Dimensions are strange things. They are talked about in physics and in woodwork, in science fiction and in tales of the supernatural, in religion and in quantum cosmology. Yet it is quite hard to know what we mean by ‘dimension’. Can we travel to other dimensions? Does my washing machine have more than three dimension? Does God exist in another dimension? Physics suggests that existence has precisely 11 dimensions, some of which are tangled into little loops. Daily experience suggests that there are only three, with time thrown in as an odd sort of one. Are we all talking about the same thing? What is a ‘dimension’ and how many of them are there really?
My modern science dictionary says that a ‘dimension’ is “the number of coordinates needed to specify the position of a point in some space of possibilities”.
What this definition says (bear with me) is that if you want to specify the position of a point within a two dimensional graph you need two co-ordinates. There are two dimensions in which the point can vary its position, in other words two aspects to its space of possibilities, the x axis and y axis. This is how we know that it is a two dimensional graph. In order to specify the precise position of a point within the graph we require two measurements, one along each axis, and therefore two values to specify the point’s complete co-ordinates. This is true because we defined our graph as being two dimensional. Likewise, if we now assume that our point is a star adrift in three dimensional space, we require three measurements to fix or specify its position, since its ‘space of possibilties’ has three dimensions, that is how we define it. (Let’s ignore time for now).
Now all this is obvious and you may wonder why I am plodding through it. This is surely stuff we all know. However there is something odd about this definition. It is not phrased in everday language, not what you're average Joe would call everyday language anyway, but that is not what makes it odd. Its oddness lies in the fact that even a little thought shows that as a definition it is completely devoid of all meaning. It is a pefectly circular bit of self-reference that tells us nothing whatsoever about dimensions.
This is obvious if you consider how you would conceive of a ‘dimension’ with the help of this definition. You find that it does not help at all. In fact ‘the number of coordinates needed to specify the position of a point in some space of possibilities’ is not something you can look at or conjure up in your mind at all. It is just the number of coordinates needed to specify the number of coordinates needed to specify how many dimensions there are in some possibility space in which someone has specified how many coordinates are needed in order to specify the position of some point within it. It is not a ‘thing’ at all. It is an explanation of how to relate our words to each other when we are talking about dimensions. We must therefore discuss dimensions without the help of my dictionary.
I take a dimension to be a degree of freedom in possibility space in which values can vary. Spatial dimensions are a particular case, but hot and cold, big and small, are equally dimensions, degrees of freedom.
Leeaus's first, and my fourth, spatial dimension, is spherical. It is a co-ordinate system and therefore a dimension. However it is a very odd co-ordinate system. By one view it is a point and by another a sphere. However these are equivalent topographical metaphors, not a description. It is a degree of freedom with the mirrored mathematical properties of both a point and a sphere, not a shape in spacetime.
The idea is that if spacetime is the 3D (+1) surface of a sphere then all points on the sphere are geometrically and ontologically underlain by the sphere, and have more than three spatial co-ordinates.
(Leeaus - did I get your view right?)