# A. SETS (RÅ©IS)

In the LDS the terms numerate (sa´khyeya), innumerable (asa´khyeya) and infinite (ananta) appear as

number-measure (sa´khyå-pramå¿a). These are minimal (jaghanya), intermediate (madhyama) and maximal

(utkâ¼²a). Terms like pit (palya) and sea (sågara) are sets of instants and universe (loka) is a set of points

(prade¼as), which are classified under the simile measure (upamå-pramå¿a).733

Instant (samaya) is indivisible and so also point (prade¹a), which are the fundamental principle-theoretic units.

Then indivisible-corresponding section (avibhågî praticcheda) forms the fundamental unit for objects like

knowledge (jñåna), operators like volition (yoga), andso on. Other sets of instants are the trail (åvali), inter

muhurta (antar-muhýrta), aeon (kalpa), and so on. Set of ultimate-particles-bound at an instant is instant-

effective-bond (samaya-prabaddha). Then set of souls (jîvas), and so on, also have been applied.

We have seen in the source material and the GJK, GKK and various commentaries, how these sets have been

handled in other that paradoxes or antinomies may not arise. Their topology have been considered through

comparability as well as divergent sequences as detailed by Jain,74 for their analytical methods as well as

history. The role of sets has its oen story.75

B. Structure (Yantra)

The trend of presentation of the Karma theory from simple to complex form may be assessed in studies from

GJK to GKK and then to LDS and KNS. The structure of the Karma theory appears as a totality, with laws of its

own. This structure contains a system of operations (karanas) whose combinations transform one object or

element into another. It has also a self-regulatory property. These three are the characteristic properties of a

structure. [Vide papers by Gershenowitz., H. (1983, BR in the “Exac Esiences in the Karma Antiquity”) for

comparison]. In the LDS, the dynamical theory of karma appears as mathematical structures through numerical,

algebraic and geometric figures. The structures are thus represented through various types of matrices, for

example the state-matrix is a triangular-matrix (triko¿a yantra), which is ultimately to of matrices, for example the

state-matrix is a triangular-matrix (triko¿a yantra), which is ultimately to be reduced to a null matrix of karma-

bound matter-particles through various passages of transformations.76

It may also be noted that the organism-structure has been shown to consist of several sub-structures which

comprise of the cosmological structure, linguistic structure, psychological structure and various others ultimately

having a constructive role to develop an organism in its capacity of the knowledge-set. Actually the aim is to

attain ultimately the supreme existential set of indivisibilble-corresponding-sections of omniscience (kevala

jñåna), which is the supreme construction set in the divergent sequences of various types of existential and

constructive sets passing in the description of various stations (sthanas) all relevant to the development of an

organismic knowledge.77

C. Systems Concepts

The technological method of information-processing and decision-making has make it possible to deal with

various types of systems, their behaviors, transformations, controls, etc., similarly in the GJK, GKK and LDS,

KNS, the karma system theory has been dealt with various types of way-ward-stations (marga¿å-sthånas) and

control-stations (gu¿a-sthånas).78 As we have already seen that cause-effect relations have been maintained

through asrava, samvara, nirjara which form inputs and out puts of the karma system. A stage reaches in the

bios or organism, when it becomes a goal-seeking system, or a control-system when it plays the role of

decision-making for attaining more and more of knowledge through various controls, as the above two types of

stations.

The above theory of karma system was thus applied by the Jaina School to their own situations where scientific

and management achievements were concerned.

D. Symmetry Concepts

In the theory of elementary-particles, symmetry concepts have played a fundamental role, through group theory.

Yang and Wigner spelt out that group theory lies in the basic postulate of quantum theory that the quantum

states of a physical system form a linear manifold.79 We have seen in the Digambar Jaina School, how various

types of sets have been formed as quantum states, through indivisible-corresponding-sections, instants

(samayas) and points (prade¹as) and the quanta of Yoga and Ka¼åya operators.

