by Manikant Shah

Some time back D.P. Agrawal had posted a story on this website, "Ancient
Jaina Mathematics: an Introduction," mainly based on George Gheverghese's
*The Crest of the Peacock: Non-European Roots of Mathematics*. There were
some critical comments on it. This time I am giving a summary of L.C. Jain's
essay. LC Jain is a well-known mathematician and an authority on Jain Maths.
(The main source and LC Jain's address are given at the end). This I hope would
encourage informed discussion.

There are various criticisms levelled against the present Euro-American Western
Science characterized by particular-ism and reductionism that began with Descartes,
Newton, Copernicus and Galileo in the 16th-17th century, as it tries to set
the standards for knowledge in other fields. It is being realized more and more
that the claims of science that it alone provides valid knowledge and that such
rational knowledge began only in the 17th century are turning out to be baseless.
This type of science had mainly attacked the religious knowledge around the
world as irrational, illogical and full of superstition. But now it is slowly
beginning to emerge that science took a very myopic view and that knowledge,
religious or otherwise, could not be condemned in its entirety for it contained
lots of knowledge that was as good as any science. In this light L.C. Jain looks
into the canonical writings of the Jain religion. Jainism is one of the world's
oldest living religions, as old as the *Vedic* religion. It is generally
identified with Vardhman or Mahavira, a contemporary of Gautam the Buddha Mahavira
was only the 24th Tirthanker in a long series of inspired prophets or seers
(Tirthankers). The Jains believe in conquering the flesh in order to attain
that supreme purity which leads to infinite knowledge, infinite happiness and
infinite power, very similar to what other religions of India have also sought.
Although Jainism and Buddhism both accept the authority of revealed knowledge
and transcendental experience, they deny the authority of the Vedas, particularly
the ritualistic portions. So they are considered as unorthodox in Indian religions.
In time the Jain religion developed into two different branches, the Digambars
and the Shvetambars. Digambars insist on extreme asceticism and do not even
care to cover their bodies taking *ambar*, the sky to be the natural cover.
The Shvetambars however put on white robes to cover their bodies.

L.C. Jain is the Honorary Director of the Acharya Vidyasagara Research Institute,
Jabalpur. His specialization is applied mathematics and Jain mathematics. His
works include publications in unified field theory, history of Indian mathematics
and history of sciences. In his paper *Scientific Ideas in Jaina Canonical
Writings and Traditions*

(* In Thought and Culture in India (from c.600 BC to c. AD 300*), he mentions
that the total Jaina canonical writings are in twelve *angas* (portions).
The twelfth *anga* is regarded as the most voluminous, difficult to understand
and intricate, comprising the mathematical theory of *karma*. This knowledge
is termed traditional (*agama*). It is also called as *sruta* (heard
or scriptural). The word *purva* refers to the knowledge that existed in
India even before Lord Mahavira. Out of the fourteen *purvas* contained
in the twelfth *anga*. The Digambara Jaina School has only a small part
from the second and the fifth *purva*.

Jain points out that the theory of *karma* could be compared with this
century's developments in systems theory and cybernetics. Both the schools,
Shvetambara and Digambara, developed scientific ideas, not only through intuition
but also through logic, conforming to their cosmological and *karma* theories
through post-universal, mathematical details. They had geographical, astronomical,
cosmological units along with *karma* theoretic units; and the Digambars
had symbolico-mathematical units in addition. LC Jain in this paper tries to
look into the scientific ideas in the Jain canons under the following heads:

- Mathematical Ideas
- Topological Ideas
- Cosmological, Astrological and Geographical Ideas
- Ideas about Material particles and
- Systems theoretic and Cybernetic Ideas.

LC Jain tells us that Mahaviracarya (c. 9th century AD) who belonged to the Digambara Jaina School spoke highly about mathematics. According to Rangacarya his classification of arithmetical operations is simpler and his rules fuller than those of Bhrahmagupta. He also gives a large number of examples for illustration and exercise.

David Eugene Smith said that the works of Brahmagupta, Mahaviracarya and Bhaskara might be described as similar in spirit but entirely different in detail. For example, all of these writers treat the areas of polygons, but Mahaviracarya is the only one to make any point of those that are re-entrant. All of them touch upon the area of a segment of a circle, but all give different rules. The shadow problems, primitive cases trigonometry and gnomonic, suggest a similarity among these three great writers, yet the writings of Mahaviracarya are much better than those of either Brahmagupta or Bhaskara. Mahaviracarya gives correct rules also for the area of a triangle as well as of a quadrilateral. He also inferred that if a sum is to be distributed to none, the sum remains unaffected.

E. T. Bell has remarked that the rule of signs became common in India after
their restatement by Mahavira in the 9th century. The first clear recognition
of imaginaries was Mahavira's extremely intelligent remark that, in the nature
of things, a negative number has no square root.

