``Imagine that the cosmos is completely empty save for one single human hand. Is it a left hand or right hand? Since there are no intrinsic, measurable differences between enantiomorphic objects, we have no basis for calling the hand left or right. Of course if you imagine yourself looking at the hand, naturally you will see it as either right or left, but that is the equivalent to putting yourself (with your sense of handedness) into 3-space. You must imagine the hand in space to be completely removed from all relationships with other geometrical structures. Clearly it would be meaningless to say that the hand is left or right as it would be to say it is large or small, or oriented with its fingers pointing up or down.''[4, p. 110]The problem arises when a human body (with severed hands) appears in this fictional cosmos. The hand will only fit either the left side or the right side. If one assumes that it will only fit the left side, then one sees the paradox in this: the hand (if it fits the left wrist) must have been a left hand before the body appeared. In other words, the concept of left-ness and right-ness existed even in a universe devoid of all other relational objects. Kant asserted that this would imply that space itself had an absolute structure.
A paradox exists that may lend credence to this discovery of Kant's, but Gardner unravels this paradox by flattening the hand and applying human 3-space perspective. If we were to have a flat hand (let us say that it is transparent so that we can use our 3-space view and look at its mirror image without difficulty) as the only object in the universe it does not have any asymmetrical properties, for it is the only thing in this universe. If we introduce a similar flat human, with two missing hands, with the qualification that ``left'' is the side associated with the placement of the heart in the human body, then we place the hand where it fits and determine its ``left'' or ``right''ness. However, since this human is flat and transparent, then we can just use our 3-space privilege and get the opposite results. The hand that we just placed on his left wrist, is now a right hand! In sum, neither the hand nor the body has changed any of its properties for us to get to this conflicting conclusion. It's simply that their relations to each other in two space are changed.[4, p. 110]
James Van Cleve points out a similar analogy regarding the circumvention of 2-space constraints via 3-space use to 3-space constraints via 4-space use. If ``tokens'' of the letter `p' and `q' were constrained to 2-space sheets of paper, none of the letters could be manipulated within 2-space to occupy each other's space. However, if we flip either letter through 3-space and bring it back down to 2-space, then we can convert the `p' to a `q' (or vice-versa.) If there were a four dimensional space, you could take the 3-space asymmetrical object (the hand, for example) and ``flip-it'' through 4-space so that the left hand becomes the right (or vice-versa.) This analogy would tend to favour the externalist view that ``left-ness'' or ``right-ness'' is purely relative; this externalist view would be true iff2 four dimensions were proven to be right.
In support of Kant's views, however, there is the physical phenomena of the fall of parity: some laws of nature are sensitive to the distinction between right and left. This would seen to support the view that left-ness and right-ness is an absolute in the universe. [3, p. 114] Hence, we are right back where we started: if the fourth dimension is true, then we can prove the externalist view of the world. Otherwise, the only thing that is solid now, is an absolutist view of the world, due to the very real fall of parity.