- 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 constant value, and was simply indefinite. Even the Greek philosophers, as well as the great mathematicians up to 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 construct an infinity greater than another infinity.
- 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.
- Formation of sets in two classification types: the existential (sat) sets(rå¹is) and the constructional sets.
- They also found methods for analyzing their values as cardinals or ordinals.
- 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).
- The existence or construction of an innumerate lying between the finite and infinite.
- 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 important concept is that of an indivisible-corresponding-section (avibhågî praticceda) of controls as those of knowledge, yoga, moha and so on.
- 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 the second purva and from the fifth purva, out of the 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 symbolism. 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:
- 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 something 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.
- 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 up to 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.
- 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.
- 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.
- 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.
- 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 principle 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.”