Mathematics Signifies accuracy and the lack of arbitrariness. Therefore, a philosophy which is not based on the arbitrary will of any `Power’ would need mathematical sciences to explain various happenings of micro and macro level in this cosmos on the basis of the universal laws of nature. Thus, the mathematics is not a luxury but a necessity in the Jaina philosophy.

In general, we see the necessity of mathematics in describing the concepts of non-violence, Karma-theory and the autonomy of the functioning of the universe. In this connection, here we have also noted the importance of modern statistical and quantum mechanical concepts which are present in the Jaina Mathematics.

One may ask-why did Jaina Acaryas spend their time and energy in developing and describing mathematical concepts in the Jaina philosophy ? When we try to find the answer of this question then we would see that they wanted to convey the concepts useful for good life. They knew that for good life one needs an understanding of the truth, a trust in the truth and to follow the truth. To achieve these goals Jaina Acaryas might have felt the necessity of mathematics in following ways :

(a) To describe the criteria of leading good life by way of non-violence by distrubing minimum number of lives.

(b) To describe the autonomy of the universe or cosmos to convey the truth that each soul is creater of its own destiny. Further, to lessen the unnecessary fear they wanted to let us know that all small and big events of the universe happen in an orderly and lawful manner, i.e., the universal laws are always obeyed.

(c) To peovide statistical information to support the journey of a traveller on the path of spiritual progress.

(d) To create the accurate framewok of logic and to establish the consistency of various concepts and ideas.

In the following sections, we shall illustrate these points by taking some specific examples. During such sttempts, we shall also see that vatious concepts of modern mathematical sciences such as Physics, Statistics, Quantum Mechanics are visible in ancient Jaina Mathematics.

Many friends ask me, “Why don’t you eat things like potatoes ? Potatoes have life but so do other green vegetables’ The answer of such a scientific question needs a detailed description of Sadharana Vanaspati as described in Gommatasara.1 There it is explained that the number of lives disturbed for the same food value (calorie and nutrients in modern language) is very large in vegetable like potatoes. Therefore, to minimize the violence on should avoid such food. The point is that without quantitative or mathematical words one can not understand this point.

Non-violence does not limit to disturbing the minimum number of lives. Jaina Acaryas have also classified living beings in various categories (see Ref. 1, 2) based on the number of senses possessed. It is expected even from a householder not to eat the food that has higher form of life, i.e., life having two or more than two senses called Tras-Jiva. Further, a person is advised not to consume the things which adversly affect the harmony of body and mind. All such criteria lead to the exclusion of things like alcohal, meat, eggs, addictive things, etc. from one’s life.

No any super being is required to admisnister this cosmos. The universe is being run by the universal laws. Each constituent of a substance follows definite laws. In modern science every movement of matter and energy of every change is based on the appropriate law of gravitation, the law of conservation of energy, the quantization rules etc. Similarly, Jaina Acaryas also consider the activities of all constituents of this universe governed by the universal laws.

The law of gravitation was operating before the descovery of the law of gravitation. Similarly, even if all the universal laws that govern the universe can not be described in words but the nature flollows these laws. We know very well that when a glass of water falls on a stone then it results into hundreds of its pieces of different size and shape at different locations.

A scientist would say thet though a computer may not solve the equations of motion to predict this breakage of the glass but every thing took place according to the definite laws. It is almost impossible to solve the equations of motion for approximately 1024 glass particles, but the natural computer in herently present with each partcle solves such equations in a fractions of a second and the event takes place in an `orderly’ way.

To us this event appears as disorder but if you look at the laws of Physics and Chemistry then we would find that no any law has been broken and every thing happened in the **`lawful’** way. Jaina Acaryas also say same for all Dravya including Jiva. For Jiva the applicable laws are called the laws of Karma or Karma Theory. In nutshell, the picture of the universe and the universal laws as provided by Jaina Acaryas may be described as follows :

(i) Jiva (infinite) | (ii) Pudgala (infinite) |

(iii) Dharma (one) | (iv) Adharma (one) |

(v) Akasa or space (one) | (vi) Kala (innumerable). |

The volume of the whole space is infinite but the space that contains Dravya other than space is limited and is called Lokakasa ; beyond Lokakasa there is only space (not other five kinds of Dravya) and that aprt od space is called Alokakasa. Lokakasa has a definite shape and size, its volume is 343 cubic Raju and its height is 14 Raju.

