Jain Studies And Science: Numerate, Innumerate and Infinite

Mahapragya, while giving a discourse on the arithmetic of the Vedic Era, mentioned about the specialty of the Sirsaprahelika, the maximum number, described in the Jain canonical texts. Yajurveda, 17/2 mentions up to Mahasankha in which 20 digits are included. An intermediate number of ten kharabas is also prominent, which is obtained by raising 10 to the power of 12 i.e. 10¹². In that mathematical text, the counting progress as ikai (unit), das (ten), shata (hundred), sahasra (thousand),...., kharaba, das kharabas,.... shankha, das shankha, mahashankha. Whereas, in the Jain ancient literature, the biggest number, called Shirsaprahelika, comprises 54 digits followed by 140 zeros. The maximum number, therefore, contains 194 digits. However, in another text, the Shirsaprahelika is recognized as a number having 250 digits. By any means, the biggest number as quoted in the Vedic era, is very small in comparison. The modern sophisticated mathematics also acknowledges the importance of large numbers such as Shirsaprahelika and has regarded it as an important discovery. We shall therefore, discuss the interesting part of Jain Arithmetic regarding the maximum and the minimum numbers.

In the Jain Texts, there is description of the numerate, the innumerate and the infinite in defining the time periods. The ultimate finest unit of time has been called as 'Samaya' (instant). The period from an instant (Samaya) up to the Shirsaprahelika is calculable, thus all numbers in between are called numerate. It is interesting to find that among the numbers smaller than the Shirsaprahelika, an intermediate number, Eighty Four Lacs, has been given special importance.

In deriving the biggest number, first the counting up to a number of eighty-four lacs was done. This number was named 'Purvanga'. After this, eighty-four lacs have been multiplied by eighty four lacs, i.e. eighty-four lacs have been squared. The number thus obtained was named 'Purva'. When the number 'Purva', is multiplied by eighty-four lacs again, the result was called iTrutitanga\ In this sequence, there are twenty eight such places. Progressing in this manner, the ultimate number,'Sirsaprahelika' can be denoted as follows:

(8400000)²⁸ or (84xl05)²⁸

Although the time-period up to the 'Sirsaprahelika' is sufficient for all practical purposes, the calculable and numerate periods beyond this limit are described using simile (upama), such as - pallayopama, the sagaropama, the avasarpini, the utsarpini. They are employed in mentioning the longevity of bios in various life-forms. The simile measure of time has been further described later in this chapter. The periods of time, beyond measures have been treated in terms of the 'innumerate' and the 'infinite'. It is interesting to note that the numerical building blocks beyond eighty-four lacs have been obtained by multiplying the preceding number with 84 lacs, for example,

84 lacs years = 1 purvanga
84 lacs purvanga = 1 purva
84 lacs purva = 1 trutitanga
............................................................
............................................................
84 lacs chulika = 1 shirsaprahelikanga
84 lacs shirsaprahelikanga = 1 shirsaprahelika

The decimal system has been used in the arithmetic of the Vedic and the Jain Schools. Both are similar to a large extent, except for the fact that in Jain arithmetics the numbers beyond 84 lacs are derived by repeated multiplication of the number 8400000 (eighty-four lacs). That is, the further increment is not in units but en block by eighty-four lacs.

In this reference, the description of the number of hellish bios in the Sthananga-sutra is observable. Mahapragya had enough audacity to deliberate on this hitherto untouched topic. However, there are two lines of thoughts which have slight difference in their respective treatments.

1. Shvetambara tradition

Here, three technical terms have been mentioned, which need detailed discussions -

• Kati
• Avakatavya
• Akati

1. The meaning of the word Kati is - how many? Here the numbers starting from two and increasing up to the largest numerate are all indicative of 'how many'.
2. The meaning of Akati is innumerate (finite, but uncountable) and infinite.
3. The meaning of Avaktavya (inexpressible) is the digit 'one'. On the basis of the Uttaradhyayana and the Anuyogadvara aphorisms, it has been mentioned that the number 'one' has not been regarded as the calculation number. In the Jain canonical texts, the minimal calculation number has been recognised as 'two'. Its exposition is necessary, and that has been given in the following pages.

