Jain Journal 1990 10

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

Page 52

92
found in the world. The mathematical operations developed to handle transfinite numbers was the greatest achievement of Jain theoreticians. Centrality attached to the operation of base 2 indexing and logarithm for cardinality determination and the ordering reminds one of genaralised continum hypothesis of Geroge Cantor. Why and how did Jains evolve base 2 indexing and logarithm operations for handling transfinite needs to be investigated in future. What formal intuition led them to such a step? Besides theoretical consequences of characterising real continum by three distinct kinds of infinitesimals unlike modern outlook which thinks continum to be populated by one kind of infinitesimal, needs to be pursued further. Perhaps it implies radical revision of the mathematical concept of measure and hence modern measure theory.
JAIN JOURNAL
In conclusion we would say that it is definite that the Jains developed mathematics of transfinite before 800 AD though basic ordering of physical actual infinities could have been developed as early as 300 BC. There remained in Jain tradition a confusion about whether asamkhyāta is finite or infinite. This was a confusion of interpretation by Jain thinkers rather than inherent in the content of transfinite mathematics and physics of the Jains. The confusion comes because of interpretation of calculation of which according to one scheme is finite and according to other schemes is infinite.
As far as content of transfinite mathematics go there exists no difference between asamkhyāta and ananta class of numbers and they are handled similarly but in difference from samkhyāta class of numbers. And the difference between two classes of transfinites is due to ontological considerations. Jaina theory of infinities is radically different in context and genesis from the modern western theory of transfinite numbers. It has an explicit ontology and is not reducible to auto idos unfolding out of primitives, like the notion of set or actual numbers.