**MATHEMATICOSCIENTIFIC IDEAS**

1. The foremost idea appears in the
abolition of the philosophical infinity and creation of calculation

infinity, ga¿anånanta,(DVL:, vol.3, p.
11) which seems to have a parallel treatment with the proper infinities

of Georg Cantor (round about 1964 A.D.).
Prior to Georg Cantor, infinity in philosophy had no fixed or

costant value, and was simply
indefinite. Even the Greek philosophers, as well as the great
mathematicians

upto the ninth century, had a certain
horror infiniti, and they could not admit its existence in their
scientific

observations, in form of some actuality
or reality. Cantor showed how to contruct an infinity greater than

another infinity.

2. The setting up of
comparabilities(alpabahutva)(DVL,VOL.3. p.322) between various types of
sets

consisting of numerate, innumerate and
infinite members. Their magnitude is denoted by denoting their order

of smallness or largeness.

3. Formation of sets in two
classification types: the existential (sat) sets(rå¹is) and the
constructional sets.

They also found methods for analysing
their values as cardinals or ordinals.

4. Introduction of divergent sequences
locating finite and transfinite sets. Thus they had solved a great

modern problem of topology (Gr. topos: a
place, logos:a discourse).

5. The existence or construction of an
innumerate lying between the finite and infinite.

6. Postulation of monads of objects,
either indivisible or ultimate, as continuum or discrete structures.

For example, the indivisible time
instant (samaya), indivisible space point (prade¼a)defined as the space

occupied by an ultimate particle of
matter (paramå¿u), space as a continuum, time particles (kålå¿us) as

discrete structures filling up the whole
space continuum. Paryåya is an event pertaining to a fluent, happening

at every instant in every one of its
controls (gu¿as). The most impotant concept is that of an

indivisible-corresponding-section (avibhågî
praticceda) of controls as those of knowledge, yoga, moha and so

on.

7. Correspondence of an indivisible time
instant with the minimal, maximal and intermediate values of

velocities. Accelerations or
displacements.This creates a more generalized space-time structure than
the

Minkowskian or Einsteinian four
dimensional space-time structure, corresponding to the generalized
group of

transformations.

The Digambara Jaina School possesses a
small part each from th second purva and from the fifth purva, out of
th

fourteen purvas contained in the twelfth
anga, regarded as the most voluminous, difficult to understand,
abstruce

and intricate, comprising of the
mathematical theory of karma. This theory, as the word purva indicates,
seems

to refer to the knowledge that existed
in India even before Lord Mahavira and its ideas could be compared with

the last century’s developments in
system theory and cybernetics. This school developed its seemingly

scientific ideas not only through
intuition (based on geometry) but also through logic (based on
arithmetic and

algebra), conforming to their number and
simile measures introduced to their cosmographical and karma theories

through a huge data base prepared
through many types of units for a quantitative pursuit. In addition,
these units

and their units expressed the measures
through symbols, very small in number for an easy manipulation but

being sometimes cofused for an
expression denoting more than one quantity. The Digambara school, as
per

records, started writing on their karma
theory in the second century A.D., and the Svetambara school started in

the fifth century or sixth century A.D.
The latter made no efforts in developing heir theory through symboloism.

As such we shall confine our studies
here in the texts on the karma theory in the Digambara
Jaina School alone.

These scientific ideas may or may not be
scientific facts yet they will be found to build up a naïve

mathematicoscientific model with a
motivation either for historicity or for comparative studies in the
karma

theories.

**
**PROSPECTS OF STUDIES IN JAINA SCHOOL OF SCIENCE

For the purpose of exploring the
symbolic, mathematical, set-theoretic, system-theoretic and
cybernetical ideas

in this traditional knowledge of the
DJCM, interlinked and interlocked texts with the Gomma²asåra Jîva

Kå¿ða(abbr. GJK), Gomma²asåra Karma
Kå¿ða(abbr. GKK), the K¼apa¿åsåra (abbr. KNS), the

Tiloyapa¿¿attî(abbr. TPT), The
Trilokasåra (abbr. TLS), as well as the ¬a²kha¿ðågama(abbr. SKG), the
Kasåya

Påhuða Sutta (abbr. KSP), the
Dhavalå(abbr.DVL), the Jaya Dhavalå(JDV), and the
Mahådhavalå(abbr. MDV)

which belong to the Kara¿ånuyoga and the
Dravyånuyoga Groups of mathematical study of
Karma philosophy in

the DJSM, covering several volumes, have
to be gone through.. The unified study is a Herculian task, more so for

its scattered scientific and symbolic
material. This work may be approached by a modern scientist for being

acquainted with the its history as a
subject of an exact science which was regarded as a theory of all
things.

