Tractatus: Being Thinkable

As it was mentioned in a previous article on causality, some laws such as the principle of sufficient reason are necessary for people. Wittgenstein says in Tractatus (6.361) that only connexions that are subject to law are thinkable. Without laws, we could not even describe the world (in the sense of 6.341), even though art could make things manifest.

Can we compare a process with 'the passage of time'? Wittgenstein says (6.3611) that we even cannot say there is such thing called 'the passage of time' (no matter whether there is such thing, or whether it makes itself manifest). We can describe the lapse of time only by relying on some other process. Time as the temporal dimension with orientation (from the past to the future) is just something in physics to describe the world. As Einstein says, time is relative. By relying on some other processes such as the rotation of the Earth and its revolution around the Sun, we call it 'the passage of time'.

Something exactly analogous applies to space (6.3611). All events occur in the so-called space-time continuum. People say that a metal bar will expand or contract in response to a change in its state such as temperature and pressure. Without such changes, neither of expansion and contraction (which exclude one another) can occur. People think that it is because there is nothing to cause the one to occur rather than the other, while Wittgenstein says (6.3611) that it is really a matter of our being unable to describe one of the two events unless there is some sort of asymmetry to be found. An asymmetry such as an increase in temperature is then regarded as the cause of the occurrence of expansion and the non-occurrence of contraction. It is we who call such an asymmetry a 'cause' to describe the world.

Another example is the motion of an object. The movements of an object in different directions at the same time are mutually exclusive events. The orientation of the three-dimensional space is something in physics. As Einstein says, position is relative. Motion can only be observed and measured relative to a frame of reference, such as the Earth, the Sun, or the Milky Way Galaxy. 'Orientation' is just something in our thought to describe the world. Gravity is one of the asymmetries that we regard as the causes of motion. Nevertheless, Newtonian mechanics describes it as a force, while general relativity ascribes it to the curvature of space-time. All the same, they are just different theories - different descriptions of the world.

As a comment on 6.3611, 6.36111 gives a purely spatial example: a right-hand glove could be put on a left hand if it could be turned round in four-dimensional space. In our world, it is the orientation that makes the completely congruent right hand and left hand not to coincide. However, we can describe the direction of an object only by relying on the position of some other object.

Furthermore, we also see that the ways in which connexions are thinkable depend on the laws they are subject to. Rotation in four-dimensional space is thinkable only in mathematics. How can we imagine such a four-dimensional space, not to mention a rotation in such a space? What Wittgenstein mentions are just analogies in the lower dimensional spaces. As an analogy, we can imagine the rotation of the right hand around a cross-section. During the rotation, only this cross-section as the rotation 'axis' remained still in our three-dimensional space, so that we could see only the cross-sections of the veins, the flesh, and the bones, while all other parts of the hand disappeared (from our three-dimensional space). If there was light in the four-dimensional space that projected the three-dimensional shadow of the hand to our three-dimensional space, then we could see the hand (its shadow) shrinking toward the cross-section of rotation 'axis', becoming flat, then inflating from the cross-section in the opposite direction, and finally becoming identical with the left hand. That is all we can think of.

By the way, String theory claims that our universes has more than four dimensions (space-time). However, these other dimensions are so small (highly curved), not only that the right-hand glove cannot be turned round, but also that we are not even aware of them. Still it is merely a theory - a description of the world.

I cannot think beyond my thought, but I can only remind myself of my own limitation.