Logical Analysis of `Minimum' and `Maximum' in Jainism

Logical Analysis of `Minimum' and `Maximum' in Jainism
Samani Chaitypragya

Assistant Professor, Jain Vishva Bharati University, LAdnun-341306 Rajasthan

ABSTRACT '''The present study aims at defining, descrbing and logically analysing the concept of minimum and maximum in Jainism. It hypothises that the concept of minimum and maximum is relative. Through method of logical analysis of the basic four questions related to the concept minimum and maximum as described in Jaina texts and vatious illustrations it concludes that the concept of minimum and maximum in Jaina philosophy is relative.'''

Introducation :
The Concept of infinity itself is riddled with paradoxes according to moore.1 They are as follows : The paradoxes of infinitely small, the paradoxes of infinitely big, the paradoxes of one and many, the paradoxes of thought about infinite. All these paradoxes of infinite occur if we consider infinitely small and big as absolute. Jaina theory of relativity predominantly plays its key role in almost all the aspects of its philosophy. It is also true in the case of Jaina Mathematics. Jaina canons have a detailed theory of Aithmatics and Computations. The unique concept of Jaina mathematics is that neither minimum nor maximum is zero in Jainism. In fact the concept of zero does not exist in Jainism. The number or counting starts from one and not from two. The minimum number technically called jaganya sankhya is valued as two and the maximum technically called as ananta is or infinte. The present study is an effort to study and search the logical basis for these contentions. It deals with four basic problems of number theory in Jainisms as follows : 1. Why minimum counting starts from 2 ? 2. Why``O'' is not a number ? 3. Can infinte be classified ? 4. Is Absolute infinte possible ? Before solving the above problems, let us have a glimpse of Jaina theory of numbers in brief. The numbers in Jainism are divided into three categories : 1. sankhyata, (Numerable) 2. asankhyata, (Innumerable) 3. ananta, (Infinite). The numerable numbers are again divded into three catogories : minimum, intermediate and maximum. The innumerable numbers are also divded into three subsclasses : low-grade, self-raised and innumerable-innumerable. Each of the three innumerable sub-calsses is again divided into minimum, intermediate and maximum. The infinite is also divided first into three sub-classes : low-grade, self raised and infinite-infinite and then first two sub-classes, infinite-infinite, is minimum and intermediate only, there is no maximum in the case of infinite-infinite. The chart no. ` from that which is2 is of a help to have a overview of Jaina way of classification of numbers.

Chart 1 Classification of Numbers in Jainism
The classification given in the chart-1 does not stop here, each of them is again divided into three kinds; minimum, intermediate and maximum except numberable. Thus from this classification the minimum number is (countable) `sankhya', jaganya sankhyata, (Minimum Numerable) which has been valued as `two'. Digambara and Swetambara texts differ in the classification of numbers. The swetumbara texts such as Thanam3 the classes of numbers are as follows : kati - 2 to numerable akati-innumerable to Infinite avaktavya - one Here kati refers to Numbers, akati refers to beyond numbers, avaktavya connotes not being a sankhya, as ``one' is not a sankhya, one is a vaktavya. The Digambaras classify the numbers as follows4 krti - 3 and more nokrti - one avakravya - two With all these differences, there is yet a general concept acceped in the tradition that, states that minimum countable number is two and 0 is not a number and infinite is relative in nature.

1. Why Minimum Counting Starts From 2 ?
`One is not a (countable) `sankhya', this is a common concept in Jaina text. As sankhya means to count. The counting starts from 2. Thus the minimum number remains 2, As seen in Anuyogadaraim and Lokaprakasa the reason given for this varies from text to text. The Anuyogadaraim5 text argues that 1 is number but not in the category of counting. That is to say it is not a countable number because it is not practically counted in day to day life, Laka[rakasa6 on the other hand contends that since the square of the 1 does not increase the number i.e. (1 x 1 = 1) as in the case of two (2 x 2 = 4, 3 x 3 = 9) and more thus it is not considered as (countable) `sankhya'. That can be explained as follows : kriti - ax2 - x2)2 > x2 3 and more nakriti x2 ~ a > x + y ) ay is one or more than one — one avaktavya neitherapplkable  z Thus avaktavya remains the minimum countable number.

2. Why ``0'' Is Not A Number ?
Zero is not considered as a number in Jainism. It is common in both the traditions that they donot consider zero a number. The existence of an entity, cannot be in zero. This absence of existence is an impossible event, never to happen in this world. The modes might change, but `existence' is blessed with eternality. It is not possible, that `what is' becomes `is not'. If one loses its existence, they cannot come to being again. By this process the world will become zero one day. This is an unreal dream. The dispute zero can also be analyzed from a scientific view. A simple scientific value of heat could be a good example, to explain the unit of zero-value, quoted by Mr. Gelada ``Generally we start from zero degree Celsius and start measuring. But when it is frozen at 00C, does it mean the heat is totally absent in it ? This is not so. Thus we use the absolute scale of heat which is Kelvin' According to this ice possesses 2730K heat. 0K means the ultimate zero. In the field of science, a postulate has been given that there is nothing equal to 0K. It is not a reality''7 The concept of minimum in Jainism is relative. In order to understand how is the concept of minimum is relative in jainism we must understand what relative and absolute minimum is. The figure below no. 1 clearly depicts what is meant by relative minimum.

Relative and Absolute Minimum
The lowest point over the entire domain of a function or relation is absolute. Note : The first derivative test and the second derivative test are common methods used to find minimum values of a function and maximum in Jainism which are relative as in following figure. Thus the concept of minimum being relative can be deduced. Minimum depends on the object for which it is assigned as depicted in Table no. 2.

