Prologue of Mathematical Sciences in the Karma Antiquity

Prof. Laxmi Chandra Jain

Prologue of Mathematical Sciences in the Karma Antiquity

PROLOGUE

Doctrines of karman occupy the space between natural and moral philosophy. They mediate between determinism and free will. Tensions between the two principal components of karman theory, a deterministic philosophy of nature or karmic causality and intention-based ethics, are in evidence throughout the history of Indian philosophy. In Jaina philosophical texts, expounding the most sophisticated deliberations on karman, the word karman, action, is also used in different ways. Historically, Jaina karman theory seems to have developed from a voluntaristic theory of action, concerned with the relationship between subjective choices and objective consequences of action for both subject and object, to a system theory of the working of karman, conceived from an entirely objective, quasi-scientific, point of view. Here, the word karman designates both an objective process and its result, karmic matter or karman pudgala, the seed for renewed action of the same kind. In modern bio-cybernetic terms, the system theory of karman can be interpreted as a boundary maintaining mechanism, governed by an internal control-hierarchy. In the Jaina case, the constitutive goals of the control-hierarchy of the karmic system are not individual preferences, but scripted religio-cultural values. The difference between the voluntaristic and the system theoretical approach is significant. A system does not act. It functions. Ethical considerations, concerning right and wrong conduct, are external to the system theory of karman which describes the inner working of karma in purely theoretical terms.

The earliest surviving Jaina texts on karman, the ‘Seniors’ among the Śvetāmbara Āgama texts, generally dated fourth or third century B.C.E., are concerned exclusively with ethics, while later Digambara and Śvetāmbara texts of from the beginning of the first millennium C.E. abound with cosmological speculation and technical philosophical and mathematical detail. The first fully-developed system theories of karman are in evidence in the Digambara “Siddhānta” and in rudimentary form already in late-canonical Śvetāmbara scriptures, such as Samavāya, Viyāhapannatti, and Ārya Śyāma’s (c. 79 B.C.E.) Paṇṇavaṇā, whose contents overlap in parts with the much more elaborated Ṣaṭkhaṇḍāgama of the Digambaras Puṣpadanta and Bhūtabali (c. 100-200 C.E.) as MĀLVAṆIYĀ (1971: 223-231) showed. The later karma-theoretical works of the Digambaras and the Śvetāmbara works Kammapayaḍi (Karmaprakṛti) of Śivaśarmasūri, Paṃcasaṃgaha (Pañcasaṃgraha) of Candrarṣi Mahattara, and the Karmagranthas of the thirteenth century Devendrasūri, all composed in Prakrit, are largely predicated on these earlier scriptures, albeit more systematic with new details added in particular on the mechanisms underlying the guṇaṭṭhāṇas (guṇasthāna) and their mathematical quantification, apparently imported from named, now lost, Puvva (Pūrva) texts such as the Kaṣāyapāhuḍa (Kaṣāyaprābhṛta). Both Śvetāmbara and Digambara texts containing system theoretical approaches of karman claim to be based on the Pūrvas, the reportedly oldest and hence most authentic Jaina texts, or on parts thereof, such as the lost twelfth Aṅga, the Diṭṭhivāya (Dṛṣṭivāda). Because some of the concepts elaborated by the later Karmagranthas are already in evidence in the canonical compilations, SCHUBRING (1935/2000: 74-6, 187, 322) was tentatively prepared to consider the possibility of the existence of an extinct common source, while ALSDORF (1973: 2) remained “not convinced”, because “in contents and style, they are typical products of later scholasticism, far removed from the much simpler language and spirit of the old canonical texts”.

