The Four-Dimensional Manifold of World-Points

He always does this. No, he doesn't.
She always says that. No, she doesn't.
It always repeats itself in exactly the same way. No, it doesn't.

Hermann Minkowski formulated a useful method of representation that enables us to translate structurally 3-D dynamic events into structurally 4-D static, higher-order abstractions.

Let x, y, z be the rectangular coordinates of space, and t denote the time. Subjects of our perception are always places and times connected. No one has observed a place except at a particular time, or has observed a time except at a particular place. Yet I still respect the dogma that time and space have independent existences each. I will call a space-point at a time-point, i.e., a system of values x, y, z, t, as a world-point. The manifold of all possible value systems of x, y, z, t, shall be denoted as the world. I boldly could draw four world-axes with a chalk upon a table. Even one axis drawn consists of nothing but quickly vibrating molecules, and besides, takes part in all the journeys of the earth in the universe; and therefore gives us plenty occasions for abstractions. The greater abstraction connected with the number of 4 does not cause the mathematician any trouble. In order not to allow any yawning gap to exist, we shall suppose that at every place and time, something perceptible exists. In order not to say either matter or electricity, we shall simply use the word substance for this something. We direct our attention to the substantial point located at world-point x, y, z, t, and suppose that we are in a position to recognize this substantial point at any other time. Let dt be the time element corresponding to the changes dx, dy, dz of space coordinates of this substantial point. Then we obtain (as a picture, so to speak, of the perennial life-career of the substantial point), a curve in the world, the world-line, whose points can unambiguously be connected to the parameter t from +infinity to -infinity. The whole world appears to be resolved in such world-lines, and I may just anticipate, that according to my opinion the physical laws would find their most perfect expression as mutual relations among these world-lines.

x,y,z,t, -- not x,y,z plus t



Thanks to Minkowski's language of new structure (introduced by him in 1909), we can intelligently order events of our world experiences with respect to four dimensions:

x (Right-Left)
y (Forward-Backward)
z (Up-Down)
t (Before-After)

Everything happens in a unique place, at a unique time. If we neglect to factor this significant consideration into our evaluation of individual occurrences, then we cannot prevent the harm that results from our unfortunate errors of judgement and our delusional views of reality.