Jain Journal 1990 07

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

Page 18

JULY, 1990
From the above five relations it is clear that property of induction is generally valid for samkhyāta numbers but is invalid for middle asamkhyāta and middle ananta numbers. This is significant as such relations will hold only if middle asamkhyāta and ananta are regarded as transfinite numbers. Property of induction seems to hold for us and uAA numbers but this is not a case as us and uAA are each a class of only one number hence relation (2) and (4) are extraordinary. A. N. Singh32 had regarded relation (2) and (4) as a fallacy in Jain theory of transfinite numbers, if
e existed one such theory. It needs to be noted that this so called anomaly is because of particular relation that Jains posited between jpa and us and between jpa and uAA. Anuyogadvāra sūtra33 explicating meaning and value of jpA gives an analogy of a platform which when being filled by hog-plums (anvala) reaches a point when a hog-plum is left which can not be contained in that platform. In a sense nth hogplum spills out of a bounded set. Number jpA is analogous to this nth hog-plum which spills out of a bounded set of asamkhyāta numbers or a platform. And the number of hog-plums which can be contained in the platform are (n-1) or highest samkhyāta is (jpA-1 )= us. Thus jpA=us +1 and jpa=uAA+1 have a special semantic significance other than the mathematical property of induction and hence there is no real anomaly. It is quite significant that the property of induction does not hold for all kinds of asamkhyāta and ananta numbers suggesting strongly that these classes of numbers are transfinite numbers radically distinct from samkhyāta (finite) numbers where property of mathematical induction holds good.
Even if induction does not hold for asamkhyāta and ananta numbers Jains evolved mathematical theory of operations which holds good for all the three primary categories of numbers. These were :
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A. N. Singh (1942). He regarded relation (2) & (4), as anomalous in the context of number relation given in Table I and not in the context layed in these five formulas. Anuyogadvara sutra (sutra 508 or p 648]/se jahanamae mance siya amalaganam bharie, tattha ege amalae pakkhitta seavi mae, anneavi pakkhitte seavi mae, anneavi pakkhitte seavi mae.evem pakkhippamanenam pakkhippamanenam ho hi seavi amalae jamsi pakkhitte se mamcae bharijjihii je tatth amalae na mahii/ '(suppose) there is a platform filled with log-plum fruits when one hog-plum fruit is placed, that also is accomodated. Yet another is placed, that also is accomodated ; when (hog-plums) are thus placed again and again, there will be a (last) hog-plum which being placed, the platform will be completely) filled (and there will be another hog-plum which will not find any place there). Number of a hog-plum which can not be accomodated will be (a).