How can a theory of action, combining subjective and objective perspectives, moral and causal constraints, and a system theory, privileging the observer’s perspective, be integrated without contradiction? A glance at the transition from action-theory to general systems theory in modern sociology illustrates the problem. Originally, action was conceived as a teleological means-ends relationship, oriented toward the realisation of a value chosen by a competent agent. To circumvent the subjectivism underpinning this conception, which does not account for the internalised norms and values orienting individual action, Talcott PARSONS created a model of action as a boundary maintaining system, perceived from an observer’s rather than an agent’s point of view. He argued that a single action, the elementary unit of an action system, can be analysed as a functional system combining four variables. That is, every action can be conceived as a system formed by the interplay of the four components or functions: values, norms, aims and resources. The action system, construed in this way, is but a special case of a living system, with norms and values integrated as functional sub-systems, rather than overarching controls. This model and PARSONS’ subsequent cybernetic system theory of action opened new ways of thinking about elementary actions and combinations of actions, social systems, without prejudicing the ontological question of what an action IS, and where it begins and ends:

‘Just as the units of a mechanical system in the classical sense, particles, can be defined only in terms of their properties, mass, velocity, location in space, direction of motion, etc., so the units of action systems also have certain basic properties without which it is not possible to conceive of the unit as “existing”. … It should be noted that the sense in which the unit act is here spoken of as an existent entity is not that of concrete spatiality or otherwise separate existence, but of conceivability as a unit in terms of a frame of reference’ (PARSONS 1949 I: 43f.).

Critics of cybernetic theories of action, such as HABERMAS, pointed to the problem that the system perspective abstracts from the agent of action altogether:

‘In the concept of an action system, actors disappear as acting subjects; they are abstracted into units to which the decisions and thus the effects of action are attributed. In so far as actions are viewed in terms of their internal analytical structure and conceived of as the outcome of a complex joint operation among the specific subsystems, actors are merely circumscribed by the places they can occupy - in each instance under different aspects - in the four subsystems’ (HABERMAS 1981/1987: 235).

Because instrumentalist and structural-functionalist perspectives cannot be integrated in a seamless theoretical whole without reducing one to the other, HABERMAS argued, the variation of viewpoints is the only viable solution. His proposition to systematically vary actor and system perspectives was chided as ‘eclecticism’ by critics. Yet, it is interesting to note that ancient Jaina philosophers reacted in a similar way to comparable theoretical problems. In response to the conundrum of integrating proto-scientific karman-theoretical explanations of the objective functioning of the cosmos (Karman Theory B) with earlier subject- rather than karma-centred soteriologies (Karman Theory A), Jaina philosophers developed a distinct perspectivist epistemology (anekāntavāda) and logic (syādvada).

The comparison of Jaina karman theories with modern sociological theories of action and social systems sharpens the eye for the idiosyncracies and the blind spots of both approaches. The most striking formal similarity is that both the developed Jaina karman theory and social theory break with ordinary common sense perception by positing the real or theoretical existence of unobservable entities and processes as explanations for the changes in the phenomenal world. In this sense, both models are essentially theoretical. Yet, they are predicated on different ontologies. Social theory and karman theory explain different phenomena, or rather different aspects of the same phenomenon (action or systems of action). Jaina thinkers are not concerned with the opposition of individual and society, nor with social ontology or social theory. They are interested in exploring the opposition of jīva and ajīva, and their modes of interaction, to build up a model of the shared components and mechanisms determining the fate of the individual soul and the cosmos. The sociological quest of explaining the mechanisms underlying the coordination of actions of more than one individual was never addressed in Jaina literature which generally privileges a subject- or soul-centred perspective. Given Its premises are: Karman accrues exclusively to the individual. It can not be transferred or shared. The individual soul alone is responsible for its own karmic fate. From this perspective, social action appears to be at best a source of problems. Social interaction is fraught with danger for the individual, something to be renounced if at all possible. According to Max WEBER (1922/1972: 11), “action is social insofar as … it takes account of the behaviour of others and is thereby oriented in its course.” Because it is oriented towards the sensitivities of others, Jaina karman theory is also a theory of social action, albeit a one-dimensional one, a negative social theory, exclusively focussing on the avoidance of injury of others and hence of one-self. Jaina karman theory considers the existence and behaviour of others, even of animals and plants, but only indirectly under the aspect of non-violence, not from the point of view of cooperation; despite the often invoked dictum of Tattvārthasūtra 5.21, parasparopagraho jīvānām, “souls render service to one another,” which modern commentators interpreted either as an import from Hindu scriptures or as an out of place common sense view of Umāsvāti himself.

