Shatkhandagama Pustak 04

Author(s): Pushpadant, Bhutbali, Hiralal Jain, Fulchandra Jain Shastri, Devkinandan, A N Upadhye
Publisher: Jain Sahityoddharak Fund Karyalay Amravati

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could not be counted in terms of any known numerical denomination. This view-point is further supported by the fact that the Jains works fix the duration of a time-instant, and so the number of time-instants in a Kalpa (Avasarpini and Utsarpini) must be finite, as the Kalpa itself is not an infinite interval of time. According to this latter view the Jaghanya-parita-ananta (which according to definition is greater than the aggregate of time instants) is finite.
As already pointed out, the method of one-to-one correspondence has proved to be the most powerful tool for the study of infinite cardinals, and the discovery and first use of the principle must be ascribed to the Jainas.
In the above classification of numbers I see a primitive attempt to evolve a theory of infinite cardinal numbers. But there are some serious defects in the theory. These defects would lead to contradictions. One of these is the assumption of the existence of the number c1, where c is infinite and a limiting number of a class. On the other hand, the Jaina conception that the vargita-samvargita of a cardinale (i. e., e°) would lead to a new number is justifiable. If it be true that the Utkrsta-asamkhyata of the early Jains literature corresponds to infinity, then the creation of the numbers of the ananta class anticipated to some extent the modern theory of infinite cardinals. Any such attempt at such an early age and stage in the growth of mathematics was bound to be a failure. The wonder is that the attempt was made at all.
The existence of several kinds of infinity was first demonstrated by George Cantor about the middle of the nineteenth century. He gave a theory of transfinite numbers. Cantor's researches in the domain of infinite aggregates, have provided a sound basis for mathematics, a powerful tool for research, and a language for correctly expressing the most abstruse mathematical ideas. The theory of transfinite numbers however, is at present in an elementary stage. We do not as yet possess a calculus of these numbers, and so have not been able to bring them effectively in mathematical analysis.
A. N. Singh, D. Sc., Lucknow University.