The four characteristic properties of bonding, state, rise, etc., as we have seen, lie in the four types of structures:

configurations (prakâtis), particle-numbers (prade¹a-rå¹is), energy (anubhåga), and life-time (sthiti). Charges

appear as unctuousness (snigdhatva) and anti-unctuousness (ruk¼atva), measured in quantum levels of

integers, odd and even, from zero of infinity.80

Now one has to concentrate on the operations in the karma-theory, under which the above intrinsu properties

change into one another of one types class, or remain invariant. The Digambara Jaina School had a very deep

study of this type of symmetry-operations, and deal with various types of creation and annihilation of matter in

bond associated with the above intrinsu and other properties, and knew well as to when they remained invariant

or otherwise. At present it cannot be said, how far they were acquainted with the group-properties of such

operators and transformations, but they knew as when, and what remained invariant or changed under what

operation in the time-series of instants.81

The Bråhmî and the Kharo¼²hî alphabets and numerals has also not been solved. Although late, some of the

ancient artha-sa´dâ¼²is, may give a connecting link about the source. [Vide Das gupta, c.c., 1950 BR in the

“Exact Sciences in the Karma Antiquity”].

Thus a study into the semantics of the sa´dâ¼²iis essential from various aspects. From the language used in the

commentaries of GJK, GKK, LDS and KNS, it appearance of various signs and symbols in most of the

sentences, for various operations to be performed along with mutual relations and their sequence.82 Signs for the

constellations (nak¼atras) also appear in the TPT, in words, along the Zodiac.83 Place value system has not

only been used in writing big numbers, but also in methodical process of subtraction and addition of terms into

certain factors of an expression.84

E. Sign And Symbol (SA¤DÂ¬¢I)

In considering cognitive processes and communication processes, which are linked, reciprocal and interlocked,

as well as semiotics the role of signs has been given importance. In the Digambara Jaina School symbolism is

seen developed as numerical (a½ka), algebraic (artha) and geometrical (åkåra).85 Abbreviations have also

been used. These are classified under samdrsti or sahanani. Although these appear in their full form, developed

in the period after 15th century A.D., traces of their existence appear in earlier works. It is a problem of history of

science to see as to when these were developed, because the earlier source material is so deep and complex in

their Dhavalå and Jayadhavalå commentaries, that it does not appear possible that their teaching in the centuries

after Vardhamåna mahåvåra could have been without use of signs. The problem regarding earlier script, prior to.

F. Cybernetic Contents

Norbert Wiener86 appears to have started a unique study of control and communication in the animal and

machine, published in the name of ‘Cybernetics’ (1948). The names of his predecessors, A.M. Ampere, Pascal,

Leibnitz, Gauss, Fareday, Darwin, may be associated with the concept. Mathematical apparatus may be

associated with the names of Cantor, Russell, Whilehead, Hilbert and Weyl. Then Shannon, Turing, Pitts,

Rashevsky, Rosen blueth, Bush, Von, Neumann, Gold stine, Mc Culloch, Lorente, Weaver, and so on, along

with Wiener may also be associated with the endeavour.87

Cybernetics has turned to the study of self-regulating systems and to self-organizing systems for its insights.

We wish to see how far the Karma theory contains these insights.

1. Algorithm

The Labdhisåra gives the solution to the problem of rising above the first-control-station (gu¿asthåna) for a bios.

The mathematical apparatus useful in this adventure is given in the following form:88

(a). Time series as lapsing of indivisible instants from ab-aeterno to ad-infinitum, with which events (paryayas) of

various fluents (dravyas) correspond, for each of their controls (gunas).

(b). Triangular matrix with columns of arithmetico-geometric sequences of number of bound ultimate-particles

and levels of their energy, respectively, denote the state (sattva) in the time-dependent picture as functional-life-

time-structure (karma-sthiti-racanå).

