The following is a summary as well as a commentary on Chapter 7 of Tertium Organum by P D Ouspensky.
What really are the dimensions of space and why are there three of them?
Ouspensky says:
We really know that all three dimensions are in substance identical, that it is possible to regard each of the three dimensions either as following the sequence, the first, the second, the third, or the other way about. This alone proves that dimensions are not mathematical magnitudes. All the real properties of a thing can be expressed mathematically as quantities, i.e., numbers, showing the relation of these properties to other properties.
According to Ouspensky, if Mathematics does not represent something real then it is not mathematics. Mathematics cannot express the difference between dimensions, because no such difference really exists.
Ouspensky says:
As a matter of fact this is what provided Hinton with a basis for his theory of tessaracts, or four-dimensional solids – a4. But this is sheer fantasy, because, in the first place, the designation of dimensions by powers is purely conventional. All powers may be represented on a line. Let us take a 5-millimetre segment of the line a. Then a 25-millimetre segment will be its square, or a2; and a 125-millimetre segment will be its cube, or a3.
According to Ouspensky, the designation of dimensions by powers (one dimension – a, two dimensions – a2, three dimensions – a3, and so on) is purely arbitrary. Hinton’s metageometry has no relationship to either Euclidian or Non-Euclidian geometry.
Geometry is an artificial construction for the purpose of solving problems. The axioms of a given geometry, express the properties of a given space. Euclidean and Non-Euclidian geometries depend on the idea of surfaces in space.
For Lobachevsky, as a geometrician, a surface was merely a means for the generalization of certain properties, upon which one or another geometric system was built. He never looked at it from Kant’s perspective of reality or unreality.
In reality we have only locations defined by objects and directions of travel defined by energy. We may imagine surfaces in space, but that is by convention.
A line in time is not a geometrical line. It is an abstract notion. If the fourth dimension is time, it cannot be looked upon as a geometrical dimension.
In order to answer the question about the three-dimensionality of space, we need to establish, whether it is a property of the world out there (objective), or a property of consciousness (subjective).
If we assume that three-dimensionality of space is objective, and that it bears within itself the conditions which allow us to establish its relations to higher space through the method of analogy, we do not arrive at the answer to the causes of the three-dimensionality of space.
Ouspensky concludes,
… we must see whether this idea of the three-dimensional extension of the world with its properties is not the outcome of certain properties of our own mentality.
Basically, Ouspensky is rejecting the idea of objective four-dimensional solids (theory of tessaracts), because it is unreal. He is suggesting that we need to examine the possibility of three-dimensionality of space being subjective, as proposed by Kant.
