DESIGN FOR A BRAIN 11/4 



been solved, there is evidence (S. 20/10) suggesting that in some 

 typical cases the number will be of the order of 2-^, where N is 

 the number of variables. There seems to be little doubt that if 

 a Homeostat were made with a thousand units, practically every 

 field would be unstable, and the chance of one occurring in a 

 lifetime would be practically zero. We thus arrive at much the 

 same conclusion as before. 



The second case to be considered is that of the system that has 

 the transformation on its states formed at random, so that every 

 state goes equiprobably to every state. Such systems have been 

 studied by Rubin and Sitgreaves. Among their results they find 

 that the modal length of trajectory is \/n, where n is the number 

 of states. Now if the whole is made of N parts, each of which 

 can take any one of a states, then the whole can take any one of 

 a N . This is n; so the modal length of trajectory is Vo N , which 

 can be written as (\Zo) N . Again, if we fill in some plausible 

 numbers, with N = 1,000, we find that the length of trajectory, 

 and therefore the time before the system settles to some equili- 

 brium, takes a time utterly beyond that ordinarily observed to 

 be taken by the living brain. 



11/4. The three functions given by the three calculations are all 

 of the exponential type, in that the number of trials is proportional 

 to some number raised to the power of the number of essential 

 variables or parts. Exponential functions have a fundamental 

 peculiarity: they increase with deceptive slowness when the 

 exponent is small, and then develop with breath-taking speed as 

 the exponent gets larger. Thus, so long as the Homeostat has 

 only a few units, the number of trials it requires is not large. 

 Let its size undergo the moderate increase to a thousand parts, 

 however, and the number of trials rushes up to numbers that make 

 even the astronomical look insignificant. In the presence of this 

 exponential form, a mere speeding up of the individual trials, or 

 similar modification, will not bring the numbers down to an 

 ordinary size. 



11/5. What had made the processes of the last two sections so 

 excessively time-consuming is that partial successes go for nothing. 

 To see how potent is this fact, consider a simple calculation which 

 will illustrate the point. 



150 



