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Philosophy of Space-Time in Early Jaina Thought: Quantification as a Means of Knowing

Abstract

The aim of this paper is to bring to light the determining role played by quantification for those writing early Jaina doctrine. Jaina philosophy of space-time attempts to fix in a finely elaborated numerical structure the model of the and the whole system of reality. In discussions about the sophisticated mathematical practices characterising Jaina sources, the achievements realized by highly elaborate formulae have been variously emphasized. An aspect that has been thus far disregarded by historiography is the reasoning behind mathematical modes of enquiry. Therefore, there is a need to investigate the relationship between the way Jaina thinkers expounded philosophical issues and the use of quantifying procedures. Everything imaginable that could be quantified was computed: the numbers of individuals among different kinds of beings in the various regions of the three worlds, the extent of their lifespans, the volumes of space occupied by living beings of various shapes, and other matters. I shall argue that in early , mathematics is a fundamental expedient of philosophical enquiry. Textual evidence from religious-philosophical works will be presented to demonstrate that, for Jainas, quantification responds to a “quest for order”: the intellectual and spiritual ambition to recognize in structured and predictable patterns.

Mathematical Modes of Enquiry Within Early Jaina Thought

‘I often say that when you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind […].’ (William Thomson, Lord Kelvin 1891: 80)

Although Lord Kelvin made this remark in a very different context, it is noteworthy that he expresses high esteem for measurement primarily because it produces knowledge. Of course, he refers to a kind of measurement based on empirical observation. In Jainism, it is the attributes of things –their properties and relations– which are measured, rather than the things themselves. There are a few questions which I have developed since the beginning of my research on early Jainism. A first concern regards the development of mathematical reasoning. In this respect, it is well-known that geometry is the earliest recorded branch of mathematics in , as in the Vedic period complex geometric techniques were needed for the construction of the altars and fire-places described by the brāhmaṇas for the performance of their rites. The link between geometry and ritual suggests that mathematical accuracy was considered of the utmost importance in the context of Hindu rituals.1 This knowledge is found in the Śulbasūtras, which are a part of the Kalpasūtras (more particularly of the Śrautasūtras) and attached to the as Vedāṅgas. Although outwardly religious texts, the Śulbasūtras are manuals for the geometrical construction of the Vedic altars and are the most ancient treatises expounding mathematical procedures and propositions in a systematic order. 2 This development of mathematics is, however, related to early instances of Vedic Hinduism. What can be said about Jainism? In my view, historiography has failed to understand the reasoning beyond the use of calculations which much characterize early Jaina-non mathematical works, by which I mean texts not dedicated to mathematics as a separate scholarly discipline. Mathematical methods have been so far investigated by scholars exclusively in terms of formulae. In the attempt to fill this gap, this paper provides

1 An earlier study of Śulbasūtras mathematics is Datta [(1932) 1993]. For an edition and annotated English translation of the four major Śulbasūtras, see Sen and Bag 1983. 2 The oldest layers of Baudhāyana’s text, the earliest among the four Śulbasūtras, seem to go back to sometime between the seventh and fifth centuries BCE.

an analytical framework to guide research questions related to mathematical enquiry within Jaina thought. Early Jainism works with vast numbers for describing the physical layout of the Universe and their conceptions of time and space, on which the theory of and liberation depend upon. Everything imaginable is quantified and computed: for instance, the numbers of individuals among different kinds of beings in the various regions of the three worlds, the extent of their lifespans, the volumes of space occupied by living beings of various shapes, the total volume of space in all the worlds, and other matters. In my view, quantification –intended as the act of calculating and assigning numbers to things –is for the Jainas a means of knowing. Such a form of cognition perfectly suits Jaina analytical categories, and its systematic paradigm for elucidating karma, rebirth, and salvation. I argue that for those writing early Jaina doctrine quantification represents an expedient to grasp and explain the laws of Universe and the human predicament therein. Since early times, Jainism has shown an approach to comprehending the immanent structures of reality –here understood as the state of things/beings as they actually exist and as they will be– by means of measurement and quantification. This reflects, in my opinion, the epistemic act of a distinct philosophical enquiry which proceeds by means of both qualitative methods of classification and quantitative methods of computation, and whose purpose is to achieve prediction and causal explanation. It may seem provocative to claim that philosophical enquiry and mathematical methodology are deeply intertwined within Jainism, yet it is a point worth insisting upon. The Sthānāṅgasūtra, a Prakrit text included into the Śvetāmbara canon, is an encyclopaedic work divided into ten chapters (sthāna) dealing with doctrine, practice, mythology, and so forth. 3 Interestingly, this gives a list of mathematical topics that were studied at that time: parikarma (arithmetical operations); vyavahāra (procedures or practices ); rajju (probably denoting geometry); rāśi (solid mensuration); kalāsavarṇa (fractions); yāvat-tāvat ( equations, algebra); varga (square, quadratic equations); ghana (cube, cubic equations); varga-varga (biquadratic equations), and vikalpa (permutations and