(c). Column-matrix or a multiple there of, as instant-effective-bond, or its multiple in the similar arithmetico-

geometric sequences, denote the input (åsrava) of karma-bound ultimate-particles.

(d). Similarly the row-matrix or a multiple there of, as instant-effective-bond or its multiple, denote the or the rise

and disintegration (udaya and nirjarå).

(e). The above alongwith time-lag (abådhå) data, are distributed in 8 primary or 148 secondary types of functionals

(karmas), according to rule.

(f). The elements of the matrices are nisusus (ni¼ekas) associated with configurations (prakâtis), number of

matter-particles (prade¹a), energy (anubhåga) and life-time (sthiti). There is various types of creation and

annihilation of these nisusus (ni¼ekas) owing to various operations according to a set of rules.

2. Operations And Feedback

For attaining 13 control stations (gu¿asthånas) first control-station, the volition (yoga) and affection (ka¼åya)

operators associated with their mathematical representations are required to be regulated for effecting the state

(sattva) matrix, of a bios.89 It may be noted that configuration (prakâti) and mass-number (prade¼a) bonding for

an instant depend on volition (yoga) and the energy (anubhåga) and life-time (sthiti) bonding depend on affection

(ka¼åya). The cycle of bonding, ever, has been going on through the first control alone, along the wheel of births

and rebirths. In suitable circumstances, described in the text, rise above the first control, to attain the fourth

control, for the first time, three operations, in mathematical representation of sequences of transforms

(pari¿åmas), elevated through the efforts of the bios itself, creates a self-regulating process, for the acquirement

target. Note that these three types of operations are to be fed back again and again according to the information

gained from the out-put. These are a. the low tended operation (adha³ pravâtta kara¿a)

b. the unprecedented operation (apýrva kara¿a)

c.the invariant operation (anivâtti kara¿a).

The mathematical effects in the state-matrix (sattva-yantra) owing to the last two above causal operations are

shown mathematically through the following in full details:

a. geometric-series (gu¿a¹re¿i)

b. geometric-transition (gu¿asa½krama¿a)

c. life-time-split-destruction (sthiti-kå¿ðaka-ghåta)

d. energy-cutting (anubhåga-kha¿ðana).

In the above arithmetrical and geometrical sequences, down traction (apakar¹a¿a) and injection (nik¼epa¿a) of

ultimate-particles, as well as their minimal and maximal over-installation (atisthåpana), and other processes

happen to be in the state-matrix (sattva-yantra). These are all timed and well-defined through comparability

(alpabahutva). The feed back operations are described as mentioned earlier.

3. Self-Regulation And Self Reproduction

Both of the functions depend upon the eight-fold-way of the functionals (karmas), represented mathematically as

shown earlier. Phases (Bhåvas) of five types, of the bios have the fundamental role to play in this connection,

which are either dependent upon or independent of the functionals (karmas).

Communication of information for adoption of controls corresponding to various situations of bond, rise,

termination of karma configurations (prakâtis), at various way-ward (margana) stations (sthånas) has already

been detailed earlier.

4. Linguistics

Exposition of the theory of functionals (karmas) in The Digambara Jaina School has been through three types of

symbolism: numerical, algebraic and geometric forms. Place-value notation has been used for writing of

numbers, subtraction and additions of quantities corresponding to factors. All these have formed fundamental

tools of expressions which are mathematical. The details of language and its function is contained in the theory

of knowledge, where scriptural knowledge is concerned. This is also expressed through mathematical

expressions.90

5. Calculation Mathematics

The mathematical contents of the Labdhisåra has been detailed earlier, versewise. There are several methods and

procedures adopted by The Digambara Jaina School to handle them. A.N. Singh91 as well as B.B. Datta92 has

given them in details. Some methods given in the GJK and GKK have not been exposed so far. Similarly details

have been given in the LDS and KNS regarding manipulation of the various types of Karma data in form of

matrices.93 All these deserve place in the history of science.