The main difference between the two approaches is one of ontology and of method. Both instrumentalist and functionalist theories of social action, whether conceived as closed or open systems or as autopoetic systems, differ from corresponding Jaina perspectives by not taking refuge to metaphysical assumptions such as the ‘law of karmic retribution’. Reciprocity is not treated as a given but as something to be explained. Jaina doctrines simply posit the ‘law of karman’ (karmavāda). That is, they assume that both the intended and the unintended consequences of action automatically rebound on the agent exactly proportionate to the degree of violence committed. Yet, they do (and can) not explain its mysterious inner workings. In both the Śvetāmbara and in the Digambara version of Umāsvātī’s classical Tattvārthasūtra and in the later Karmagranthas the existence of living beings, the injuring and the injured, is entirely bracketed. Only the objective effect of an action on the acting karmic system itself is described in terms of an abstract feedback mechanism. The solipsistic imagery of the embodied soul’s attraction and bondage of further invisible material particles floating throughout the cosmos due to the karmic activity of the passions (kaṣāya) contrasts with the focus of the early Āgamic karma theory on (the avoidance of) violence (arambha) committed by a tangible agent in different situations. Yet, despite the existence of two different if not competing interpretations of karman as such in Jaina literature, the karmic law of reciprocity, interpreted in a moral-ontological sense, is nowhere put forward as a model whose explanatory power is to be tested and refined by a process of falsification. It is taken as a given mechanism in re which can be dogmatically projected or inferred but not rationally understood by non-omniscient beings, as GLASENAPP (1915/1942: xix) pointed out:

“The task of the Karmagranthas is to expose completely a dogma but not to prove it. That is why we find in them a full enumeration of the different kinds of the karman, of the states of the soul, the degrees of their development, etc., but we do not hear why any of this is thus and not otherwise.”

Jaina karman theory was not designed to address social phenomena. What about natural or cosmological phenomena? To what extent is Jain karman theory compatible with the modern natural sciences? Johannes BRONKHORST (2000) produced one of very few studies addressing the question of the relationship between teleological and non-teleological, i.e. naturalistic, karman theoretical explanations of the world. He noticed similar tendencies in the history of modern science and in the history of Indian philosophy to move away from teleological explanations to non-teleological explanations:

‘Indian thinkers were confronted with essentially the same problem, and were at times ready to take drastic steps in order to arrive at explanations of processes which at first sight seem to be goal-oriented in non-teleological terms’ (ibid., p. 122).

With regard to Jainism, he came to the conclusion that, in contrast to the non-dualistic philosophical systems of Hinduism, arguing that the world at large is determined by the collective actions of all its inhabitants, in Jaina philosophy (and to some extent in Sāṃkhya and Yoga philosophy) ‘the working of karma is confined to the soul to which it sticks. Other occurrences in the world have to be explained through other causal processes.” In Jaina karma theory “None of the eight kinds of karma … is claimed to have an effect on the world at large, and none have an effect on the creation of the world’ (ibid., p. 119). In contrast to Śaṅkara, for instance, classical Jaina cosmology distinguishes between purely physical forms of causality, such as attraction and repulsion of atoms, and karmic forms of causality, Yet, because most tangible entities, including the elements, are conceived as living beings, that is, as composites of a soul and karmic matter, from a Jain point of view, the object of modern natural sciences, dead matter, can hardly ever be experienced, except for a short period after an act of killing. The Jain concept of pure matter (pudgala) is for all conventional analytical purposes a theoretical concept, a limiting case, as much as the notion of the pure soul. During their creative period, almost two millennia ago, the Jain life-sciences were preoccupied with the classification and analysis of the empirical world of living beings, which is the world of karman, conceived from the perspective of Karman Theory B. Yet, the constitutive Jaina problematic of the fundamental ontological dualism and the resulting question of how spirit and matter can interact through the mysterious workings of the law of karman are not explained by Jain thinkers, but accepted qua belief in the supernatural insight (kevala jñāna) of the Jinas.