3 The Śvetāmbara canon is a voluminous body of texts in Prakrit which were put into writing during the fifth century. On Jaina Canonical literature, see Glasenapp (1999: 109 ̶124), P. S. Jaini 1979, and Kāpadīā 1941. The tenth volume of Encyclopaedia of Indian Philosophies edited by Malvania and Soni 2007 and the fourteenth volume edited by Potter and Balcerowicz 2013 provide a comprehensive account of of Jaina literature.

combinations).4 It is noteworthy that these subjects and the technical terms used by the Jainas were adopted by later mathematicians despite their religious beliefs. That a well- established tradition of investigation into mathematics was deeply rooted within early Jaina thought is evoked in a passage of the Gaṇitasārasaṃgraha by Mahāvīrācārya’s words (1.9– 16).5 Interestingly, here the author states the relation between calculation and knowledge. Using a poetic language and mentioning concepts related to Jaina philosophy, Mahāvīrācārya conveys his point: mathematical procedures are indispensable for understanding worldly affairs, carrying out astronomical computations, and comprehending religious matters as well: ‘In all those transactions concerning worldly, Vedic or [other] similarly religious affairs, calculation (saṃkhyāna) is of use. In the science of love, in the science of wealth, in music and in the art of drama, in the art of cooking, similarly in medicine, and in things like the knowledge of architecture, in prosody, in poetics and poetry, in logic, grammar, and such other things, and in relation to all [other] characteristics of the [various] arts: the science of computation is highly appreciated. In relation to the movements of the sun and other heavenly bodies, in connection with eclipses and the conjunctions of planets, and in connection with the tripaśna and the course of the moon ̶indeed in all these it (i.e. computation) is applied.6 The number, the diameter and the perimeter of islands, oceans and mountains, things such as the measure of Prakīrṇaka,7 the extension of Indraka and Śreṇībaddha, [and of the habitations] of the Bhavanas, the Vyantaras, the habitants of the world of Light (Jyotiṣkas), the Heavenly Beings (Kalpavāsis), and the Nārakas (Hellish Beings):8 ̶ all these are known by means of computation. Things like the abodes of human beings, the of their lives, their eight

4 See Plofker (2007: 59). 5 For an English translation of the Gaṇitasārasaṃgraha, see Raṅgācārya [(1912) 2011]. For a study on the mathematics of this text, see Datta and Singh ([1935] 1962); Morice-Singh 2015; Petrocchi 2015 and 2016; Plofker (2007: 162–171), and Srinivasiengar (1967: 70–78). 6 Tripraśna (lit.”three questions”) is the name of a chapter in astronomical works, dedicated to the determination of direction, place, and time of heavenly bodies. 7 Raṅgācārya ([1912] 2011: 2–3) misinterprets this passage, probably not being familiar with Jaina terminology. In fact, Indraka, Śreṇībaddha, and Prakīrnaka are three types of heavenly abodes. Interestingly, in this passage Mahāvīrācārya refers to terms found in early Jaina scriptures, such as chapters 3–4 of the Tattvārthādhigamasūtra. This work summarises the main principles of Jain belief. 8Kalpas are celestial habitations of devas. See Tattvārthādhigamasūtra 4.19, 23–24.

attributes,9 [the length of] their traveling [to holy places] and other such things, [the amount of] their aggregation and such other things ̶[the knowledge of] all these are dependent upon computation (for their measurement and comprehension). What is the purpose of such a discussion? Whatever there is in all the three worlds, made of moving and non-moving beings ̶ all that cannot exist indeed as apart from measurement.’10

