3.1 Euclidean Dimension
A space is a collection of entities called points. Both terms are undefined but their relation is important: space is superordinate while point is subordinate. Our everyday notion of a point is that it is a position or location in a space that contains all the possible locations. Since everything doesn't happen in exactly the same place, we live in what can rightly be called a space, but points need not be point-like. Any kind of object can be a point. Other geometric objects, for instance, are totally acceptable (lines, planes, circles, ellipses, conic sections) as are algebraic entities (functions, variables, parameters, coefficients) or physical measurements (time, speed, temperature, index of refraction). Even so-called "real" things can be points in a space: people are points in the space of a nation's population, nations are points in the global political space, and telephones are points in the space of a telecommunications network.
Any space that can be conceived of also has a characteristic number associated with it called a dimension. From the time I began the serious study physics and mathematics up until I discovered chaos and fractals, I had what I thought was a complete definition of dimension. My definition of dimension (which I had assumed to be the definition) was the number of real number parameters needed to uniquely describe all the points in a space. Thus, the real number line is one-dimensional as it only takes one real number (parameter) to describe each real number (point). Dimension is invariant so that a plane, for example, requires two parameters in rectangular (x, y) or polar (r, θ [theta]) coordinates. Other suitable examples come to mind. The set of lines in a plane is two-dimensional as describing any one of them uniquely requires two parameters: the slope and y-intercept (m, b) or the x and y-intercepts (x0, y0), for example. The set of all circles in a plane is three-dimensional (two for the coordinates of the center and one for the radius) and the set of all conic sections in a plane is five-dimensional (trust me).
This situation is great for physicists, and other scientists too, as it enables them to mine the wealth of mathematical knowledge about curves, surfaces, and all the rest and apply them to the rigorous study of the natural world. It takes four numbers to adequately describe the thermodynamic state of a region containing a gas: pressure, volume, temperature, and amount of material. If we limit ourselves to a fixed amount of material and assume that the gas is ideal, we can reduce the system down to two dimensions: pressure and volume (a reasonable assumption believe it or not). Thus, anything that any mathematician has ever done in a two-dimensional space could, at least in principle, be used in the study of ideal gases. The work done by a gas as it expands isothermally turns out to be a problem in finding the area under the curve y = a/x. Luckily, for the physicists of the nineteenth century, a mathematician had already determined the solution.
What about a point? Well, a single point is easy. It takes no numbers to uniquely identify a single isolated object. If we have only one thing, we don't need to discriminate between it and anything else so a point is zero-dimensional. What about two points? This space should be one-dimensional according to our definition as describing any part of the space requires one number. The same would be true for a space of twenty points or twenty million points. Any countable set (finite or infinite) would require one parameter to adequately describe all of its points. Does this now make a countable collection of zero-dimensional objects a one-dimensional space? I wouldn't like this to be the case. A countable set of points is very different from an uncountable set (like the real numbers) and their dimensions should reflect this difference. Maybe we have an out. My original definition said that the dimension of a space is given by the number of real parameters it takes to identify the points. A countable set can be adequately paired with the whole numbers in a manner that would assign each a unique coordinate. I do not find this distinction very satisfying, however.
What is the dimension of the space of acceptable telephone numbers? Since each telephone can be uniquely located with one number it should be one-dimensional. In actuality, however, a phone number is an area code plus an exchange prefixed on to a four-digit number making the whole sequence three-dimensional. To further complicate matters, an exchange is really a pair of coordinates. Think about old movies where a character would bark into the receiver, "Operator! Give me Klondike 5 - 1234." If Klondike and 5 are an ordered pair then current phone numbers are really four-dimensional. Such complications further reduce the validity of the parameter counting method for determining dimension.
Another way to think of dimension is as the degrees of freedom available within the space. Our choice of directions must be orthogonal, however. Motion in a particular direction cannot be expressed as the finite combination of infinitesimal motions in the other directions. It should not be possible to move up by the finite combination of extremely small left and forward motions. If it is, then we have over-described the dimension of the space. Physical space (euclidean three-space) is three-dimensional because there are three independent directions that objects within the space can move: up/down, left/right, and forward/backward. The surface of the earth, on the other hand, is two-dimensional as we are only free to move in one of two directions: left/right and forward/backward. Any vertical motion is the result of moving in the other two directions. Under these constraints, a countable set of points is now zero-dimensional as we have zero degrees of freedom. It is not possible to move through such a space from one point to another without leaving the space.
What about an x-shaped space (╳) composed of two line segments? Locally such a space is one-dimensional everywhere except at the intersection of the two segments where it becomes two-dimensional. How should we handle such cases? Should we go with the minimum value as the true dimension? This would make the space one-dimensional, which feels more natural. If we did, however, then the union of a point and a filled square ( · █) would be zero-dimensional. This does not feel natural. The first example (╳) should be one-dimensional and the second ( · █) two-dimensional. So far, my commonsense definitions are not agreeing with my commonsense answers. I guess it's time to crack open the books and see what the real mathematicians have to say.