Functional Geometry and the Determination of Pattern in Mosaic Receptors 391 



The next stage of thought might be to perceive a relationship between two 

 different g's, 



^./'i V^i"+i = RQ 



m 12 



and so on to stages of any order. 



This operator-form of the equation seems to be the simplest familiar form 

 that expresses all the required relationships. It suggests that Q can be regarded 

 as a characteristic value or output value of the relationship operator when the 

 latter connects the state-functions P. Or g is a symbol of the relationship 

 between P^ and P^. The equations suggest the possible usefulness of a formal 

 calculus of abstraction. 



In the first equation, if either P^ or P^ or q is changed, Q is different and 

 generally vanishes. A little reflection will show that it is typical of thought, 

 as it is of such operator equations, that there is only a restricted class of pairs 

 of P's that have any relation to each other. The ^'s are sharply limited at the 

 same time as the P's; catalogues of the possible ^'s have been made by various 

 philosophers. And there is only a restricted class of realization signals, Q, 

 in any case. If the equation is to have a non-vanishing value, it imposes simul- 

 taneous restrictions on all four variables. Some /*'s may never show any 

 congruence. Some ^'s may operate forever in vain. And the Q's may take one 

 or two values so repeatedly that they become independent of what particular 

 P's are present. 



The significant thing here is that these same equations would also describe 

 many of the processes and structures in an address-determining network. 

 So, in the functional determination process, q could be the functional geometry 

 displacement operation, P^ and P2 the geometrical surfaces or the geometrical 

 patterns of excited cells before and after the operation, and Q the signal of 

 self-congruence. 



In the velocity-component cell of Fig. 4, q could be the delay operation, 

 Pi and P2 the pulse patterns in two of the input channels, and Q the coincidence 

 signal output. In the neuroanatomical structure of the same hypothetical 

 cell, q represents the delay line or lines, P^ and P, the input cells of the previous 

 stage, and Q the cell itself or its output axon or other output processes. 



At a higher stage of the network, we might imagine that each of these 

 structural elements in a particular neuron may also be connected to a verbal 

 motor stage, with the relationships among these structural connections again 

 represented by the same equations; and probably this structural parallel would 

 be repeated in the language structure among the words themselves. 



An address-detennining system therefore seems to be necessarily a symbol- 

 creating system. Whether regarded from the process aspect or the signal 

 aspect or the structure aspect, a relationship or pattern of patterns in each case 

 becomes represented by a symbol. 



The close parallel between neuronal transmission and logical operations 

 has been discussed for many years and is of great importance in the com- 

 putational and logical performance of decision nets. But the present equations 

 and hypothetical models suggest that analogy-perception, 'closure', 'insight' 

 and other apparent 'extrapolations' from the known — in short, thinking — 



