Jain Journal 1990 10

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

Page 46

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JAIN JOURNAL
With these three fundamental dimensions of measurement it becomes possible to pronounce measure of every conceivable thing. Once notion of actual infinity is founded at ontological level and units of measurement derived from infinitesimals of global actual infinities and further developing of detailed standard of measurement the stage was set to obtain detailed measures of entities of the world. All entities were actual infinities in terms of three distinct infintesimals but a numerable (samkhyāta) counting of these actual infinities was possible. In fact this exactly was the Jain notion of samkhyāta numbers. Unlike asamkhyāta and ananta numbers the countable numbers samkhyāta were not founded ontologically. Their existence is a posterior existence as they are based on counting actual infinities. Countability of physical actual infinities is an interesting aspect of the Jaina theory of infinities.
But if all entities in the world are actual infinity it is quite a task to develop consistent theory of cardinal hierarchies as it should not violate the order of things as given to us in the external world. Mathematics of transfinite as well as detailed arguments of cardinality of global actual infinities was developed just to fulfill this task by Jain theoreticians. It was a self-imposed project of Jains to develop this hierarchy so that the world of experience is not violated and on the other hand is comfortably reckoned with.
Since notion of actual infinity and notion of infinitesimal elements were evolved on the basis of ontology and measure theory, even the mathematics of actual infinities was rooted on the base of ontology and measure theory. Difference between asamkhyāta and ananta was made on the basis of ontological reasoning at the same time these classes of actual infinites were mathematically related transfinite numbers. Detailed mathematical typology of asamkhyāta and ananta were done to account for different physical actual infinities as well as for the sake of mathematical consistency. Mathematical reasoning was developed and adopted solely to relate physically existent actual infinities. It is an externality which gives it a content and not the intrinsic mathematical order as is the case in the modern theory of transfinite numbers, though the intrinsic constraints are not violated. It is only in this context that Jain mathematics of transfinite numbers can be understood.
Most consistent and well developed mathematical reasoning of transfinite numbers is to be found in that section of Dhavala which is concerned with locating cardinality of actual infinity of mithyādrsti jiva of hell in accordance with different standards of measurement. Attempt was to locate cardinality in alpabahutva order (order according to being