Mathematics and Reality

Abstract

Let us imagine that we have been travelling on a great journey to some far-off world. We shall call this world Tor’Bled-Nam. Our remote sensing device has picked up a signal which is now displayed on a screen in front of us. The image comes into focus and we see (Fig. 3.1): What can it be? Is it some strange-looking insect? Perhaps, instead, it is a dark-coloured lake, with many mountain streams entering it. Or could it be some vast and oddly shaped alien city, with roads going off in various directions to small towns and villages nearby? Maybe it is an island - and then let us try to find whether there is a nearby continent with which it is associated. This we can do by ‘backing away’, reducing the magnification of our sensing device by a linear factor of about fifteen. Lo and behold, the entire world springs into view (Fig. 3.2): Our ‘island’ is seen as a small dot indicated below ‘Fig. 3.1’ in Fig. 3.2. The filaments (streams, roads, bridges?), from the original island all come to an end, with the exception of the one attached at the inside of its right-hand crevice, which finally joins on to the very much larger object that we see depicted in Fig. 3.2. This larger object is clearly similar to the island that we saw first - though it is not precisely the same. If we focus more closely on what appears to be this object’s coastline we see innumerable protuberances - roundish, but themselves possessing similar protuberances of their own. Each small protuberance seems to be attached to a larger one at some minute place, producing many warts upon warts. As the picture becomes clearer, we see myriads of tiny filaments emanating from the structure. The filaments themselves are forked at various places and often meander wildly. At certain spots on the filaments we seem to see little knots of complication which our sensing device, with its present magnification, cannot resolve. Clearly the object is no actual island or continent, nor a landscape of any kind.