9. Concluding Remarks

It is now evident that the Digambara Jaina School paid special attention to the study of the karma theory and

developed it through mathematical manoeuvre in their own way. This tradition in writing seems to have started

some where in the first century A.D., ranging up to the eighteenth century A.D. Most of the fundamental work

appears to have been done in the South India, where Mahåvîråcårya compiledhis Ga¿ita såra sa´graha, and

perhaps most of the methods and procedures based on the mathematical contents of the Ågama might have

given him an urge for the compilation.91 Through the present project, the whole material relevant to that of The

Labdhisåra, and that of The Labdhisåra itself, has been systematically compiled, for an easy access and survey

of this unified work of about eighteen hundred years of a continuous tradition of The Digambara Jaina School,

which had been all along distinguishing, isolated, unexposed for several years after the publication of the Ga¿ita

såra sa´graha.

This project may bring to the notice of scholars, not only the mathematical contents of the Labdhisåra and of its

relevant texts, but also the way in which it was applied to the theory of Karma. It is up to the scholars to see

what methods and procedures adopted in these texts and their commentaries, interlinked together, might have

been originated or developed in this school, pursuing the same model of karma theory all along.

References

Vikalå tahaå kasåyå indiya¿iddå taheva pa¿uo ya /
sa´kha taha patthåro pari ya²²a¿a ¿a²²ha taha samuddi²²ha´/
the pañca payåråpamåda samukktta¿e ¿eyå//35//
Cf. GJK (E), p. 27. Cf. also GJK, I, pp. 69.

“savvepi puvva bha½gå uvarima bha½gesu ekka mekkesu /
melantitti ya kamaso gu¿ide uppajjade sa´khå //36//”
Cf GJK (E), p. 28 Cf. also GJK, I, p. 64.

“paðhama´ pamadapamå¿a´ kame¿a ¿ikkhiviya uvarimå¿a´ ca /
pi¿ða´ paði ekkeka´ ¿ikkhitte hodi patthåro //37//”
Cf. GJK (E), p. 29. Cf. also GJK, I, pp. 65-66.

“¿ikkhittu bidiyametta´ paðhama´ tassuvari bidiya mekkekka /
pi¿ða´ paði¿ikkheo eva´ savvattha kåyavva //38/”
Cf. GJK (E), pp. 29-30. Cf. also GJK, I, pp. 67-68.

do¿¿ivi ga´tý¿a´ta´ ådigade sa½kamedi paðhamakkho //39//”
Cf. GJK (E), p. 30. Cf. also GJK, I, pp. 68-69.

“paðhamakkho anta gado ådigade sa¿kamedi bidiyakkho /
Cf. GJK (E), p. 31. Cf. also GJK, I, pp.70-71.

“sagamå¿eh´ vibhatte sesa´ lakkhittu jå¿a akkhapada´ /
laddhe rýba´ pakkhiba suddhe ante ¿a rýba pakkheso //41//
Cf. GJK (E), p. 31. Cf. also GJK, I, pp. 71-72.

:sam´²håvidý¿a rýva´ uvariîdo sa½gu¿ittu sagamå¿e /
ava¿ijja a¿a½kidaya´ kujjå emeva savvattha //42//
Cf. GJK (E), p. 32. Cf. also GJK, I, pp. 73-74.

Note: For attempts at combinations in China, cf. Needham and Ling (1959- BB), vol.3, pp. 139, et seq. For

attempts in India, cf. Bose, etal (1971-BB), pp. 156 et seq. Cf. also, ibid., pp 162-163, for attempts by The Jaina

School, in praståranayanopåya. Cf also “praståra-ratnåvalî”, op. cit.