It is acknowledged that the Gaṇitasārasaṃgraha is the first known independent treatise on mathematics written by a Jaina to have survived. What are then the earlier textual sources of mathematical ideas in Jainism? Differently from ancient Greece, in the Hindu context mathematics developed apart from philosophy. Is this also true in relation to the history of mathematical thought in Jainism? By addressing these questions, my intention is to understand what mathematics represented to those writing Jaina doctrine. Between the closing phase of the Vedic period and the middle of the first millennium CE, mathematical ideas were thoroughly elaborated by Jaina scholars. A striking amount of calculations related to concepts of cosmology, philosophy, and religious beliefs occurs in works expounding early Jaina doctrine.11 It will take us much far afield to discuss in details the nature of calculations found in Jaina non-mathematical works; for obvious reasons of space, in this paper I limit myself to mention some of them.

9 See Tattvārthādhigamasūtra 8.4. 10 All the translations presented in this dissertation are my own, unless otherwise stated. 11 The post-canonical Jaina literature texts of both the and Śvetāmbara traditions are called anuyogas (“expositions”), although this is a term mainly used by . Among the four anuyogas is karaṇānuyoga or the exposition on calculations and techniques. It deals with technical matters and include texts on astronomy, cosmology, and karma.

Quantification and the Jaina Mechanistic Portrayal of the Universe

Understanding the Jaina worldview and karma theory means understanding the Jaina philosophical theory of cosmological structures.12 The traditional Jain conception of the Universe is extremely important in religious terms for all Jaina sects.13 These broadly agree on theories of cosmology but the two main groups of Śvetāmbaras and Digambaras differ in certain areas.14 Both sects support the religious authority of the Tattvārthādhigamasūtra (henceforth TĀS) by Umāsvatī (ca. fourth–fifth centuries CE), which is a key Jain text with a long section on cosmology; this underscores the crucial part of cosmological theory in the basic tenets of Jaina belief.15 The details of the Jaina Universe can be overwhelming;16 the numbers and the multitude of descriptions and classifications used to describe its extent, its time cycles, and the number of beings are all extremely large.17 The Jain universe, a complex structure of multiple continents and encircling oceans, is thought of in terms of dimensions and quantities of units.18 This Universe is "a self-replicating composite" (Granoff 2009: 56). The Universe () consists of three divisions, situated one above the other: ūrdhvaloka or the “upper world”, madhyaloka or the “middle world”, and adholoka or the “lower world”. The first is the abode of the celestial beings, the second of men and other creatures, and the third of hellish beings.

12 In Jainism, many mathematical ideas developed out of cosmological theory. A study on sequences and progressions is by R.C. Gupta (1992a: 69–94). 13 Interestingly, the sixteenth and seventeenth centuries seems to have seen a revival of interest in cosmographical texts and the appearance of large painted representations of the cosmos. Dundas 2007 suggests that this interest in cosmographical texts may represent a response to new geographical texts and astronomical knowledge brought to India by Muslims and Christians present at the Mughal Courts. 14 I limit myself to mention various sources rather than to distinguish Digambara and Śvetāmbara’s views. 15 In TĀS 3, 4, and 5, one of the most concise, comprehensive, and earliest accounts of occurs. See the English translation is by Tatia 1994. 16 Explanations and images of the Jaina cosmos are found in Caillat and Kumar (1981) and Granoff 2009. See also Kirfel 1920. For more recent Jaina representations of the cosmos, see Hegewald 2000. 17 In this section, I rely on the description of the Jaina theory of the cosmos found in Bossche 2007, Cort (2001, 2009), Dundas 2005, Granoff 2009, and P.S. Jaini 1979. 18 Jainism considers the universe to be without beginning and without end; it is without the characteristics of permanence, origination, and perishing. It was made by no one and is supported by no one. It is self-produced and, moreover, remains in space without support.

