________________
130 First Steps to Jainism
inexpressible (avakta) module 5. And when x = 9, x = 2,3 or 9, so y may be 3 or 4, as well as .'
Thus for a full enumeration of functions modulo m we need a table with m+1 places corresponding to the residues 0, 1, 2.. , m-1, and . In each place we can set one, or any number, of these symbols, but we must set at least one. So each place can be filled in 2"+'-1 ways, for each of the m+1 symbols can be present or absent, except that all cannot be absent. Thus the total number of functions modulo m is (2"+'-1)"+1, for example 62, 523, 502, 209 if m=5, as compared with only 120 biunivocal functions, and 3125 univocal.
Now consider the simplest of the finite arithmetics, namely arithmetic modulo 2. There are only two elements, 0 and 1. Electronic calculators are based on this arithmetic. These machines are so designed that each unit, as the result of any instruction, will be active (1) or inactive (0) at any given moment. And it is possible, in principle, to predict whether it will be active or inactive. That is to say ambiguity is avoided, and the machine is designed to operate in terms of univocal functions. Nevertheless it is possible to provide such a machine with an instruction to which it cannot give a definite answer. It is said that some such machines, when given an instruction equivalent to one of the paradoxes of Principia Mathematical, come to no conclusion, but print 101010.... indefinitely. Clearly a machine could be designed to print u in such a case. It is obviously possible to design a machine which would print "O or 1" in response to the instruction x'-x=0. A machine with the further refinement suggested above would respond "0, 1, or " to the instruction (x'-x) cos x = O(mod 2)". Such a machine could give any of 7 responses, namely :
0,1, 1, 0 or 1, 0 or 9, 1 or
0 or 1 or .
These are the saptabhanghinaya with the omission of the syllable syad.
I now pass to an example where the saptabhanghinaya is actually applied in scientific research, and which I suspect is not far from what was in Bhadrabahu's mind. In the study of the physiology of the sense organs it is important to determine a threshold. For example a light cannot be seen below a certain intensity, or a solution of
Jain Education International
For Private & Personal Use Only
www.jainelibrary.org