9. “igiviti ca pa¿akha pa¿adasa pa¿¿arasa´ kha vîsatåla sa²²hîya /

sa´²haviya pamada ²hå¿e ¿a²²huddi²²ha´ ca jå¿a ti²²hå¿e //43//

Cf. GJK (E), p. 33. Cf. also GJK,I, pp. 74, et seq. Cf. also various books and research papers on combinations,

noted in the BB and BR, in the “Exact Sciences in the Karma Antiquity” for comparing the method and contexts.

10. “igiviti ca kha caðavåra´ khasolaråga²²ha dålacausa²²hi´ /

sa´²haviya pamada²håne ¿a²²huddi²²ha´ ca jå¿a ti²²hå¿e //44//

Cf. GJK (E), pp. 34-36. Cf. also GJK, I, pp. 75-78. A text, “Sri Praståra Ratnåvalî” compiled by R.C. Swami in

Gujarati (Sa´vat 1981). at p. 125, comparable with Bag A.K.

11. Note that here appears the indeterminate analysis. Cf. Datta and Singh, (1962, BB). Cf. also Bag, A.K.

(1977, BR) and other papers.

12. “anto muhuttametto takkålo hodi tattha pari¿åmå /

logå ¿amasa´khamidå uvaruvari´ sarisavaððhigayå //49//

Cf. GJK (E), p. 38, and also p. 42-44. Cf. also GJK, I, pp. 81-112.

13. For attempts in China, cf. Needham and Ling (1959, BB), vol. 3, pp. 137-139, found first in Chou Pei, inform

of arithmetical progression. In the Jaina School of Digambara sect, TheTiloyapa¿¿attî of Yativâ¼abhåcårya

provides many types of progressions, applied in the study of cosmography. Cf. TPG for their mathematical

details. For attempts at studies in progressive series in India, cf. Bose, etal. (1971, BB), pp. 144-145. Cf. also

Ga¿ita såra sa´graha of Mahåvîråcårya, ed. Jain, L.C., (1963, BB), pp. 20-35, and other topics dealing with

progressions. Cf. also, Bag, A.K., (1979, BB). pp. 180-187. Cf. also Shukla, K>S., (gune, 1971, BR), pp. 115-

130, about the various types of summation of series, known as Sa½kalana, varga sa½kalana, ghana sa½kalana,

sa½kalana-sa½kalana. 14. Cf. GJK, I, pp. 207-250. Cf. also GJK (E), pp. 20-28. In the Tiloyapa¿¿attî , apart

from the construction of various types of ordinals and cardinals, there may also be seen various, types of series,

arithmetic, geometric, mixed. Spefically, the comparability of areas of successive rings of islands and seas is

worth deep study. Cf. TPG for this purpose. Cf. also Jaini, J.L. (1918.BB), appendix B. Cf. also M. kumar (1969,

BB), pp. 93 , et seq.

15. “pajjattîpa²²hava¿a´ jugava´ tu kame¿a hodi ¿i²²hava¿a´ /

antomuhutta kåle¿ahiyakamå tattiyålåvå //120// Cf. GJK (E), pp. 84-86. Cf. also Chakrabarti, G.G. (1934, BR) for

treatment of fractions. Other may also be seen aswell a, the DVL (vols. 3-4).

16. “såma¿¿å ¿eraiyå gha¿a a½gula bidiyamýla gu¿aseðhî /

Cf. GJK (E), pp. 101-102. Cf. also GJK, I, pp. 282-284 for ancient symbolic representation, Cf. also TPG, p. 46.

17. “seðhî sýî a½gula ådimatadiyapada bhåjidegý¿å /

såma¿¿a ma¿usa råsî pañcamakadigha¿asamå pu¿¿å //157//”

Cf. GJK (E), p. 103-104. Cf. also GJK, I, p. 286. Symbols given are . Further symbol for the latter is 42 = 42 = 42 =. The

latter is the badala cubed. Note how short the ancient symbolism was.