From a modern scientific perspective, predicated as it is on the diremption of nature and culture, both early and classical Jaina models of karman merge natural philosophical and moral considerations in an unclear way. Probably the closest analogy of Jaina karma theory in modern (life) science is the theory of genetic modification. But modern science has still a long way to go until phenomena as subtle as those theorised by the ancient Jaina philosophers can be addressed in a methodical way. Rather than merely associating ancient Jaina conceptions metaphorically with the jargon of modern science, Yuvācārya MAHĀPRAJÑA (1992) and Jethalal ZAVERI and his son Muni MAHENDRA KUMĀR (1991, 1992) pointed out at least some tentative ways in which a theory of karma that is compatible with natural scientific methods could be developed experimentally in a creative way.

To explain how the act of killing effects the killer, early Jaina texts recurred to popular hydraulic metaphors of invisible moral fluids, conceived as physical substrates, flowing in and out of the embodied souls, as Hermann JACOBI (1914: 468, etc.) and others observed. Two disciples of Dalsukh MĀLVAṆIYĀ, Krishna Kumar DIXIT (1973: 6), who commented on Viyāhapannatti (Viy.) 1.8.68 [91b], and Suzuko OHIRA (1994: 5-8), who commented on Āyāra 1.3.2.3 and Sūyagaḍa (Sūy.) 1.1.1.3 and 1.10.21, argued that, historically, the Jaina notion of karman emerged from the concept of revenge, vera (Skt. vaira, from vīra), of which it is but an objectified form. In the earliest strata of the Āyāra and Sūyagaḍa, and in the Viyāhapannatti, the word vera seems to be used to designate the ‘energy discharged by a victim’s soul’ in form of a stream, soya (Skt. srota) (AS 1.3.4.143f.). The word soya is rarely used in later texts. In Viy. 1.8.2 [91b], the expression vereṇaṃ puṭṭhe, ‘being touched by the revenge’ of the victim, is used. In Uttarajjhayaṇa 4.2, and 6.7 and Dasaveyāliya 9.3.7, the word vera is employed more in the ordinary sense of hostility and anger, not in the sense of retribution. In the early sections of the same two texts, the word kamma (Skt. karman) is understood as karmic matter (kamma puggala, Skt. karma pudgala), rather than as action in general, as in Ācāra I. It is worthwhile citing DIXIT’s (1973: 5f.) speculative reconstruction of the evolution of the early Jaina karman theory at length:

‘The Bhagavaī’ treatment of the problems of karma-doctrine has its own value. In this connection a peculiar verbal usage of the text deserves notice. Thus when it intends to say that a person commits a kriyā (kriyām karoti) it sometimes says that this person is touched by this kriya (kroyayā spṛṣṭaḥ) [91b]. Certainly, the phrase “touched by kriyā” used here is somewhat odd but it seems to have been patterned after a popular phrase of those times. For in the dialogue considering the case of one person killing an animal and another person killing this person himself we are told that the first person is touched by the enmity of the animal (mṛgavaireṇa spṛṣṭaḥ), the second person touched by the enmity of the first person (puruṣavaireṇa spṛṣṭaḥ) [91b]. Now the modern anthropologists tell us of the primitive people who believe that when a person commits a crime against another person this crime hounds the first person as long as it does not bring upon him an appropriate disaster. And in all probability such a belief was prevalent among that circle of Indian populace which was accustomed to the phrase “touched by the enmity of so and so”. This in turn became the starting point for the Jaina authors developing their doctrine of karma which in its essence is but a refined version of the belief in question. The first step in this connection must have been to speak of the technical concept “kriyā” instead of the popular concept “vaira”. Then the idea must have occurred to those Jainas that if kriyā is to touch a person it must be something tangible, and thus came into existence the concept of kriyā treated as a physical entity. Soon, however, kriyā qua a physical entity came to be designated karma and one began to speak of a person committing a karma (karma karoti) or a person being touched by a karma (karma spṛṣṭaḥ). Lastly, the search was made for an active voice usage expressing the same idea as “karmaṇā spṛṣṭaḥ”, and the phrase “karma badhnāti” (binds down a karma) was the outcome.

Here we reach the stage represented by the classical Jaina authors who in this connection exclusively employed the phrase “karma badhnāti”. But the noteworthy thing is that in Bhagavatī the phrase “karma badhnāti” is a relatively rare occurrence; for here the moral usual phrase is karma (or kriyām) karoti (or prakaroti) [prakaroti: 51b, karoti: 52a, 52b, 63b, 79a], occasionally karmaṇā (or kriyayā) spṛṣṭaḥ) [91b]. All this makes it sufficiently clear that in Bhagavatī what we here having before out eyes are the beginnings of the specific Jaina version of the doctrine of karma - of which version there was little trace in the oldest texts.’