Jains meditate on the structure of the Universe as much as on transmigration, meaning the circulation of the soul through the Universe. For Jainas, meditating on the Universe and understanding how it works (the interconnectedness of its structures, the soul moving through the cycle of birth) is crucial to progressing spiritually on the way to salvation. In TĀS 9.7, it is said that there are twelve types of contemplation (anuprekṣā). Among the twelve anuprekṣās, there is the “contemplation on the Universe” (lokānuprekṣā). He who practices contemplation in this way is enabled to practice moral virtues and consequently to free the soul from the bondage of karma.19 The main purpose for the study of cosmology is to understand that humans live in a very small fraction of the cosmos, and that to be born human means to have the possibility to spiritually progress toward liberation, a condition reserved only to human beings. This is one of the reasons why concepts and explanations related to cosmological issues are also found in religious and philosophical texts. In Jainism, the main way of communicating cosmological ideas has been through written texts (scriptures, cosmological treatises, and narratives), and visual art,20 which was developed from the earliest times. Visual tools, such as cosmological paintings, convey messages on the stark reality of existence,21 so as to “wake up” people to the ignorance that causes vast suffering.22 As Cort (2009: 44) observes: ‘The wise Jain who views a painting of the cosmic man or a map of the middle realm where humans live, who contemplates the immensity of the ever-repeating , and who studies and understands the large number of bodies in which a soul may find itself experiences a combination of cognitive understanding and existential feeling that simultaneously evokes wonder and fear.’

19 On the Jaina foundational theory of karma, for instance, Glasenapp 1942, P.S. Jaini 1980b, and Johnson 1995. Recent studies are by Appleton 2014 and by Wiley 1999, 2000, and 2003. 20 In the essays edited by Granoff 2009, one finds images of stunning works of art produced by and for Jainas. 21 Some are miniatures in manuscripts of cosmological texts, other are monumental paintings in the walls of temples and monasteries. 22 Jain story literature of the eighth century onwards proves that portable paintings of the universe were used to teach cosmology (see Cort 2009: 43).

In this context, the Jain term saṃvega denotes both a fear of saṃsara and a joy for the understanding of the salvific message of . The experience of viewing the depiction of the cosmic man operates on many levels: aesthetic, intellectual, and at a deeper level a spiritual experience with the commitment of transforming change one’s own life so to attain the most and best thing: a human life. Numerous textual evidences indicate that the quantification was for Jainas an expedient enabling an exact description of reality, and all its aspects, in mathematical form. Numbers and mathematical procedures occur frequently in Jaina non-mathematical texts. For instance, these are found in the Sarvārthasiddhi by Pūjyapada (sixth century CE), who is the first scholar to write a commentary on the TĀS, and a few centuries later, the Dhavalā of Vīrasena (ninth century CE) presents computations much elaborated while commenting upon the Ṣaṭkhaṇḍāgama, a Digambara work dealing with the nature of karma. In the Gommaṭsāra, numerical data and detailed calculations provide minute descriptions of matters related to karma and liberation.23 This work covers the most important concepts of Jaina philosophy and lays down the ways and means to liberation.24 This Prakrit text expounds the main teachings of Jaina philosophy and is traditionally attributed to Nemicandra Cakravarti (eleventh century CE?), who is also the author of the Dravyasaṃgraha, another authoritative text.25 The Gommaṭsāra consists of two parts: the Jīvakāṇḍa (henceforth GSJK) and the Karmakāṇḍa (henceforth GSKK). The first part deals with the natural characteristics of the souls (jīvas) and the means and stages of their development, while the second part describes the obstacles producing the bondage of karma that must be removed. What are the implications of all of this for our understanding of quantification and its relation to conceptualizations of space-time within early Jainism? It can be hardly ignored that objects and other phenomena become conceivable and thus real if we can measure them, if we can establish relations among them. Relations give rise to order and prediction; what is perfectly ordered can be absolutely predictable. As Granoff (2009: 55) observes, the Jaina world is “a world of perfect order”. I claim that in Jaina thought, mathematics represents an approach to knowing which is able to establish models for

23 On the formulae found in this text, see Datta 1935 and Jadhav 2015. 24 See the edition and English translation by J.L. Jaini 1927. 25 Text and English translation by Ghoshal 1917.

understanding the Laws of the Universe and the human predicament. In a such context, quantification responds to a quest for order. What I mean is that for Jainas, quantifying procedures embody the possibility of obtaining knowledge of some sort about the objects of reality they investigate. In this way, the Universe becomes real, knowable, and measurable. Numbers are accurate means in as much as they are able to grasp the reality of idealized depictions, as no process of discerning measurements or verifying their accuracy is involved. Does quantification fit with the traditional standard ways of knowledge (jñāna)? 26 I would not say that quantification is part of the Jaina epistemological model; it is not a matter of the nature and scope of knowledge, and whether it is legitimate or not, but rather a form of cognitio by which the rational descriptions produced impel a soteriological system. Quantification is hence as an intellectual, learning process stressed at its explanatory power and by which acquiring understanding on the human condition in the Universe, which is further processed for realising the spiritual truth. In Jainism, numbers and calculations are part of space-time geometries and representations providing a constructive and ontological framework and a moral imperatives as well as the vast time scales of karmic fruition and the rarity of human birth impel what Cort calls the “aesthetic shock”. He underlines (2009:36) that: ‘the Jain who understands the nature of reality sees that the cosmic man, the standing figure who presents such a striking image, represents a world filled with ‘countless souls [who] wander without knowledge’. Progressing spiritually requires understanding and meditating upon these cosmological theories’.27

