ON THE 'WORLD' OF MINKOWSKI 671 

not affect the strip at all but it is rather a section cut out of the X axis. In actual experience, it is only the strip as a manifold of world points which has physical reality, and not the cross sections, which, as we see, are not equal on different axes. The 'contraction' is not a change in 'physical reality' but merely a consequence of our way of regarding things. We see that the notorious argument as to whether the 'contraction' is 'real' or 'apparent' is based on a misunderstanding. Born gives an excellent example. If we slice a cucumber in different directions it is fallacious to argue that the smallest slice which is perpendicular to the axis is the 'real' one and the larger oblique slices only 'apparent'. Similarly in the Einstein theory, a rod has various lengths according to the motion of the observer.
One of these lengths, the static length, is the greatest, but it is no more 'real than any other. Similar remarks can be made about 'time'.
Attention should be given to one extremely important semantic point concerning the Minkowski fourdimensional world. We already know that for our nervous systems the passing from dynamic to static, and vice versa, is a most vital structural problem. The first step of this translation has already been given in the notion of the 'variable'. The calculus carried it a step further. In the Minkowski world we reach the complete solution of the problem.
As Keyser points out in his Mathematical Philosophy, we had two verbal methods of dealing with 'time'. One was the method of Newton, the method of the structural importation of 'time'. From the objective dynamic world of the lower order abstractions, 'time' is imported into the static world of the higher order abstractions. We import it with 'motion', we say things 'move'. Such language is structurally unsatisfactory, even on the earlier level of our development. It hampers analysis, and is contrary to the structure and function of the human nervous system. It breeds tremendous metaphysical impasses, and is ultimately based on semantic disturbances due to identification.
If we introduce dynamic, shifting entities into static higher order abstractions, rationality is impossible and we drift toward mysticism.
A very real semantic problem appears here. We want to give the best possible account of the structurally dynamic world around us; yet our higher order abstractions are structurally static, and for their proper working they must use static means. Here seemed to be an impasse which for milleniums had defied solution. 'Philosophers' of different schools were preaching and teaching that we should never be able to be 'rational' and understand this world and ourselves. Antiintellectual schools began to flourish, to the bewilderment of all.
The issue, after all, was simple, the moment some one discovered and stated it. We did not need to change either the world around us or ourselves;