There are many terms that are in some manner related to only Jain mathematics.
LC Jain enumerates some of them as:

Parikamma (Parikarma):

LC Jain tells us that besides certain other meanings,Parikarmaalso means a special type of mathematics or calculation (samkhyana). Mahavira mentions that it includes eight types of fundamental mathematical operations, which are:pratyutpanna,bhagahara,varga,vargamula,ghana,ghanamula,samkalitaandvyutkalita. These were used in thekarananuyogagroup of works and in Karma theory.

Rasi:

L. C. Jain in one of his articles on the topic, 'Set theory in Jaina School of Mathematics,' enumerates that the synonyms ofrasiaresamuha,ogha,punja,vrnda,sampata,samudaya,Pinda,avasesa,abhinnaandsamanya, which are related to sets. The units constituting sets aresamaya(indivisible, instant),pradesa(space occupied by an ultimate particle),avibhagi praticcheda(indivisible-corresponding section),varga(variate),samayaprabaddha(instant-effective-bond), and so on.There are two types of

rasiin general. One is thesattatmaka(existential) and other is theracanatmaka(constructional)

Raju:

The literal meaning ofRajuis 'rope'. HereRajuis taken as a unit of measure. Sevenrajusmake ajagasreni(universe line) which is the unit of cosmological measure, the base of the universe being a square with a side equal to onejagasreniThe height of the universe is twojagasrenis, onejagasrenifor the lower universe and remaining for the upper. In between is the middle universe with a width of onerajufrom one end to another, containing islands and ring oceans, each having a width two times the preceding one.

Yojana:This term represents a linear length or distance measure. It has a set theoretic base also since it is related to

angulaand in turn topalyasets. Theyojanais meant for geographical, astronomical and cosmological measures and is of 4krosasor 768,000angulas.Yojanais as mysterious as the Chinese 'Li' for different authorities have given different values to it in terms of modern measurement units. Arajucan be equivalent to innumerableyojanas.

Kalasavarna:

Mahaviracarya has described this as fractions of six sub-types calledgunana,bhaga,varga,vargamula,ghana,ghanamula,summation, on the basis of these operations.

Yavat-tavat:

The use of this denotes those limits to which the quantities are to be extended, or some simple equation is to be framed. Its meaning is also taken as, 'as far as . . . . upto that. . .' It is used in multiplication and summation of series.

Palya(Palla):

LC Jain says that the literal meaning ofPalyais a pit in right circular cylindrical form, which is used for filling up and exhausting it for the process of constructing instant-sets.Palyais of three types:angulaoryojana,vyavahara,uddharaandaddha, which are instant-sets, given in years.

Avalika(avalia):

It means a trail and denotes an ordered set of instants (samayas).

Apart from the above *Ardhaccheda*, *Vargasalaka*, *Vargita-samvargita*,
*Sthana* and *Vikalpa* etc. are some other words related to Jain mathematics.

The Jaina school proceeds to define abstract structures of sets through concrete
structures of islands and oceans in the middle universe (*madhya loka*),
the first island being the jambu of 1 lac *yojana* and the subsequent series
of alternate ocean rings and island rings appearing alternately, the subsequent
ring having twice the width of the preceding ring. The widths of the rings form
a geometrical progression and their various types of (inner, middle and outer)
diameters are compared along with the areas of the rings. The largest diameter
is subjected to logarithm to base two, getting the number of bisections of a
*Raju*, the cosmological unit of distance, which is related with the number
of astral bodies set in the islands and oceans. With the help of this topological
cosmology, R C. Gupta has explained how the first un-enumerable is produced
through a long and tedious process.

Virasenacarya has mentioned similar topological situations through various
types of *vikalpas* (abstractions). The word *vikalpa* (abstraction)
is used for the theory of *bhanga* *vidya*, the method of permutations
and combinations, with geometric *kuta* structures, and for constructing
a number greater than the cardinal or ordinal of a set. The terms used in arithmetical
and geometric progressions are: *adi*, *mukha*, *vadana*, *prabhava*
for the first term, *caya*, *uttara*, *visesa* for common differences,
*gunakara* for common ratio, *gaccha*, *pada* for a number of
terms, and *sankalita* *dhana* for the sum. The sums are also classified
as *adidhana*, *madhyadhana* and *uttaradhana*.

In the texts of the *Karananuyoga* group the geographical maps are wheel
maps or T – O maps. There is symmetry in the depiction of north and south regions,
as well as in proportionate divisions of mountains and rivers. The symmetry
is still held for such maps of the rings of islands and seas beyond the Jambu
Island. There is the flat conceptual earth on which the distances are measured
in *yojanas*. The geometry of the map is mainly of a circle and a straight
line. There is a variety of formulae used with area of a circle as A, *d*
its diameter,* r* its radius, *s* as arc of its segments whose height
is *h*, *c* is the chord and *p* the perimeter or circumference.