Suzuko OHIRA (1994: 5-8) subsequently argued that the word vera was used in the sense of ‘revenge’ already in Āyāra 1.3.2.3 and Sūyagaḍa (Sūy.) 1.1.1.3 and 1.10.21, but ceased to be used in later texts. She speculated that the vaira theory was taught by Pārśva and then transformed by Mahāvīra into the karmavāda doctrine evident in AS 1. Klaus BRUHN (1993 II: 48) noted, in sum, that the Jaina theory of karman is ‘neither a consistent whole nor a concept which tries to explain human behaviour in its entirety’. Further, ‘we notice not only high-level syncretism, but also the dynamism of popular beliefs and the parallelism of karma and fate’: ‘in ethics, emphasis is sometimes on the act and sometimes on the attitude; finally, Jainism as a whole, oscillates between ahiṃsā legalism and kaṣāya soteriology’ (ibid., p. 21). Jaina karman theory, it is argued, assumed its classical form after interpreting karmic matter in terms of the imported atomistic philosophy of the Vaiśeṣika tradition. BRUHN and BUTZENBERGER (1994: iv) summarised the principal features of currently prevalent academic interpretations:

‘… If we ignore some very primitive statements we can probably say that there was a clear shift of emphasis from popular ethical thought (offence, suffering = atonement, restoration of the original state of the offender) to abstract soteriology. The latter term stands in this case not for a doctrine of salvation in its usual sense, but for an esoteric current in Jaina dogmatics which is not merely detailed and technical, but likewise in its basic character far removed from the common notions about action, retribution, and redemption. The typical “later karman theory” is of course later than the last period of the canon and also later that the Tattvārthasūtra. … (iii) … ‘Presuming quite unspecific assumptions on moral retribution, the development started with explicitly investigating the nature of this retribution, and with considering it to be a stream of subtle matter afflicting the individual. A second major event seems to have been the idea of describing matter according to the latest atomistic theories of natural philosophy, and thus being able to explain the otherwise disturbing circumstance that karmic matter is invisible: except if they constitute a considerably large conglomeration like a pot, etc. (ghaṭādikārya), atoms are invisible (apratyakṣa).’

The question, how the rebounding effect of an action on the agent can be explained in other than naturalistic moral-ontological terms has hardly been addressed in ancient or modern literature. From the perspective of an individual, a point of departure for a comparative phenomenology of ‘guilt’, or generalised feeling of ‘obligation’, may be offered by the Jaina doctrine of lessā (Skt. leśyā), colouration of the soul, which could be compared, for instance, with HEIDEGGER’s (1927/1962) notion of mood (Grundstimmung).

How can momentary actions have consequences even after long intervening periods of time? Wilhelm HALBFASS (1992: 300ff.) showed that in the Brāhmaṇic tradition the Pūrvamīmāṃsā concept of apūrva, potency, and the Vaiśeṣika concept of the adṛṣṭa, unseen, were proposed to explain storable causal potencies which attach to the agent of an (sacrificial) action (kriyā), and hence account for the efficacy and power of sacrifices. According to the ’magico-ritualistic world-view’ of the Mīmāṃsā philosopher Kumārila in his Apūrvādhikaraṇa of the Tantravarttika, the agent is ātman, the soul of the sacrificer who pays for the sacrifice, and the positive consequences of his sacrificial action - the apūrva or dharma - remain in form of personal dispositions (saṃskāra). Śaṅkara later related the non-karmic apūrva theory to karman in his Brahmasūtrabhāṣya. In contrast to the Mīmāṃsā tradition, the Vaiśeṣika concepts work within the framework of a karman theory which accounts for both positive and negative consequences not only of sacrifices but of all actions. The classical Vaiśeṣika author Praśastapada uses the term adṛṣṭa to designate dharma and adharma, merit and demerit. The Jaina perspective rejects the notion of an invisible mystical force (adṛṣṭa) altogether, and in the second stage of the development of Jaina karman theory instead posits the notion of invisible karmic matter which attaches itself to and binds the soul, producing certain causal conditions. This is a purely theoretical conception of karman, far removed from its original meaning of the term. In contrast to the observable, more or less violent, actions of an individual, unseen karmic particles and processes of karmic transformation are mechanisms which can be modelled conceptually and mathematically, but are unrelated to human psychology or ethics. SCHUBRING (1935/2000: 323) commented: “it appears that here the Jains have cultivated psychology without soul.”