Among Jainas, the main purpose for the study of cosmology is to understand that humans live in a very small fraction of the cosmos, and that to be born human means to have the possibility to spiritually progress toward liberation, a condition reserved only to human

26 In TĀS 1.9–1.31, it is said that there are five kinds of knowledge (jñāna). According to TĀS 1.9, these are: sensitive (mati), scriptural (śruta), visual (avadhi), mental (manaḥ paryaya), and perfect (kevala). Knowledge (jñāna) is obtained by means of pramāṇa and naya. 27 The three worlds of the Jaina Universe (the lower, the middle, and the upper) are usually depicted in a diagram known as the “Cosmic Man” (lokapuruṣa). The stylised body of a human figure is divided into three parts, each standing for the one of the three worlds. Three layers of air surround the three : the first layer is humid, the middle dense, and the outer rarefied. Beyond these, there is the alokākāśa.

beings. This is the reason why in religious and philosophical texts concepts and explanations related to cosmological issues play such a significant role. My argument is that for early Jainism, calculations seem to unfold a possibility of discovery, a reliable method of truth-preserving able to present an intrinsically simple and mathematically elegant, unified picture of reality. In the Jaina mechanistic portrayal of the Universe, mathematics is functional to the explanation of how karma and rebirth work which indirectly, through the karma doctrine, supports the understanding of proper conduct (samyag caritra). Jaina worldview is based upon a systematic mode of organizing reality from which the realm of religious experience unfolds. If you can measure, the experience becomes almost sensory, immediate. The elaboration of mathematical means to express and predict the measurability of a property –concepts such as time, space, the lifespan of beings– enables the identification of numerical laws. This makes it possible to reproduce universal, eternal intelligible structures that determine a system of laws of ideal constructions. It is evident that Jainas ‘strive to give a precisely calculated account of the structure of the cosmos as well as of the various phenomena which they observe in it or deduce to be there’ (Caillat and Kumar 1981: 34). Jain cosmology is, in fact, based on mathematics, both calculations and geometry formulae. As Plofker (2007: 69) put it ‘the Jain concept of the thoroughly quantified cosmos was not meant to mimic empirical perceptions but to transcend them in a world that allowed the fusing of math and myth’. Numbers and mathematics underlie symmetry and repetition, so that ‘to know one part is to know the whole’ (Granoff 2009: 55). Hence, coming back to my earlier concern, while in the Hindu context, before becoming an independent discipline mathematics was the chief matrix of astronomy and was linked to calendric knowledge for the performance of rituals, it can be asserted that in early Jainism mathematics was part of cosmological speculations and a fundamental expedient of philosophical enquiry. Therefore, a central topic for further research is deepening our understanding of the content and scope of mathematical knowledge in relationship to other areas of Jaina thought. The pursuit of investigation into these topics should be motivated not only by the need to clarify the epistemic function of quantification, but also by the prospect of contributing to other areas of philosophical discussion concerning Jainism.

Time as a Substance

Jainism displays a notion of time which is atomistic.. How such a notion of time is related to the analysis put forward in this paper? I have thus far suggested that, in Jainism, the comprehension of the order of reality, its structures, relations, change, and process appeal to a type of discourse which proceeds by means of quantification. In this respect, I argue that the Jaina theory of , its pragmatism, and the realistic view of karma represents the matrix on which detailed descriptions based on numbers were conceived. With respect to this, in order to understand the relation between philosophy and mathematics, Jaina view on the ontological nature of time is a fundamental issue; this concerns speculations regarding the nature of matter, being, becoming, as well as the duration of karma and the various processes which karmic matter undergoes. Time is a substance, although it has no corporeality, and consists of an infinite number of atoms which never mix up. According to Jainism, five of the six “substances” () which constitute the whole Universe are corporeal (astikāya).28 Time” (kāla) is not astikāya;29 it has existence, it is a substance, but it is not corporeal. All the six entities of reality undergo countless modifications, having a permanent substantiality unchanged. TĀS 5.38 ̶39 states that a substance persists through its own “qualities” (guṇa) and “modifications” (paryāya). Jīva has the nature of cetana (“consciousness”) and upayoga (“perception”), while ajīva has structure, colour, taste, touch, smell, and sound.30 Moreover, the characteristic of a substance is “being” (sat); it possesses "becoming”