In all the available Prakrit texts a similar treatment of the heavenly bodies
is given. There is a *Yuga* of 5 years as in the *Vedanga jyotisa*,
with details of the motion of the sun and the moon (those of the planets having
been lost in course of time). The motion of the sun and the moon has been arrived
at by taking the real sun and the counter sun, and the real moon as well as
the counter moon, placed exactly diametrically opposite. Similar treatment has
been given to the planets, stars and constellations as well. Jain has shown
this model establishes the following principles:

1. Recognition of a five-year yuga

2. Description of thetithisandnaksatrasthrough constant-set (dhruva-rasi) technique

3. Mathematics and prediction as a means

4. Rules for calculation of astronomical events

5. The consideration of solstices and equinoxes.

Time is measured as a function of shadow length related with solstices. Seasons
could be computed or determined through gnomonics. Some more concepts have been
elucidated by Lishk about the phases of the moon, eclispses determination through
colour frequency and their periodicity, lunar occultation with Citra (*Chatratichatra
yoga*). So far as the cosmological ideas are concerned, the universe is finite
with a volume of 343 cubic *rajus*, being *ab-acterno* and *ad-infinitum*,
with an empty non-universe or empty space extending in all directions to infinity.

The Jaina concept of *pudgala* may be compared with the concept of matter
in the physical sciences to the extent that it is conceived as the fluent (*dravya*)
in the sense of stuff of which the material universe is constituted. Interaction
processes of fusion (*Pud*) and fission (*gala*) or integration and
disintegration are also conceived in Jainism. The contents of the universe have
been studied with regard to *dravya* (fluent), *ksetra* (quarter),
*kala* (time) and *bhava* (phase), the contents being specifically
the infinite number of pudgala ultimate particles, each endowed with control
(*guna*) and mode or event (*paryaya*), each being permanent, non-living,
extensive, physical, corporeal, concrete, active, changeable, some of them capable
of becoming *karma* to be assimilated with the soul for fruition.

Regarding the changing nature of matter, Jaina metaphysics explains that matter
goes both *adi* and *anadi parinamas* (effects). Its *adi parinamas*
are found in the transformation of its colour, taste, smell, touch, space, union,
division, motion, sound and neither heaviness (gravity) nor levity, while its
*anadi parinamas* are experienced in its *dravyatva* (fluentness),
*murtatva* (tangibility) *sattva* (existentiality) etc. According
to Jaina *Agamas*, matter is eternal and constant in number (of particles)
existent in the past *samayas*, present and future *samayas*. Matter
has *samsthana* (shape or figure), *akrti*, being active with origination,
decay and permanence. The bodies of bios are made up of *pudgala*, the
*pudgala* being existent in the grossest, gross, gross-cum-fine, fine-cum-gross,
fine and finest, forms from the point of view of grossness and fineness, perceptibility
and imperceptibility, tangibility and intangibility.

The ultimate particle (*Paramanu*) is that which is indivisible, that
which cannot be cut or pierced, and that which cannot be annihilated by water,
fire, etc.

LC Jain points out that Modern Systems Theory is a development of only about a century and Cybernetics also has its origins in the 19th century. Like their geographical, astronomical and cosmological and mathematical models, the Jaina sources have models for their karma system theory as well. The model of the systems theory is itself based upon certain subsystems.

The characteristics deciding whether an ultimate particle or group of such particles is in a karmic state are four as follows:

1. Configuration (

prakrti) group which is dynamical.

2. Point (pradesa) group, showing the number of particles. In a configuration

(Kinetic energy)

3. Lifetime (sthiti) showing how long the karmic state concerned with a group

of particles would last

4. Recoil impartation energy (anubhagu) [Potential energy]

The configuration (*prakrtis*) is of eight following types, further subdivided
into 148 types.

1. Knowledge screening (

Jnanavaraniya)

2. Vision screening (Darsanavaraniya)

3. Captivating (Mohaniya)

4. Incapacitating (Antaraya)

5. Genetic Coding (Nama)

6. Inheriting (Gotra)

7. Pathogenic (Vedaniya)

8. Lifetiming or Ageing (Ayu)

Jain concludes by driving home the fact that the mathematical model of karma
theory was developed as a unified theory in which many scientific ideas found
their place. Mathematics provided the data in units of various types, having
domains and ranges from finite to innumerate and infinite magnitude. Sets were
introduced as existential and in order to give their measures or norms, construction
theories were developed. Apart from the model of astronomy other models are
abstract, unless they are put to experimental verification, the theory of *karma*
remains in the status of theoretical significance, at present. Although it seems
to be scientific, perfect from the point of view of logic and intuition yet
it requires practical verification also for being established as a real model
with concrete causality relations found in the nature of things as in modern
science.

(Note: the essay is based on the source given below. For further enquiries and discussions, interested readers should contact Prof. L.C. Jain, Director, Vidyasagar Research Institute, Pisanhari Hills, Jabalpur, India)

L.C. Jain. 2001. Scientific Ideas in Jaina Canonical Writings and Traditions.
In *Life, Thought and Culture in India (from c. 600 BC to c. AD 300)*.
Volume I Part 2. PHISPC: Centre for Studies in Civilization, New Delhi. Pp.
813-844.