(b) There are definite laws of combination of Pudgala Paramanu. Following Sutra describe some of these laws5.

**स्रिग्ध—रुक्षत्वाद्वन्ध:** Snigdha ruksatvadbandhah [33]

**न जघन्य—गुणानाम्** Na jaghanva gunanama [34]

**गुण साम्ये सदृशानाम् **Gunasamye sadrasanam [35]

**द्वयधिकादि—गुणानां तु **Dvyadhikadigunanamtu [36]

**बन्धेऽधिकौ** **परिणामिको च ** Bandhe adhikau parinamikau ca [37]

The meaning of these Sutra (not described here) is clear but it is difficult to interpret these sutra in terms of modern scientific concepts due to the fact that Pudgala Paramanu is the ultimate entity of matter or energy and is not yet realized in moden science. The atom defined in Physics and Chemistry is very big entity and is a combination of many Pudgala Paramanus.

Even Karmic dust of Karmana Vargana is smaller than the atoms and molecules dscribed in Physice/Chemistry. The modern science is progressive and, may be, one day it can raech to a micro-micro level of Karmana Vargana (when ? after 50, 100 or 200….. years? Nobody knows).

At least one point is clearly conveyed by these Sutra : There are definite laws that govern the matter, i.e., nothing is random or arbiturary.

(c) In the following verse weitten by Acarya Kundakunda6 we note that nobody can create or destory any particle or Jiva in absolute sense. Nobody can add any new attribute in any Dravya.

**दव्वगुरस्स य आदा, ण कुणदि पोग्गलमयम्हि कम्मम्हि।तं उभयमकुव्वंतो, तम्हि कहं तस्स सो कत्ता।**

This verse makes says that nobody is creator of any thing in absolute sense and, therefore, in this cosmos the number of lives and the number of Dravya and Pudgala Paramanu can neither decrease nor can increase. Its interpretation in the modern scientific language is simple, it is the law of conservation of energy as well as the law of conservation of each kind of Dravya.

(d) There are definite Karmic laws. For example the following Sutra describe the cause of influx of Asatavedaniya Karma that brings pain and sufferings.

**”’दु:ख—शोक—तापाक्रन्दन—वध—परिवेव नान्यात्म—परोभय—स्थानाम्यसद्वेद्यस्य**

Dukha soka tapakrandana vadha paridevanatmaparobhayasth-ananysadvedyasya (6.11).

This Sutra says that aciities/thoughts of suffering, sorrow, guilt, destruction (killing), crying and weeping towards others and/or self lead to bondage with Asatavedaniya Karma that results into more pain and sufferings.

Similarly, the following Sutra gives the cause of influx of Satavedaniya Karma that brings pleasant solutions.

**भूत—व्रत्यनुकम्पादान—सरागसंयमादि योग: क्षांति शौच मिति सद्वेयस्य।**

Bhutavratyanukampadana saranga sanyamadi yogah ksantih saucamiti sadvedyasya (6-12).

This Sutra says that the compassion, kindness and charity towards living beings including pious persons, actions of discipling mind and body, forgiveness, purity, etc. with some motives lead to bondage with Satavedaniya Karma that results into worldly pleasures and happiness.

For more Kamric laws one can refer to Gomatasara karmakanda.9

(e) For the liberation also there is a definite process, In very logical way it is described by Acarya Kundkund in the following verse.10

**अहमेक्को खलु सुद्धो, य णिम्ममओ णाणदंसणसमग्गो।ताम्हि ठिदो तच्चित्तो, सव्वे एदे खयंणेमि।।**

**Ahamekko khalu sudhdho, ya nimmanao nanadansanasamaggo****Tamhi thido tatcitto savve ede khayam nemi 73**

All these points illustrate that all events happen according to the definite laws of nature and the mathematics is required to describe and explain such laws.

Statistical information available in the Jain literature shows the level of scientific maturity of Jain Acaryas. For example, consider the following information. The number of souls liberated is 608 in every month and 8 Samay. Modern experts of Statistics can digest such concept but an ordinary lay person would say that why this number is same again and again. How is it possible ?

But in modern era of science, an expert knows that for an ensemble of very large number of consstituents under a definite set of conditions various outcomes are almost definite. As an examples, the percentage of molecules evaporated per second from a liquid pot under given conditions remains same. At this point one can again ask that the process of liberation does not happen same in all Yuga or Kala, in Pancama Kala (Kali Yuga) it is almost zero then how is it possible that this number remains 608 even now ?