18. “talalîna madhuga vimala´ dhýma silågåvi corabhaya meru /

ta²aharikha jhaså honti hu må¿usapajjatta sa´kha½kå //158//

42, p. 104. Here the use of Ka²apayådi system has been made.Cf. also GJK, I. pp. 286-287. The same number

has beeen quoted here in the katapayadi system from right to left as follows in JTP (kar¿å²avâtti):

“sådhýraråja kîrtere¿å½ko bhåratî vilola³ samadhi³/

gu¿avargga dharmma nigalita-sa´khyå vanmånave¼u var¿akramta³//

cf ibid p. 287.

19. “ti¿¿isaya sa²²hi virahidalakkha´ dasa mýla tåðide mýla´ /

¿ava gu¿ide sa²²hi hide cakkhupphå sassa addhå¿a´ //170//”

42, pp. 108-109. Cf. also GJK, I, PP. 299-300.

20. Note use of here. This is connected with astronomy of the sun in Jambý island cf. GJK, I, pp. 299-300.

21. “åvali asa´khabhåge¿a vahida pallý ¿aså yaraddhachidå /

bådara tepa¿i bhýjala vådå¿a´ carima såyara´ pu¿¿a´ //213//

Cf. GJK (E), p.128 Cf. also GJK, I, pp. 347-349.

The next verse give manipulation with these expressions on form their differences. Then the 2 is raised to the

expression as exponent.

22. “din¿¿icchedenavahida i²²hacchedehiå payadavirala¿a´ bhajide /

Cf. GJK (E), pp. 129, 130, 131. Cf also GJK, I, pp. 391, et seq. Rule of these sets (trairå¹ika) has been

extensively applied in the works on karma theory. Cf Gupta, R.C., (Dec. 1974 BR). Cf. GJK, I, p. 352.

23. “ekka´ samayapabaddha´ bandhadi ekka´ udedi carimammi /

gu¿ahå¿î¿a divaððha´ samayapabaddha´ have satta´ //254//

Cf. GJK (E), p.148. Cf. also GJK, I, pp. 406, 407.

24. “¿avari ya dusarîrå¿a´ galidavaseså umetta²hidibandho /

gu¿ahå¿î¿a divaððha´ sa´cayamudaya´ ca carimamhi //255//

Cf. GJK (E), pp. 148-150. Cf. also GJK, I, p. 408.

25. This is the simplest numerical representation of the state matrix of karma, consisting of a lapering structural

matrix corresponding to the life-time of a nisusus (ni¼ekas). Here two types of regressions are involved the

regression and arithmetic-regression.

26. “rýý¿a¿¿o¿¿abbhatthavahidadavva´ tu carimagun¿o davva´ /

hodi tade dugu¿a kamå ådima gu¿ahå¿idavvotti //”

Cf. GJK, I, p. 397.

cf. ibid., p. 398.

Cf. ibid., p. 398.

29. Cf. Jain. L.C. (1979 BR). Compare this structure with the axiomatic constant structre theory of general

system theory: CNCTEMHBIE NCCJIEAOBAHNA (1971, BB), pp. 128-152, (year book).

30. It may be remembered that this is the usual unperturbed sequence when volition (yoga) function and other

operations are constants. As these factors vary, there is change as described in the Labdhisåra. The process

becomes complicated.

31. Cf. GJK, I, pp. 415, et seq.

32.A historical remark may not be irrelevant here. Remarks are from Capra as follows,” Thus the bootstrap

philosophy represents the culmination of a view of nature that arose in quantum theory with the realization of an

essential and universal interrelationship, acquired its synamce content inrelativity theory, and was formulated in

termsof reaction probabilities in S-matrix theory. At the same time, this40. “savvasamaso niyama ruvahiya

kandayassa vaggassa / bindassa ya samvaggo hoditti jinehim niddittham //330//”

Cf. GJK (E), p. 192. Cf. also GJK ,II, pp. 555-556, and the following for details.

