The topology of visual appearance

Abstract

There has been a long, and at times acrimonious, debate in the philosophy of perception over the question of whether depth is seen 'directly'. Berkeley notoriously held that it was not.1 Much depends, of course, on how one understands the notion of 'direct' or 'immediate' (visual) perception, assuming indeed that one admits such a notion at all. Berkeley's notion of indirect perception is essentially an inferential one: something is perceived indirectly just in case it is in some sense only 'inferred' (perhaps only unconsciously) from what is perceived directly.2 Direct perception is accordingly non-inferential. Rather than enter this debate, I want to show that there is a clear sense in which 'visual appearances' are only tworather than three-dimensional; the epistemological significance of this is another matter. I shall deal first with the monocular case and then with the binocular. Consider, then, the following simple 'experiment' in monocular vision, to perform which we require a small loop of thin wire, A, and a small length of flexible thin wire, B. Suppose x and y are any two objects which visually appear to be separated (e.g., they might be two distant houses, one situated to the left of the other from where the observer stands, or they might be opposite corners of a matchbox viewed at arm's length). Then it must be possible to position the loop A in such a fashion that if the two ends of B visually appear to touch jc and y respectively, then B must visually appear to intersect A (see figure 1). But this just reflects the following theorem of the topology of two dimensional surfaces: if x and y are any two separate points on a surface, it must be possible to draw a closed curve A on the surface in such a fashion that any open curve B on the surface whose end-points coincide with x and y must intersect A. From this we may conclude that in monocular vision, at least, there certainly is a use of the notion of 'visual appearance' according to which the spatial properties that objects visually appear to have obey a two-dimensional geometry.3 (This is the sense in which, for instance, one may say that a small coin viewed face-on at arm's length appears the same size as the moon as seen from the earth.) Of course it does not