CHAPTER XXXVIII ON THE 'WORLD' OF MINKOWSKI
Moreover, the really fundamental things have a way of appearing to be simple once they have been stated by a genius, who was in this case Minkowski. (431) G. Y. RAINICH
We have already freely used the structural term 'dimension' and only hinted at its meanings. Before we approach the Minkowski world we must summarize roughly what for our purpose we should know about dimensions.
There is nothing mysterious about the term 'dimension*. First of all, the dimensionality of a manifold is not inherent in the manifold as such. It is a characteristic of order and so of structure. A manifold can be ordered in different ways, so that it follows that one manifold may have different dimensionality, depending on how we order it.
A manifold which has linear order and structure is called one-dimensional. A two-dimensional manifold is then a linearly ordered manifold of linearly ordered manifolds,.
Usually we speak about our 'space' of daily experiences as a three-dimensional manifold, but this is true only with reference to points, and not true with reference to lines or spheres. The manifold of all spheres in 'space' is, for instance, a four-dimensional manifold; so also is a manifold of lines.
Let us explain the line-dimensionality of our 'space' in terms of lines. A line can be given by two points - one, let us say, in the floor of our room, the other in the ceiling. Each of these points is given by two co-ordinates; it has two degrees of freedom; and so our 'space' is a four-dimensional (2x2) manifold in lines. This means that to distinguish any line in our 'space' from any other line we would have to have four data. Similarly, if we deal with spheres, a manifold made up of spheres requires four data, three for locating the centre and one giving the radius of the sphere. The above examples, of course, do not exhaust the structural possibilities.1
The term 'dimension' does not apply solely to what we call 'space'. The term applies to any manifold which we can order in some particular way. Manifolds or aggregates abound everywhere in our lives. The domain of colours, for instance, is a manifold; and so is the domain of tone, or of remembrances , . No manifold in itself has any dimensionality. To ascribe dimensionality to the manifold we must first order it and the number of its dimensionality, or its ascribed or discovered structure, may differ according to the principle of ordering used.
In discussing dimensionality we have two purposes. First, to dispel the semantic fright about this simple term; and, second, to suggest means for visualization, which for our purpose are of great neurological importance.
When we say that the world is structurally a four-dimensional manifold, we mean only that according to our experience and the structure of our nervous