41. Cf. GJK (E), vv. 384 et seq., pp. 211, et seq. Cf. also GJK, II, vv. 384 et seq., pp. 628 et seq., for details.

42. “tivvatama tivvatara tivva asuha suha taha manda /

mandatare mandatama chatthanagaye hu patteyam //500//

Cf. GJK (E), v. 500, pp. 251-252. Cf. also GJK, II, pp. 701-702.

43. “atragrhitasya samdrstih sunyam misrasya hamsapadam

grhitasyankah, anantavarasya dvicarah / tat samdrstih”

See the above for symbolism. Further explanation is as

“agahida missam gahidam missamagohidam tahevagahidam ca /

58.Cf. LDS. p. 1.

59. Bha²²åraka Jñånabhý¼a¿a, belonged to the school of original organizer Kundakundåcårya, Vide V.

Jaharapurkar, Bha²²åraka Sampradaya, pp. 201-202.

60. Ke¹avavar¿î was the disciple of Abhayacandra Siddhånta-Cakravartî.

61. “eva´ dar¹ana mohak¼apa¿å tippa¿am /”

trayastrim¹atsa´khyåni pravacananusåre¿a vyåkhyåtåni /”.

63. Bha²²åraka Sampradåya, op. cit., p. 183.

64. Cf. LDS, pp. 30-32. Cf. also K. C. Shastri, vol. 1, 1975, pp. 463-482 for further details.

65. In the LDS, intro., the editor feels that in might also be the out come of a team work by certain

Bha²²årakas.

66. Cf., Jain, L.C. (July, 1977), pp. 10-23.

67. On the basis of this ¢oðaramala compiled the Ðhu´ðhårî commentary on the K¼apa¿åsåra or the chapter on

annihilation of character-charm (cåritra moha-k¼apa¿å).

68. For full details of life and works of ¢oðaramala, vide Bharilla, H.C., Pa¿ðita ¢oðaramala : Vyaktitva aura

kartâtva, (Thesis for Ph. D., Indore University), Jaipur, 1973.

69. Note that this symbolsm is differente from the method of application of areas. Vide Szabo, A. (1974, BR),

which was used in the DVL as well as KSP.

70. The sign for negative has been discussed by several authors. Cf. Datta and Singh (1962, 66), pp. 14-15 and

other articles in BR.

71.Cf. remarks of A.N. Whitehead, “There is also another sort of language, purely a written language, which is

constituted by the mathematical symbols of the science of algebra. In some ways, these symbols are different

to those of ordinary language, because the manipulation of the algebraical symbols does your reasoning for you,

provided that you keep to the algebraic rule. This is not the cast with ordinary language. You can never forget the

meaning of language, and trust to mere syntax to help you out. In any case, language and algebra seem to

exemplify more fundamental types of symbolism than do the cathedrals of Medieval Europe.” Vide F. S. C.

Northrop and M. W. Gross, (1953, BB), p. 533, On symbolism, its meaning and effect.

72. Russell, B., History of Western Philosophy (1957, BB), p. 53. Cf. also Berka, K. (1977, BR).

73. Cf Rucker (1982-86), chapters 1 and 2 on infinity” and on “all the numbers”.

74. Cf. Jain, L.C., (1973, 1976, 1977, BR).

75. The abstract set theory was developed in the nineteenth and twentieth centuries by Bolzano (1781-1848),

Cantor (1845-1918), Frege (1848-1925). Antinonies were published by Russell andBulari Forti, Heijencort and

Richard. Various foundations were laid by Zermelo-Fraenkel, Von-Neumann-Bernays-Godel Russell-Whitehead,

Morse-Kelley and Quine, and so on. (cf. Mouton, 1972, pp. 1-34). The Digambara Jaina School applied the set

theory to their Karma theory, through various foundational means.

76. Cf. ASL. The karma structures have a great period of stability, as also of very low periods of life-time. These

could be compared with modern set of atomic structure and molecular structure.

