We have overlooked the possibility of a suspended map capable of being folded and unfolded in a different orientation. This solution would no doubt eliminate many of the difficulties discussed above, but, even if technically different from the folding map of the third category, it would prove physically more cumbersome. It would in any case involve the same paradoxes of folding that arise with this third type of map, and would be open to the same objections.
2.3 Transparent Map, Permeable, Extended, and Adjustable
Let us imagine that such a map, drawn on transparent and permeable material (gauze, for example), is spread out over the surface and is adjustable.
In any case, once the map has been drawn and spread out, either the subjects remain on the territory beneath it, or they climb on top of it. But if the subjects were to prepare the map while it is above their heads, not only would they be unable to move, because every movement would alter the positions of the subjects that the map describes (unless we have recourse, once again, to an impoverished map), but further, in moving, they would cause tangles in the very fine membrane above them, resulting in serious discomfort and once more making the map unfaithfuclass="underline" it would assume a different topological configuration, producing disaster areas not corresponding to the planimetry of the territory. It must therefore be supposed that the subjects have produced and extended the map while remaining on top of it.
In this case we can adduce numerous paradoxes already considered in connection with the previous maps: the map would represent a territory inhabited by subjects who in reality inhabit the map (unless it is a summary, or impoverished, map); the map could not be consulted because each subject could examine only the part corresponding to the territory on which subject and map lie; the map's transparency would eliminate its semiotic function, since it would be functional as sign only in the presence of its own referent; residing on the map, the subjects could not tend the territory, which would deteriorate, making the map unfaithful....It is necessary, then, for the map to be capable of being folded and then reopened with a different orientation, so that every point X of the map representing a point Y of the territory can be consulted when the point X of the map lies on any point Z of the territory, where Z ≠ Y. Folding and unfolding, finally, permit long periods of time when the map is not being consulted and does not cover the territory, and thus allow the cultivation and maintenance of territory necessary to keep its actual configuration always the equivalent of the one depicted on the map.
2.4 Folding and Unfolding the Map
Certain preliminary conditions must be postulated: (i) that the reliefs of the terrain allow the free movement of those subjects assigned to folding; (ii) that a vast central desert exist, where the folded map can be stored and where it can be turned when it must be unfolded again in a different orientation; (iii) that the territory have the form either of a circle or of a regular polygon, so that the map, however oriented, will not exceed its boundaries (a 1:1 map of Italy, shifted ninety degrees, would be spread out over the Mediterranean); and (iv) that, as an inevitable consequence, the map will have a central point, lying always on the same portion of the territory that it represents.
Once these conditions have been satisfied, the subjects can move en masse towards the farthest boundaries of the empire to avoid the map's being folded up with subjects inside. To avoid potential over-crowding when the subjects are all clustered at the edges of the map (and of the empire), we must postulate an empire inhabited by a number of subjects not superior to the number of measuring units of the total perimeter of the map, the perimetric unit of measurement being equivalent to the space occupied by one subject in a standing position.
Now suppose that each subject grasps a bit of the edge of the map and begins folding it, while retreating further and further. A critical point would be reached at which the subjects would all be crammed together at the center of the territory, standing on top of the center of the map and supporting its folded edges above their heads: a situation aptly termed scrotum catastrophe, as the entire population of the empire is enclosed in a little transparent sac, in a situation of theoretical stalemate and of considerable physical and psychological discomfort. The subjects must therefore, as the folding gradually proceeds, leap instead outside the map and onto the territory itself, where they will continue folding from outside, until the final stages of the folding, when no subject remains inside the sac.
But this solution would inevitably produce the following situation: the territory would consist, once folding is completed, of the original terrain, plus an enormous folded map in its center. Thus the folded map, no longer consumable, would prove unfaithful as well, because it is known for certain that it would represent the territory without its folded self in the center. And there is no apparent reason why a map should be unfolded and consulted when it is known a priori to be unfaithful. On the other hand, if the map were to depict the territory with itself folded in the center, it would immediately become unfaithful every time it was unfolded.
It could be assumed that the map is subject to a principle of indetermination, for it is the act of unfolding that makes a map faithful whereas, when folded, it is unfaithful. In this situation the map could be unfolded whenever there was a desire to make it faithful.
There still remains, however (unless we have recourse to the partial, or summary, map), the problem of the position to be assumed by the subjects after the map has been unfolded and laid out with a different orientation. For it to be faithful each subject, once the unfolding is completed, must assume the position he had, at the moment of its creation, on the actual territory. Only at this cost will a subject resident at point 2 of the territory—on which, say, a point X2 of the map lies—be depicted exactly at point X1 of the map that currently lies, for example, on point Y of the territory. At the same time, every subject could obtain information from the map about a point of the territory different from the one where he resides—and about a subject different from himself.
Toilsome as it may be, and full of practical difficulties, this solution makes the transparent and permeable map, spread out and adjustable, the best prospect, while obviating any need to settle for a summary map. But this map, too, like the previously mentioned ones, falls victim to the Normal Map paradox.
3. The Paradox of the Normal Map
When the map is installed over all the territory (whether suspended or not), the territory of the empire has the characteristic of being a territory entirely covered by a map. The map does not take account of this characteristic, which would have to be presented on another map that depicted the territory plus the lower map. But such a process would be infinite (the "third man" argument). In any case, if the process stops, a final map is produced that represents all the maps between itself and the territory, but does not represent itself. We call this map the Normal Map.
A Normal Map is subject to a quasi-Russell-Frege paradox: every territory, plus a map representing it, can be seen as a normal set (the map does not belong to the set of objects that constitute the territory). But we cannot conceive sets of normal sets. Therefore we should think either of a not-normal set, in which the final map is part of the territory it represents (which is false, otherwise it should also represent itself) or of a normal set in which the final map is necessarily unfaithful, as explained above.