In the Digambara tradition, the word 'Kadi' occurs in place of the word Kati. Its meaning has been expressed as that of 'Kruti'. The mathematical exposition of the word Kruti is different from that in the Shvetambara tradition. Digambars, therefore, use another term 'Nokruti' in their classification of numbers. Their three terms are -

• Nokruti
• Kruti
• Avakatavya

A number which fulfils following three criteria is termed as Kruti -

1. Which increases after being squared,
2. When the difference of its square and the number itself is greater than the number,
3. When above difference is squared and preceding number is subtracted again, result must greater than the original number. This incremental pattern must continue as many times as this process is repeated.

The difference among these three terms can be understood with the help of examples -

Let us take an example of number 'one'.

1. There is no increase when it is squared
   (1)² =1x1 = 1
2. When, out of this square the number itself is subtracted, it results
   in null-
   1-1=0
3. There is no increase if the difference so obtained is squared again-
   0² = 0 x 0 = 0
   0-1 = not possible

Hence the number 'one' does not fall under the criteria of Kruti and thus, has been called nokruti.

Let us now take an example of number 'two'.

1. An increase is seen on squaring the number two, hence it cannot be called nokruti:
   (2)² = 2x2 = 4
2. On subtracting from this, the number itself, the original number is obtained:
   4-2 = 2
3. When this resulting number is again squared and the square root is subtracted, there is no increase in number:
   (2)² = 4, and
   4-2 = 2

From these equations it is found that the number 'two' is neither Kruti nor nokruti, hence it is called Avaktavya (inexpressible).

Let us take an example of number 'three'.

1. There is increase on squaring the number 'three' onwards:
   (3)² = 9
2. On subtracting the original number from this number, result is incremental:
   9-3 = 6
3. On repeating, in this sequence, the increment cumulates:
   (6)² =36
   36-6 = 30

From the above description it is known that

• The number 'one' is nokruti.
• The number 'two' is Avaktavya.
• The numbers 'three' and the onwards numbers are Kruti.

Mahapragya concedes that this difference, though vital, still establishes that in Jain arithmetic the increment in number is obtained by the mathematical operation of squaring and that the counting does not start from the digit 'one'. In both the traditions, the Shvetamber and the Digamber, 'Samkhyati itisamkhya' - that which could be divided, is a number. From this point of view, the minimum number starts with 'two'.

Now, let us dwell on the basic question, why the powers of eighty-four lacs were employed in counting of large numbers? Here could be three probable reasons for this -

1. In Jain theory, there are eighty-four lacs possible originations or classes (yoyni) of bios (living beings). The bios transmigrate again and again within these classes during the cycles of births and rebirths, till it does not become free from the bondage of Karma. Perhaps this limit had rendered the counting beyond this number as redundant.
2. In view of the digits, the number 84 has the specialty that on multiplying it again and again, the resulting numbers are such that the sum of their individual digits is always nine (9). As such the number 9 is regarded as divine and holy. For example, if we square the number 84, we get;

   1. 84x84 =7056;
      The sum of the digits of the above number is (7+0+5+6) = 18; (1+8) = 9
      Similarly,
   2. 84x84x84 = 592704;
      The sum of the digits of the above number is
      (5+9+2+7+0+4)=27; (2+7)=9
      Similarly,
   3. 84x84x84x84 = 49787136; (4+9+7+8+7+l+3+6)=45; (4+5)=9
      This pattern continues up to the'shirshprahalika' and beyond. This is just indicative possibility; no such explanation is available in Jain Agam literature.
3. In view of the author, the most probable reason could be that the mathematics in that era was more practical than theoretical. For very practical reason of reducing the numerical terms between the figure of 841acs till the shirshprahelika and extending up to innumerate, the mathematicians of that ancient period decided to increment in the powers rather than addition and multiplication. Let us look at the following table -
   Numerical terms between 1& 100 = hundred
   Numerical terms between 101 & 1000 = nine hundred
   Numerical terms between 1001& 10000 = nine thousand
   ...............................................................................................
   Numerical terms between 8400000 & shirshprahelika =?????

The entire purpose thus becomes clear. It was the ingenuity of those scholars who circumvented the cumbersome problem of unmanageably large number of terms at the higher end of the counting. So, simple unity increment or multiples were replaced with the powers which reduced the number of numerical terms between 8400000 & shirshprahelika to only 28! How practical!