To seek the answer of this question and to see the consistency we need to consider all regions (5 Bharat Ksetra, 5 Eravat Ksetra, etc. see Ref. 11) and need to note that in one region when there is fifth kala then there can be fourth Kala in an other region and so on such that over all the sum of liverated souls may be same, i.e……,

No of souls liberated = 608

here T = 6 months and 8 Samay

The purpose of this information is manifold. One one hand, it asserts that in some respects overall the universe is a homgeneous system and is operating under definite laws. On the other hand, it shows that the laws of nature or Karmic laws are always facourable to living beings : fevourable in the sense that the net effect is the progress of the living beings, i.e., the direction of flow of the living beings is of spiritual growth. It may be considered as one of the best statements that describe the bright future of all livings beings.

In Gammatsara (Ref. 9) we would find a large number of such statements with numbers. There, we would also see various numbers related with the Bandh, Udaya, Vyuchiti of vatious Karma Prakrti in different Gunasthana. All such numbers and statements may act as road map and mile stones useful to a traveler on the spiritual path.

Number of Pudhala PArmanu existing in the Lokakasa is infinite whereas the number of Pradesa of space in the Lokakasa is innumerable (not infinite). The space occupied by one Pudgal Parmanu is euqual to one Pradesa. The volume of one Pradesa is very small but not zero. These statements look inconsistent on the first reading. However, the following verse of Pravacanasara resolves this apparent contradiction.

”**‘आगासमणुणिविठ्ठं आगासपदेससण्णया भरिदं।”’****”’सव्वेिंस च अणूणं सक्कदि तं देदुभवगासं।।”’**

**Agasamanuniviththam agasapadesasannaya bhanidamSavvesim ca anunam sakkadi tam dedumavagasam 140**

The above verse says that though each Pudgal Parmanu occupies one Pradesa but one Pradesa is able to accommodate all Pudgal Parmanu. Those who are familiar with Quantum Mechanics and the concept of Bosons13 can digest this concept very well. For lay persons in this field, a simple example of light may be helpful. In one room the light photons emitted from one lamp resides but in the same room the light photons emitted from two or many lamps can also reside.

In this connection it would be interesting for a quantum physicist to note the existence of concept of Fermions in the Jaina literature. It is well known in Jaina literature that though at one Pradesh of space more than one Pudgal Parmanu can reside but at one Pradesa only one Kalanu can reside as if Kalanu is a fermion.(13)

Again one may ask what is the spiritual purpose of such information ? Answer is simple : Such an information is needed for the sake of completeness of the description of cosmos that helps in having the true and basic understanding of the universe and our place in it. However, one can also extrapolate such concepts for other types of spiritual advantage. One can learn to co-exist to-gether.

This information reveals that the coexistence of all six kinds of Dravya at one place is possible. Even at the same place more than one soul can coexist. In Siddhalaya many liberated souls coexist in the same space without interfering each other.

We have seen various aspects of the usefulness of Jain Mathematics. All these aspects are helpful in improving our life. It may be important to count the calories and nutrition value of the food. It may be important to know the cost of food at the time of buying. But it is meaningless if the food is not used for eating. In the same way the spiritual knowledge with mathematical details is highly useful but is would be meaningful only when one assimilates the truth and adopts in one’s life.

1.Acarya Nemicandra, Gomatasara Jivakanda

2.Acarya Umasvami, Tatvartha Sutra. Chapter 2

3.Acarya Umasvami, Tatvartha Sutra, Chapter 5

4.Pandit Dyanata Rai, Carca Sataka, Verse No. 12

5.Ref. 2, Sutra 5.33 to 5.37

6.Acarya Kundakunda, Samayasara, Verse 104

7.Ref. 2, Sutra 6, 11

8.Ref. 2, Sutra 6, 12

9.Acarya Nemicandra, Gomatasara Karmakanda

10.Ref. 6, Verse 73

11.Ref. 2, Sutra 3.27 and 3.37

12.Acarya Kundakunda, Pravacanasara, Verse 140

13.Paras Mal Agarwal, Quantum Siddhanta (Rajasthan Hindi Grantha Academy, Tilak Nagar, Jaipur, 1973

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Jain Mathematics