Table No. 2 Logical basis for value assigned to minimum
Nature of Objects for which			Minimum value in Minimum value is Assigned			Numerical form non existent						0 For single existent					1 For multi existent					2 from the above table and discussion it can be concluded that since the concept of minimum in Jainism is relative, it has not considerd O or 1 as minimum countable number as shown in table 2

3. Can Infinite be classified ?
The infinite has always stirred the emotions of mankind more deeply than any other question not only because the infinite has stimulated and fertilized reason as only atew few other ideas have : but also the infinite, more than any other notion is the need of clarification. The word ananta is etymologically defined in savartha siddh {8 as ``avidyamanonto yeisham te anantha'' that which does not end is infinte where as Dhavala defines it as ``jo rasi egegar7ve avijjamane, nittadi so asankhejja, jo pulta na samappai so asl ananto The set of numbers that can be reduced to empty set by removal of each of its member subsequently is known as innumerable, where that wich never gets emptied is infinite. If we compura the classification of Infinite with innumerable. The highest value according to Jainism is intermediate-infinite-infinite.

There are eight kinds of infinite
The concept of infinite (ananta is relative in Jainism. Infinite is of nine types as stated in anuyogadaraim9 which clearly stated that infinte in Jainism is mainly redative. The maximum value according to Jainism is Infinte (ananta). In Jainism Infinite is defined as that amount which does not come to an end. Relative Maximum, The highest point in a particular section of a graph.10 Relative Maximum, = local Maximum, Global Maximum = Absolute Maximum The highest point over the entire domain of a function of relation. The truly infinite does not exist.11 As Ali Enayat and Roman Kossak opines, Non absolute concept of 1 is model dependent in contrast to the is something absolute 1 which is not a set of all sets.

4. Is Absolute Infinite Possible ?
From the comparison of the classsification of infinite with ihe classification of infinite : innumerable is classified into nine types where as infinite has been in to only eight types. The ninth type maximum innumerable innumerable similar set of infinite) is absent or mising which seems to conceptually highest. value. This absence of the (maximum infinite infinite in the naturaal classification of numbers prove s the absence tof absolute infinite. Cantor12 contends that infinite is not possible, because as soon as it is symbolised, it turns to be finite. Therefore the (truly) infinite is impossible, n etc. all for some reason are finite. Inconsistant totalities which on the ground of understanding do not really exist. For ex. and sets of all sets. Thus infinte is not possible. But this must be understood only absolute infinite Sense. Because sets of all sets is the other term for Absolute infinite. More over if n, etc. be a finite, then there would be no difference between infinite and an infinite set. Hence they must be accepted as relative infinite. Wittgeinste on the other hand Factual world-(Infinite what could be said) = S Possible world-a infinte could what be shown) = A to I Logical world - a infinite what could be thought) = liu / x13 Accoding to Michel Potter14 u Every finite set is countable where infinty is countble but not finite. If the function A onto B. A is countable then B is countable. A set is said to be countably infinite if it is both countable and infinite. The set of natural numbers is countably infinite proposition x  countably infinite. For proof consider the funcation which maps any finite string n1, n2 ..........n Thus to exclude or overcome the fallacy of Law of excluded middle Moore in his the infinite proves it though a syllogism15. If some is true then no  become false There exists some infinite Therefore no infinite is false The recent concept of relative and absolute infinite would assist the Jaina conept of infinity for better comprhenstion. According to Ali Enayat and Roman Kossak' A set is absolutely uncountable when this property is not in the model but in real world can be studied by non-formal mathematics a i.e it is an infinite set that is by contrast of something that is model dependent)*'' Non absolute concept by contrast is something that is model dependent. The eight kinds of infinte explained in Jaina scriptures are relative. The ninth which goes to absolute is absent.

Conclusion
Thus the study Concludes that The concept of minimum and maximum in Jainism are relative as it is Realistic philosophy.
 * O does not exist. hence cannot be minimum
 * Since infinite is also relative there can be its types which can be proved through diagrams encompassed.
 * The existence of absolute infinite must also be admitted at least in the logical world.
 * As Ali Enayat and Roman Kossak opines, Non absolute concept of I is model dependent in contrast to the is something absolute I which is not = x/Ina = a set of all sets.

Reference :
1.A. W. Moore, The infinite, Routledge London, 1990

2.Nathmal Tatia which is, english translation of Tattvartha Sutra p. 266 c.f. Anuyoga daraim 596-603 commentary p. 332.

3.Thanam, 3.7 ed by Muni Nathmala, Jain Vishva Bharati, Ladnun, 1976 tiviha neraiya panntta, akati sancita, akati sancita, avaktvy sancita.

4.Commentary of Thanam ed by Muni Nathmala, p. 261

5.Anuyogadaraim, 574 vrtti of haribhadra, p. 109.

6.Lokaprakasa, 4. 310-312

7.Mahavira Raja Gelada, Jain Vidya Aur Vigyan, P. 219.

8.Savartha siddhi, 5.9.275

9.Anuyogadaraim, 515-19

10.Note : The first derivative test and the secon derivative test are common methods used to find maximum values of a function.

11.A. W. Moore, The infinite, routledge, 1990, 2nd 2001 p. 206

12.Cantor, Goerg, Contributions To the Founding of the Theory of Transfinite, Numvers, Trains. Philip E. B. Jourdain, New York, 1955

13.A. W. Moore The Infinite, 2001, p. 198

14.Set Theory and Its Philosophy, Oxford University Press, Oxford, 2004, p. 113

15.A. W. Moore, Infinite p. 209

16.Contemporary Mathematics, Nonstandard Model Of Arithmatic and Set Theory, American Mathematical Society, 2003, p. 2q

अर्हत् वचन जनवरी—मार्च २०१२