28 The six substances are: “soul” (jīva), “matter” (), “principle of motion” (dharma), “principle of rest” (adharma), “space” (ākāśa), and “time” (kāla). The six substances are also divided into two broad categories: jīva and ajīva (which include the latter five substances).The jīvāstikāya (soul conglomerate), pudgalāstikāya (matter conglomerate), dharmāstikāya (motion conglomerate), adharmāstikāya (rest conglomerate), and ākāśāstikāya (space conglomerate) altogether constitute the pañcāstikāyas. In the five corporeal substances, the space-points are not separate from each other, rather they are a conglomeration. 29 See TP 1.92, TĀS 5.1 ̶3.39, Niyamasāra 9, and Uttarādhyayanasūtra 28.7. However, among Jaina scriptures there is no unanimity in viewing kāla as an independent substance. The Digambara tradition reads TĀS 5.39 “kālaś ca”. The Śvetāmbara recension reads: kālaḥ ca iti eke or “Thus according to others, time is also a substance”. 30 See, for instance, Dravyasaṃgraha 2; Pañcāstikāyasāra 1.16, 124, 126 ̶127.

(utpāda), “decay” (vyaya) and “permanence” (drauvya).31 Ajīva is classified into two groups: that which has “form” (mūrta) and qualities ̶such as “matter” (pudgala)– and substances “without form” (amūrta),32 such as dharma, adharma, and ākāśa.33 They are eternal, uncreated and of enormous magnitude, having innumerable and indivisible “space-points” (pradeśa).34 “Time” does not have extensiveness because it has only one space-point, and it is not thus a conglomerate. It is a mono-dimensional category, not having spatiality or the quality kāyatva.35 In TĀS 5.22, it is said that the function of time is to assist substances in their “continuity” (vartanā), “transformation” (pariṇāma), “motion” (kriyā), “temporal priority” (paratva), and “temporal posteriority” (aparatva) of the five substances. Time is thus the instrumental cause of all modifications in the five substances. Substances exist by themselves with their modes, and time conditions the changes in things. Being the condition of these changes does not oppose its characteristic of inactivity, as it is simply an accompanying cause. The three modifications of a substance are possible only because of time. Also, time is said to be twofold: “absolute time” (niścayakāla) and “relative time” (vyavahārakāla). Niyamasāra 33 observes that time is that which helps all substances in undergoing modifications. GSJK 568 points out that time is the cause of continuity in being and all substances undergo change through the support of time. Time never alters itself; it is merely the auxiliary cause of different kinds of modifications in other substances (GSJK 570).36 “Relative time” (vyavahārakāla) is that which is known from modifications in substances and from the movements of the heavenly bodies, while “real or absolute time” (niścayakāla) is that which helps to produce changes in substances and support the other five realities in their continuity. 37 Vyavahārakāla has a beginning and an end, while niścayakāla is eternal. With the help of the relative time, it is possible to determine temporal priority and temporal posteriority of substances and events (Pravacanasāra 107). Uttarādhyayanasūtra 28.10 and the Tattvārtharājavārtika, commenting on TĀS

31 Pravacanasāra 2.50 ̶52. 32 Dravyasaṃgraha 15. 33 TĀS 3.6 explains that dharma and adharma sustain the stability of the cosmos. 34 Pravacanasāra 1.4. 35 Pravacanasāra 1.102. In 2.49, this text observes that time has only vertical extension, while the quality of kāyatva is a horizontal extension. 36 Time does not cause the changes produced in substances but indirectly supports the production of such changes. 37 GSJK 577.

