Space

Among the most elementary actions known to us are small displacements "in space". We have put the quotes, because people have accustomed to imagine that some entity, called "space" exists as a primary reality, which creates the possibility of moving from one point of this space to another. Our analysis turns this notion topsy-turvy. Only actions constitute observable reality; space is nothing but a product of our imagination which we construct from small displacements, or shifts, of even smaller objects called points. If x is such a shift, then xx -- the action x repeated twice -- is a double shift, which we would call in our conventional wisdom a shift at the double distance in the same direction. On the other hand, we may want to represent a shift x as the result of another shift x' repeated twice: x = x'x'. It so happens that we can make three different kinds of shifts, call them x, y, z, none of which can be reduced to a combination of the other two. At the same time any shift w can be reduced to a properly chosen combination of shifts x, y, z. So we say that our space has three dimensions.