We have discussed the special features of Jain Arithmetic with regard to the biggest numerate number, the importance of the number 84 Lacs and the probable reasons for squaring the 84 Lacs in expanding the numerates. These discussions establish the independent development of Jain Mathematics. There is one more aspect of Jain Arithmetic about 'zero' which is equally interesting - whatever exists may be near to or tending to zero but cannot be 'pure zero'. Entities and realities can be infinitesimally small but cannot be pure zero. This concept is another example of their practical approach towards mathematics. It is a matter of serious contemplation as to why 'zero' finds no place in the Jain Agam Literature?

Tending to Zero but not Zero

The mathematician, Euclid assigned no dimension to the finest point of the space. Although, any dimension (length, breadth and height) is obtained by joining these dimensionless points in the space. Similarly, the scientist, Newton, used to treat the finest particles of matter, the atoms, as having zero dimensions. But Einstein, while making use of the Gaussian geometry, has compared the finest point of space with other micro particles. Similarly, in Jain literature, the finest forms of the six realities (mattereals) have been compared and a definite relationship is being established. For instance, mutual equivalence has been established among the finest part of matter, paramanu - the dion, the finest part of akash, Pradesh, and the finest part of time, samaya. Hence the finest part of elementary tattvas of nature cannot be reduced to zero. They all may be defined to be infinitesimally small - tending to zero - but cannot be zero.

Let us try to know the fact seriously as to why there is no place given to the digit 'zero' in the Jain canonical literature?

As per Jain philosophy, once an entity (Tattva) is reduced to 'zero', it can not reappear into existence. In other words, no mattereal which exists in this lok can reduce to zero either in quantity or in quality. Hence 'zero' cannot have existence in the realm of Dravyas or in relation to Dravyas. No basic Dravya (mattereal) can vanish or can be destroyed. Jain philosophy, while dealing with only eternal matter or phenomenon, has disregarded the use of digit 'zero' in counting or in mathematical operation. For instance, a black parmanu (dion) on association with other parmanus may increase or decrease the nature of black colour but it will never completely loose the original property of colour i.e. colour property will not reduce to 'zero'.

We have seen earlier, number one has been excluded from the counting. Rather, it is treated as the nearest to zero or tending to zero - very near to zero but not zero. Similarly, Jains rule out the existence of a 'superior infinitely infinite' known as Asadbhav in Jain terminology. Both presumptions of non­existence of absolute zero and absolute infinity establish that the entire mathematics lies between these two extremes.

The evolutionary or deductive methodology of modern mathematics necessarily need to have 'zero' quantity (digit), because of its subsequent applications - to segregate existent and non-existent, real or imaginary quantity etc. This was not the case in olden age. As such the necessity to use 'zero' did not exist in that era's philosophy. Instead of zero, Jain philosophy considered the term 'tending to zero' as more appropriate. Such an example is found in relation to the measurement of heat by the modern scientists -

There are two units prevalent for the measurement of temperature -

1. Centigrade scale - °C (Degree Celsius)
2. Absolute scale - °K (Degree Kelvin)

When we say that water freezes at 0°C, it does not mean that the heat energy has reduced to zero. Below 0°C, negative temperatures are used such as -50°C - temperature of Antarctica. This, however does not mean negative energy. Therefore, scientists developed another scale of 0°K which means absolute zero energy at 0°K. In this absolute scale of heat, the ice temperature of 0°C is referred as 273°K above absolute zero. Accordingly, -50°C becomes 223°K. There is no negative value in this scale. Liquid gases exhibit temperatures below 273°K. In the experiments of superconductivity, the scientists were successful to achieve temperatures as low as 3-4 °K. But then, how to explain 0°K? Entity ceases to exist at that instant. 0°K is thus treated as absolute zero and there is a clear conception in the field of science that this temperature (heat) is only notional and not real. At absolute zero, no properties have any relevance in physics.

Another example from the world of science is the properties of a photon -

1. Photon is treated as a bundle of energy with zero-mass (massless) at rest.
2. Photons travel at the speed of light and acquire mass which cannot be measured due to its motion.