The following points are emphasized:

1. The nodal points of discovery and
invention in the mathematical Karma theory appears to have left
permanent

mark on the edifice of the ancient and
the medieval knowledge round about the Christian era. For example, this

research has
someting to say about classification of numbers into the numerate,
proper types of the innumerate,

as well as the proper infinities,
similar to those of proper infinities invented by Georg Cantor (after
1964 A.D.)

which found application in modern atomic
physics.There has been use of the place value notation and its use not

only in writing of big numbers but also
in factors. There is manipulation of the tables resembling the matrix
forms

of today for showing the
system-theoretic approach with state inputs and outputs in a hereditary
set up from

infinite time earlier. Then there is
development of numerical, algebraic as well as geometric symbolism in a

precise manner.

2. One may also like to follow an
ethnological path, for finding out how much a modern set up of modern
set

theory and cybernetics could grow up in
a comparatively small civilized sect of the DJSM, so deeply different

from all other schools, in so far as it
was developed with mathematical symbolism and dealing with mathematical

operations with transfinite numbers
through their developed logic for variable and constant and mixed types
of

sets. This is evident from their
measure(pramå¿a) theory of sets(rå¹is) treated through logarithms to
the base

two(ardhaccheda) and other bases
ranging upto transfinite numbers. There is also application of eight
analytical

methods to give the idea of operations
among the transfinite numbers. Then there is the method of

comparability(alpabahutva), a process to
keep the various sets according to their order of smallness or

greatness, leading to the measures of
the minima and maxima of a mathematical object which is so important in

the problems where variational
principles are employed in field theories. The same process has been
more

elaborated in the divergent sequences
locating transfinite sets through fourteen sequences of various
measures

needed in the Karma theory. Then there
are methods given for summaion of triangular and other matrices type of

tables of karma structures (vide N, app.
II) as given in the Heisenberg matrix- mechanics. Various types of
series

have also been dealt with in many
structural measures needed in showing the karma phenomena.

3. It may also lead to the study of the
contacts, influences and transmissions during the development of

exposition of the Karma theory. For
example, the travel of the place value notation, the method of
application of

areas, the role of three sets,
manipulation with fractions.

4. It may also be realized by some
scholars how the ancient and medieval records of the scientific
concepts,

methods and procedures adopted in the
Karma theory (vide M and N, app. II and III), could be of significant
value

in the modern set up of mathematical
biology.

5. Application of mathematics in
symbolic form in the Karma theory in the DJSM may also be of some
interest

to the historians of mathematics for
comparing the evolution of the set theory in the DJSM and in Europe by

George Cantor and other scholars of
mathematical philosophy. Similarly, comparison could be in the way of
the

evolution of the theory of logarithms by
the DJSM and by John Napier and Just Burgi.

6.It may further be explored whether
there lies hidden, the fundamental basis of the development in the

mathematical and
horoscopic astrology in the Karma theory. One of the connecting links
may be the indicators.

The indication of various phases of a
bios in the Karma theory are the ultimate particles in various states
of bond,

conditioned by several factors,
represented mathematically through microcosm. Similarly, indicators of
various

phases of a bios in astrology are the
heavenly bodies in various states of dynamical or kinematical
conjunctions,

moving in various types of orbits
tracing the curves which could placed in correspondence principle while

studying the geometry of life in modern
age of biotechnology. Some such facts could enhance the scope of the

studies in the history of sciences. The
remarks of Neugebauer (Science and Civilization, BB., p.171, 1957), may

be quoted here regarding all these
developments, "Though it is quite plausible that the original impetus
for

horoscopic astrology came from Babylon
as a new development from the old celestial omens, it seems to me

that its actual development must be
considered as an important component of Hellenistic science. To a
modern

scientist, an ancient astrological
treatise appears as mere nonsense. But we should not forget that we
must

evaluate such doctrine against the
contemporary background. To Greek philosophers and astronomers, the

universe was a well defined structure of
directly related bodies. The concept of predictable influence between

these bodies is in priciple not at all
different from any modern mechanistic theory. And it stands in sharpest

contrast to the ideas of either
arbitrary rulership of deities or of the possibility of influencing
events by

magical operations. Compared with the
background of religion, magic and mysticism, the fundamental doctrines

of astrology are pure science. Of
course, the boundaries between relational science and loose speculation
were

rapidly obliterated and astrological
lore did not stem - but rather promoted - superstition and magical
practices.

The ease of such a transformation from
science to humbug is not difficult to exemplify in our modern world."