77. Structuralism, does not necessarily rule out, any considerations of history, genesis, functioning, and a

subjects activities. “Any structure is always located at the intersection of a multiplicity of disparate disciplines,

so that no general theory of structures can possibley escape the requirement that it be not simply multi-

disciplinary but authanticary into disciplinary. “cf. Mouton (1972, BB), p. 54, as observed by Jean Piaget.

78. Cf. Kedrov. B.M. and Volodarsky, A.S. (1971, BR). Cf. also Systems Research Year books (Moscow), 1971,

1976-78, 1981, 1985 and various other papers mentioned in the BR.

79. Cf. Salam, :. (Mouton, 1972 BB), p. 71.

80. Cf. ASG, relevant topic.

81. This may well be seen in their topics on various types of transitions (sankramanas), and operations (karnas).

Cf. LDS, vv. 49, et seq., and vv. 68, et seq.

82. Cf. GJK, GKK, LDS, KNS and their commentaries, Cf. also ASG and ASL, cf. also Channabasappa, M.N.

(1984, BR).

83. Cf. TPT, vol. 2, ch. 7, vv. 465-467, p. 737 (1951).

84. Cf. ASG, and ASL.

85. Cf. ASG and ASL. Cf. also Dantzig, T., (1954, BB), chapter 5, on “symbols”, for development of algebra in

three stages: the rhetorical, the syncopated and the symbolic. Cf. Bhandarkar, A.S. (1954, BR Das S.R. (1927,

BR), Datta, B.B(1936, 1929, 1931, BR), Heel, J.F. (1911 BR), Ganguly, S.K. (1932-33, BR); (1982 BR). and so

on in the BR. 86. Wienar, N., (1948, or 1957 edn, BB).

87. Cf. ibid, introduction, pp. 7-39. Now cybernetics implies application of information theory to comparison of

mechanical or electrical controls with biological equivalents.

88. The ligical decisions may also be noted: (i) The maxim of the lion’s vide LDS, v. 101; (ii) The maxim to

denote the part as the whole (eka de¹a vikâta mananyavod-bhåvåtîti nyåya), vide ibid, . 262; (iii) The maxim of the

last lamp (anta dîpaka nyåya), cf. ibid, v. 379. All these appears to have mathematical implication.

89. In Piaget’s terms a mental operation is an internatizer action which is reversible: inversion and reciprocity.

The Jaina term is karana which is far wider in application, even beyond mind and each type of operator is defined

through mathematics, involving time factor also.

90. For details of a little portion of the Jñåna-pravåda, cf. KSP, vol. 1, (1974), pp. 11-138. Cf. also GJK, vol. 2,

(1979), pp. 505-680. Vide also Sikdar, J.C., (July, 1972 BR).

91. Vide the remark of Mahåvîråcårya in GSS (p. 70), “Thus the terminology is stated briefly be the great sages.

What still remains to be said should be learnt in detail from the Ågama.”

92. Singh, A.N. (1942, BB, 1949, BR, 1950, BR), Datta, B.B. (1929, BR, 1935, BR), cf. also Datta, B.B. and

Sjingh, A.N., 91962, BB), Vide also Shukla, K.S., (1979, BR), Cf. also Jain, L.C., (1958, BB, 1967, BR, 1976,

BR, 1977, BR, 1981, BR, 1979 BR).

93. Singh, A.N. (1942, BB, 1949, BR, 1950, BR), Datta, B.B. (1929, BR, 1935, BR), cf. also Datta, B.B. and

Sjingh, A.N., 91962, BB), Vide also Shukla, K.S., (1979, BR), Cf. also Jain, L.C., (1958, BB, 1967, BR, 1976,

BR, 1977, BR, 1981, BR, 1979 BR).

94. Vide Report on the research project on “The Labdhisåra of Nemicandra Siddhånta Cakravartî” (1984-1987),

submitted at the INSA.