Jain Journal 1990 10

Author(s): Jain Bhawan Publication
Publisher: Jain Bhawan Publication

Page 41

OCTOBER, 1990
81
entities of dravya-s there is a need for understanding infinitesimal elements of these grand actual infinities. This is imperative for understanding or developing relative measures of infinitely many entities in the world. Deeper probing of infinitesimal elements led to developing more precisely idea of cardinality of infinities. The investigation into infinitesimals led to what can be called mathematical inquiry. This investigation was called by the Jains alaukika ganita. Understanding of nature and relations of infinitesimals gave rise to what we have called elsewhere Jain theory of measurement. Based on the infinitesimals, mathematics of transfinite number could develop and also theory of measures of various entities in the world. Here we would give one example of how idea of infinitesimal gave rise to a mathematical method of oneto-one correspondence for ordering non-global actual infinities. 59 Jains contended that total number of wretched souls (almost unfit for liberation called mithyadrsti jīva rāļi) though infinite are more than the actual infinity representing past time. Dhavalā gives very curious proof for this with the help of a thought experiment. Dhavala says that keep all souls of mithyadrsti in one conceptual pond and all the past in other. Notice that both these ponds have infinite elements. Then remove one element from one pond and correspondingly one element from other. Keep removing one-one element from each pond. We will find that pond of elements of past gets exhausted whereas elements of mithyādrșți jiva ponds remain. As the past has ended but wretched souls remain in present. Thus mithyādrsti jiva has higher cardinality than the cardinality of time instants of past. This is because we see mithyādrsţi jiva in present time and they have been always there. If mithyādrsţi souls were less than the time-instants of past than they all would have attained liberation by now and there would have been no wretched soul at present. Jains hold that all souls intrinsically are capable of moving higher up in spiritual ladder and attaining liberation. Removing a soul from pond of mithyādrsţi is equivalent of moving up in spiritual ladder by the soul through transmigration. This is a fine example of one-to-one correspondence method for determining cardinal relation of actual infinity which was possible because of notion of elements of actual infinity. Finding cardinal relation this way is exact and leads to consistent hierarchy of actual infinities. Any consistent hierarchy of actual infinities needs mathematical theory of transfinites. The Jains developed such a theory on the basis of their understanding of infinitesimal elements of dravya-s.
** Dhavala (II1, pp. 28-32).