Jaina Mathematics and Karman Theory

Professor Laxmi Chandra JAIN’s pioneering work on the mathematical content of classical Jaina karman theoretical texts deals exclusively with the mathematically refined forms of what he calls the ‘karma system theoretical approach’ or the biocybernetic system functionalism’ of the mathematical passages in the Digambara commentaries on the Ṣaṭkhaṇḍāgama and the Kaṣāyapāhuḍa. What a unit of karman pudgala actually represents, how karmic matter (a uniquely Jaina conception) is formed, and how Karman Theory B relates to Karman Theory A does not concern these texts. Hence, the term karman is used in a special sense in the present volume, which applies the language of modern biocybernetic system theory to the karman theory of Nemicandra:

‘”Karma” has been translated as “functional” and it has a mathematical significance of two meanings: as a transformation and as an operator with lapse over time’ (L.C. JAIN, infra, p. 16/26).

The author has created a whole range of further new translations to account for the specific technical mathematical use of vocabulary that seems familiar from Jaina philosophical contexts. The author’s system theoretical interpretation of the guṇasthāna scheme informs the translations of guṇasthāna as ‘control station’, of mārgaṇāsthāna as ‘way-ward-station’, of the āsrava, saṃvara, nirjara as ‘cause-effect’ or ‘input-output relations’ of the karman system. Jīva is translated as ‘bios’ and dravya as ‘fluent’ (elsewhere also: ‘organism’). Kartā becomes ‘functor’, kartṛtva ‘function’, and niṣeka ‘nisus’. Even a trained Jainologist who is unfamiliar with this terminology, which needs to be learned first, will find it difficult to follow both the original texts and their translation through strings of English neologisms. The advantage of this creative exercise of defamiliarisation is that the conceptual differences between instrumental and functional Jaina karma theories become apparent.

Professor L.C. JAIN began his research on the terminology of the mathematical passages in the Digambara scriptures more than fifty years ago. His inquiries commenced in 1955 in form of discussions with Digambara luminaries such as Kṣullaka Manohār Lāl Vārṇī ‘Sahajānanda’, Ratan Cand Mukhtar and Dr Hīrālāl Jain, which benefited his first major publication entitled Tiloyapaṇṇattī kā Gaṇita (1957), comparing the mathematical karman theory in the Tiloyapaṇṇattī with modern system theory and the cybernetics of N. WIENER. This approach informed a lifetime of research in the history of Indian mathematics. The south Indian Digambara Jaina School of Mathematics developed a distinctive mathematical symbolism (saṃdṛṣṭi) for theorising the classical mathematical Digambara Jaina ‘system theory of karman’, or karma praṇāli siddhānta, an approach which seems to be missing in the Śvetāmbara Karmagranthas. Between 1984-1995 L.C. JAIN worked on three Indian National Science Academy (INSA) funded projects on mathematical passages in Digambara in texts. The first project, under Dr A. K. BAG, focused on the decipherment of the mathematical symbolism in the Labdhisāra (1984-87). This was followed by two projects on the Prastāra Ratnāvalī (1989-91), and Mathematical Concepts of the Digambara Jaina Texts of the Karaṇānuyoga Group (1992-95). At this time a long-standing collaboration with the Sanskritist Brahmacārinī Dr Prabha JAIN was initiated which informed the seminal work of the author, The Taō of Jaina Sciences (1992), which contains his principal analytical ideas, and the publication of the first volume on the Labdhisāra, containing a glossary of technical terms (L.C. JAIN 1994: 17-66) and summary tables of the proposed decoding of the ‘Symbolism and List of Working Symbols’, especially the geometric symbolism, or ākāra saṃdṛṣṭi (ib., pp. 67-98). Between 2003-2005 five volumes were published in the series Exact Sciences in the Karma Antiquity on the Tiloyapaṇṇattī, Trilokasāra, Lokavibhāga, Jambudīva Paṇṇattī Saṃgaho and the Karaṇānuyoga texts. The explanation of the symbolism of the Labdhisāra is given in the present first volume of the planned Labdhisāra study, to be published as part of a five volume series entitled Mathematical Sciences in the Karma Antiquity comprising the outcomes of the first INSA project on Nemicandra’s work, with a focus on the mathematical content of the Labdhisāra and its application to theory of karma. It is based on the original text edited by Gajādhar Lāl JAIN and Śrī Lāl JAIN in 1919, and the Sanskrit and Hindi paraphrases of the original Prakrit by M. SHASTRI (Bombay 1916) and the Hindi commentary by Pt. R.C. MUKHTAR (Mahavirji 1982).

