and contents. In order that this representation should be constructed, the representation itself will have to contain once more, as a part of itself, a representation of its own contour and contents; and this representation, in order to be exact, will have once more to contain an image of itself; and so on without limit. We should now, indeed, have to suppose the space occupied by our perfect map to be infinitely divisible, even if not a continuum.[1]
One who, with absolute exactness of perception, looked down upon the ideal map thus supposed to be constructed, would see lying upon the surface of England, and at a definite place thereon, a representation of England on as large or small a scale as you please. This representation would agree in contour with the real England, but at a place within this map of England, there would appear, upon a smaller scale, a new representation of the contour of England. This representation, which would repeat in the outer portions the details of the former, but upon a smaller space, would be seen to contain yet another England, and this another, and so on without limit.
That such an endless variety of maps within maps could not physically be constructed by men, and that ideally such a map, if viewed as a finished construction, would involve us in all the problems about the infinite divisibility of matter and of space, I freely recognize. What I point out is that if my supposed exact observer, looking down upon the map, saw anywhere in the series of maps within maps, a last map, such that it contained within itself no further representation of the original object, he would know at once that the rule in question had not been carried out, that the resources of the map-maker had failed, and that the required map of England was imperfect. On the other hand, this endless variety of maps within maps, while its existence as a fact in the world might
- ↑ In the older discussions of continuity, this concept was very generally confounded with that of infinite divisibility. The confusion is no longer made by mathematicians. Continuity implies infinite divisibility. The converse does not hold true.
