravity (General Relativity), and modern string theories. I want to focus on the mathematical definition of a dimension.
A dimension is just a way of parameterizing sets of possibilities. (The full set of possibilities is usually a “space.”) Dimensions are often visualized by converting them to spatial dimensions: The axes of any line- or scatter-plot count. The three familiar spatial dimensions are {forward/backward, left/right, up/down}. Alternately, you could relabel the dimensions as {latitude, longitude, and altitude} or {x, y, z}. Discussions of changes in the orientation of a ship or plane might describe motions using {pitch, roll, yaw}. TNB frames can be used to describe what’s happening near a moving roller-coaster car. If it’s necessary to describe an evolving state, like the path you last took to the grocery store, we add a fourth dimension (time) to any of the above sets of three.
A “coordinate system” defines a specific set of dimensions, including the mapping of those conceptual positions onto specific numbers. You can index positions near the Earth based on {latitude, longitude, and altitude} or based on distance {forward/backward, left/right, or up/down} relative to where you are currently. Similarly, there are other coordinate systems which can be used to reference the same space. Different systems have different purposes: You could use GPS coordinates to describe the process of tying on your shoes, but doing so would be needlessly complex. Sometimes coordinate systems have ambiguities: What replaces “up/down” in weightless conditions? The coordinates {north/south, east/west, and up/down} face trouble when you’re standing exactly at the North Pole (or South Pole). If such ambiguities are likely to show up in a problem, it’s generally a sign that you should shift to a different coordinate system.
As useful as it is to talk about spatial dimensions, there are many more possibilities which show up in science and policy discussions: The periodic table has two dimensions, arranging atoms based on the number of filled electron s sublevels and the number of electrons added since the last level was filled. (I’m fudging slightly there, click the link for more of the chemistry describing the periodic table’s structure.) Members of Congress can be placed on a grid based on how often they vote with each other. Individuals can be plotted based on their income, education, and other variables. Entire countries may be compared based on things like their crime rates and gun ownership.
Much of science (including sociology) involves plotting data in two or more dimensions, looking for correlations and teasing out causal relationships. Those processes can be complex and are fraught with their own risks, but mathematically they’re based on the concept of a dimension, no matter the subject. Rigorous analysis often involves clearly identifying the dimensions you’re dealing with and then trying to develop a feel or visualization for how your data points are arranged in those dimensions. Spatial plots can be supplemented by coloring (or otherwise marking) data according to an explicit scheme, or by imagining how a static plot evolves over time.