The symbolism employed in the mathematical passages of the Labdhisāra cannot be understood without a prior study of the Trilokasāra and Gommaṭasāra, and the Kṣapaṇāsāra. Nemicandra understood these four works as ‘summary’ explanations of the entire Digambara Siddhānta. That is, the Ṣaṭkhaṇḍāgama of Puṣpadanta and Bhūtabali, and the Kaṣāyapāhuḍa of Guṇadharācārya, and their commentaries. He conceived the Gommaṭasāra as a synopsis of the Ṣaṭkhaṇḍāgama, and the Labdhisāra and Kṣapaṇāsāra as summaries of Guṇadharācārya’s Kaṣāyapāhuḍa, while the Trilokasāra summarises the Tiloyapaṇṇatti (Trilokaprajñapti) of Yativṛṣabhācārya (5th -6th C.). The present first volume, focusing on the Gommaṭasāra Jīvakāṇḍa, and the second volume on the Gommaṭasāra Karmakāṇḍa are conceived as introductions to the study of the mathematical content of Nemicandra’s (late 10th - early 11th century) Labdhisāra, Essence of Attainment. This text represents the culmination of the work of Nemicandra, who was given the honorific title Siddhānta Cakravartin, Conqueror of all Scripture, which apparently the Digambara community conveyed only to one who mastered both the Ṣaṭkhaṇḍāgama and the Kaṣāyapāhuḍa (Kaṣāyaprābhṛta), the two main texts of the so-called Digambara Siddhānta. Because both text collections were, however, hardly accessible, both physically and intellectually, for many centuries, Nemicandra’s Gommaṭasāra, Essence of Excellence, was regarded as the most sacred scripture of the Digambara tradition; until the re-discovery and publication of the only surviving manuscripts of the Ṣaṭkhaṇḍāgama and the Kaṣāyapāhuḍa in the 20th century.

Both the Ṣaṭkhaṇḍāgama and the Kaṣāyapāhuḍa claim to be based on the lost Pūrvas of the ‘original’ Jaina canon. Generally, the Kaṣāyapāhuḍa is considered to be a later text than the 2nd century Ṣaṭkhaṇḍāgama. But L.C. JAIN considers it to be an earlier composition of the first century C.E. The Labdhisāra

The present text is a collection of diverse materials for the study of the Labdhisāra. It is divided into three parts, the first two of which are introductory:

(I) An ‘Introduction’ of seventy one pages on the mathematical content of the Gommaṭasāra and Labdhisāra texts, based on the commentaries of …, comprising five sections, an introduction and concluding remarks (see infra), establishes the necessary link between the Gommaṭasāra and the Labdhisāra.

(II) A Reprint of the original text and the 1927 English translation of the Gommaṭasāra Jīvakāṇḍa by Barrister Jugmandar Lāl Jaini and Brahmacārī Sītāl Prasād Jain, published in 1927, minus the English commentary, but with a new commentary in Hindī by ….

(III) An English rendition of the Artha Saṃdṛṣṭi, Gauge (Measure) Symbolism, a chapter in Paṇḍit Ṭoḍaramala’s 18th century Samyag-Jñāna Candrikā. This two hundred and one page long English text, presented under the title ‘Jīvakāṇḍa’ as a ‘Prelude’ to the symbolism of the Labdhisāra (the proliferation of multiple heading in the books can be disorientating), is the main contribution of the present book. It supplements both the 1927 English translation and the Hindī commentary to the latest Gommaṭasāra edition published by Bharatīyā Jñānapīṭha in 1978-1981, which offer no explanation of the symbolism and meaning of the mathematical verses in Nemicandra’s work.

