Functional Geometry and the Determination of Pattern in Mosaic Receptors 393 



sharp boundaries, dazzle patterns with the proper kaleidoscopic confusions of 

 rapidly disappearing and reappearing spots, the diabolical fields of Ditchburn 

 that refuse to be scanned — in fact, any patterns that deny the analogies that 

 the network is prepared to detect — are probably all equivalent to a structureless 

 environment in their failure to produce organization behind the retina. 



The second point becomes obvious when we consider functional geometry 

 and the profound limitations it imposes. Just as the lens-grinding machine 

 with sufficient freedom of motion necessarily produces spherical surfaces 

 no matter what it starts with, so the scanning eyeball necessarily generates 

 'external' space relations or addresses corresponding to the continuous three- 

 dimensional rotation group, no matter what is the structure of the external 

 field. Likewise at the retinal level, any scanning retina necessarily acquires 

 a unique perception for continuous lines of constant curvature and for parallel 

 lines and lines or points periodically spaced, which it can never accord to 

 patterns violating these displacement-congruences. 



These natural congruence relations may play the same organizing and 

 aesthetic role in vision that octaves and simple frequency relations play in 

 hearing. 



It is peculiar to functional geometry and it is extremely important for 

 philosophy that these necessary relations are neither given to the visual system 

 by any particular external field or experience, nor are they logically implicit 

 in the structure of the network, even when we include the analogy-detecting 

 structure. They are Q's that, like spheres, turn up invariably, no matter what 

 the P's or ^'s. They have rather the character of geometrical preconditions 

 simultaneously imposed on both the external field and the network organization 

 if any learning is to be possible. And they are not imposed by the scanning 

 operation, even though it does mediate between the field and the network — 

 any more than the spherical shape is imposed on the lens by the loose random 

 grinding machine, or on the eyeball by the muscles. They are more like a 

 priori requirements, mathematical absolutes, that determine the only kinds 

 of experience that can be organized and the only forms that learning can take, 

 if learning is to be done at all. 



Functional geometry thus may be the origin of the Kantian epistemological 

 limitations on thinking, as represented by 'the synthetic a priori categories of 

 the apperceptive dialectic' Pitts has described this as 'perhaps the most 

 fundamental problem of neurophysiology and psychology' (3). It appears 

 that much, if not all, of Kant's theory of knowledge can be translated word 

 for word into the language of inputs, structures and geometries in an address- 

 determining network. 



Functional Storage 



A neuron that has been selected by experience, so that its output signals 

 an experienced pattern, constitutes a storage of the experience — a memory. 

 The storage is not 'dead storage' but a functional link which permanently 

 changes the operation of the larger net and which remains part of it. The 

 address-determining connections therefore constitute a functional storage of 

 experience. At any instant, the net is the memory; the memory is the net. 



This kind of storage differs in an essential way from that of a prc-addressed 