Once again, the scientists say that the absolute zero mass is only notional and even at rest it only tends to become zero. These two examples establish a close analogy between science and Jainism.

Many scholars have regarded zero to designate the absence, as this does not come under the provisions mathematics, hence its description is not given here.

Number System

Quantitatively, numbers are classified in three main categories depending on their countability - numerate, innumerate and infinite. These are further divided in three sub-categories - Minimum, Medium and Maximum. Amongst these nine categories, only eight were adopted by the mathematicians and the last sub­category of maximal infinity is discarded on similar grounds upon which the zero was discarded. Their further subclasses are twenty, which are as follows -
There are three classes of the numerates -

1. Minimum (minimal)
2. Medium, intermediate, middle or intervening
3. Maximum (maximal)

There are nine types of the innumerate -

1. Minimal lowest innumerate,
2. medium lowest innumerate,
3. maximal lowest innumerate,
4. minimal average innumerate,
5. medium average innumerate,
6. maximal average innumerate,
7. minimal highest innumerate,
8. medium highest innumerate, and
9. maximal highest innumerate.

There are eight types of the infinite -

1. minimal lowest infinite,
2. medium lowest infinite,
3. maximal lowest infinite,
4. minimal average infinite,
5. medium average infinite,
6. maximal average infinite,
7. minimal highest infinite, and
8. medium highest infinite

The ninth possibility of 'maximal highest infinite' has been treated as absolute infinity and thus discarded as non-existent. This is termed as 'Asadbhav'. This presumption was premeditated as the absolute zero was considered non-existent. By simple analogy, absolute infinity is not possible; any thing can only tends to become absolutely infinite.

Infinitum and Innumerate

Various similes were employed in the process of building the counting patterns running into innumerate and further to infinite. A description of four concentric cylinders (cups) namely, Anavasthith, Shalaka, Pratishalaka and Mahashalaka is very interesting. An imaginary process of emptying the seed contents of these cups into the volumes of Jambu Island, Lavan Ocean, Dhatkikhand and so on; results in the seventeen categories of innumerate and infinites as enlisted above. The process of selecting lowest, highest and median is very similar to that employed in the modern statistics, where the smallest and largest samples of data are excluded from the series of numbers used for calculating the median or average.

The word infinite in the Jain canonical literature has been extensively used. The meaning of this word assumes slight variations with the changed context. Mahapragya defines that which does not come to an end is called infinite. This definition fits in all the references where this word is employed.

Numerate is subject to calculations and counting. Innumerate is not subject to calculations but it is used for comparison, hence it too, is not endless. Various references where 'infinite' is used are related to the characteristics of six fundamental mattereals, like -

• Jiva (bios) consists of infinite pradeshas
• Infinite dions (paramanu) combine to form quadons (skandhs)
• Group of infinite quadons makes an octon (behavioural paramanu)
• Infinite Samay (instant) elapses in an Avalika
• Space is comprises of infinite Pradesh.

Besides the eight classifications of 'infinity', words like 'uni-infinity* and 'universal infinity' do appear in the Jain texts. These are used with respect to space and time both. In space, 'uni-infinity' is used for one-dimensional lengths and 'universal infinity' is used for spatial extensions. Similarly, in time, 'uni-infinity' is used for either the past or the future and 'universal infinity' defines the entire time frame.

We find a specialty in Jain mathematics that the numerate and the innumerate have been placed together. In the Bhagavati-Sutra, a table has been given which formulates the units of time -

Saroaya	Smallest unit
Innumerate samaya	one avalika
Numerate avalika	one respiration
One breath - respiration	one prana
Seven pranas	one stoka
..................................................	...............................................
Forty eight minutes	one muhurta etc.

An outstanding fact revealed by this table is that a numerate avalika embrace infinite samay, That is, even infinite could be a part of the finite.

The number scale, as used in Jain texts, starting from numerate and ending at infinitum comprises all the constituents of modern mathematics, highlights of which are -

• Use of arithmetical progression to build the number scale up to 84 lacs,
• Use of geometrical progression to continue counting beyond 84 lacs.
• Use of statistical quantities like maxima, minima, median and average.
• Rejection of absolute zero and absolute infinity.