The ‘Introduction’ comprises seven sections of diverse nature, summarising much needed background information.

(1) ‘Introduction’

(2) About the Labdhisāra, it's authort and date.

(3) Summaries of selected verses of the Trilokasāra divergent from the Tiloyapaṇṇattī on the basis of the Sanskrit commentary of Mādhava Candra Traividya (the source for Pt. Ṭoḍaramala’s commentary) and the Hindi commentary of Āryikā Viśuddhimati.

(4) Analytical summaries of selected verses from the Gommaṭasāra Jīvakāṇḍa, on the basis of the 11th century auto-commentary Jīva Tattva Pradīpikā of Nemicandra (Karṇāṭaka Vṛtti and Sanskrit Ṭīkā), and the 18th century commentary Samyag-jñāna-candrikā written in Ḍhūṇḍhārī by Paṇḍit Ṭoḍaramala (c. 1720-1767).

(5) The ‘Exposition (Vṛtti) and Commentary of the Labdhisāra and their Authors’ discusses the problem of the authorship of the only Sanskrit commentary of the Labdhisāra, the incomplete anonymous Vṛtti, which is here attributed to the 16th century author Nemicandra [II], though the 14th century Kanarese author Keśavavarṇī is mentioned as another likely source. The only full commentary on the Labdhisāra is contained in Ṭoḍaramala’s Samyag-jñāna-candrikā, which has a separate chapter on the mathematical verses of the Labdhisāra and the Kṣapaṇāsāra, entitled Artha Saṃdṛṣti Adhikāra,, whose section on the Kṣapaṇāsāra relies on the Kṣapaṇāsāra commentary by Mādhavacandra Traividya of 1203. This chapter is the focus of the present volume.

(6) The section entitled ‘Scientific Thought Evident in the Labdhisāra’ is based on an article already published by L.C. JAIN (2005b). It comprises the following sub-sections: A. Sets (rāśīs), B. Structure (yantra), C. Systems Concepts, D. Symmetry Concepts, E. Sign and Symbol (saṃdṛṣṭi), F. Cybernetic Contents: (a) Algorithm, (b) Self-Regulation and Self-Reproduction, (c) Linguistics, (d) Calculation Mathematics.

(7) ‘Concluding Remarks’

It is difficult to comment on the mathematical or indeed philosophical content of the selected Artha Saṃdṛṣṭi passages. The work shows that the classical Digambara doctrines of karman theorised changes and transformations of karman in terms of ‘sequences and series, progressions and regressions’ of ‘successive instants (samaya)’, of ‘instant sets’ (p. 42). For the history of science, the value of decoding their mathematical, scientific and philosophical content is immense. Sensibly, the author of Mathematical Sciences in the Karma Antiquity refrains from premature conclusions, and presents the reader with the material without adding his own comments. The stated aims of the text are to furnish ‘the basis for a chapter on Indian mathematical systems theory in the larger history of ancient mathematical science’ (xiv): Its ‘ancient form was called the “Karma Theory” or “Action Theory” or “Functional Theory”’ (ib.). It is the invaluablea great contribution of the author to have explicated the symbolism and the system theoretical implications of the mathematical passages of Nemicandra’s work together with his assistant Brahmacarinī Doctor Prabha JAIN. The works of Professor L.C. JAIN confine themselves to the Digambara Jaina school of mathematics. It is unlikely that work of a similar magnitude and depth will be conducted in the near future on the mathematical contents of the Śvetāmbara Karmagranthas which also point to the lost Dṛṣṭivāda as their source.

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GKK1 Gommaṭasāra Karmakāṇḍa of Nemicandra. Edited with Introduction, Translation and Commentary by Rai Bahadur Jugmandar Lal Jaini. Assisted by Brahmachari Sital Prasad. The Sacred Books of Jainas Vol. 4. Lucknow: The Central Jaina Publishing House, 1927.

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