5.22, observe that the characteristic of absolute time is vartanā, by which the perception of the existence of a substance is observed at every moment by means of the modifications occurring in it. Absolute time consists of instants or extremely small points of time (kālāṇu). These atoms have no colour, taste, smell, or touch.38 Dravyasaṃgraha 22 and Vardhamānpurāṇa 16.35 state that innumerable grains of time reside, one in each space-point of the finite Universe (lokākaśa), like heaps of jewels. According to Jainism, the whole universe is full of these grains of time. These units never mix with one other but are always separate (thus the metaphor with jewels). In Jaina thought, reality is made up of kālāṇus, pradeśas and paramāṇus or grains of time, cells of space or space-points, and ultimate indivisible particle of matter. 39 These grains are invisible, without form and inactive, in static conditions and countless in number.40 Relative time is made up of conventional periods measured by standard units. The smallest unit of time is the samaya, which is the time taken by the smaller part of matter in going from one space-point to another. Among the largest units is the śīrṣaprahelika,41 which corresponds to 8,400,00028 years. The biggest unit of time is the mahākalpa, which consists of two aeons: avasarpiṇī utsarpiṇī are the aeons of increase and decrease, or of unwinding and rewinding of the universe respectively. Both phases, the down and up phases of the time-cycle has a similar length and similar sub-divisions. Each aeon consists of 413452630308203177749512192 × 1050 (77 digits) solar years, and each aeon has six ages.42 Another immense period of time is the vyavahārapalya. This is mentioned in the Tiloyapaṇṇaṭṭī 1.94 (henceforth TP) and in TĀS 3.6 in relation to the age of hellish beings and it is said to be the time taken when a circular receptacle of one yojana 43 diameter filled tightly with fine human hairs of the length that in a normal adult would grow in seven days, is emptied by taking out one hair every hundred year. TĀS 3.38 states that the maximum and the minimum periods of lifetime of human beings are three palyas and one antarmuhūrta. The palya is of three kinds: vyavahāra, uddhāra, and

38 TĀS 5.4. 39 The paramāṇū is sub-divided into suhuma or “subtle” and vavahāra or “practical”. 40 There is only one kālāṇu in each space point. 41 The unit of time śirṣaprahelikā is composed by 194 digits: 758263253073010241157973569975696406218968848080183296 followed by 140 zeroes. 42 The brahmakalpa of the Hindus too consists of 77 digits, but the numbers are not the same. On the relation between Jaina, Hindu, and Buddhist see Kirfel 1920 and Ohira 1994. A recent study is by Satinsky 2015. 43 A yojana is a measure of distance.

addhāpalya. The first is the basis for measuring the others.44 For instance, TP 1.94 observes that the durations of karma are determined by the time measuring unit addhāpalya. Also, innumerable vyavahārapalyas make 1 uddhārapalya, innumerable uddhārapalyas make 1 addhāpalya, 10 × 1014 addhāpalyas make a sāgara. There are six ages in each aeon. For instance, the six ages of avasarpiṇī are: 45 1) sukhamāsukhmā= 4 × 1014 sāgaras 2) sukhamā= 3 × 1014 sāgaras 3) sukhamāduḥkhamā= 2 × 1014 sāgaras 4) duḥkhamāsukhamā= 1× 1014 sāgaras minus 42,000 years 5) duḥkhamā= 21,000 years 6) duḥkhamāduḥkhamā= 21,000 years Altogether it gives 1 daśakoḍākoḍi sāgara or 10 × 1014 sāgaras. Large units of time, such as the hastapraheli = 8,400,00016 × 8414, are also mentioned in the Trailokyadīpikā by Indravāmadeva and in the Trailokyasāra by Nemicandra.46 In the context of such a discussion, it is significant that while expounding philosophy of time Jaina thinkers make use of rhetorical devises. For instance, the Aṇuogaddārāiṃ uses similes as literary devices to explain large units of time and space. For instance, the number of time-instants is said to be of two kinds: the first is the number indicated by the simile of a store and the second is indicated by the simile of ocean. In sutta 372, this work mentions a store of corn one yojana in length and breadth, and one yojana in height; the circumference is three yojanas. This store is completely filled with many koṭis (1 koṭi= 1.000.000) of hair-tips grown in seven days. The time taken for taking out these hair-tips one by one at each samaya until the store becomes empty corresponds to one uddhārapaliovama.

44 TP 1.123 ̶124 lists the digits of the palya, which are thirty-two followed by eighteen zeroes at the end. 45 See, for instance, the Sarvārthasiddhi on TĀS 3.27. 46 The Trailokyadīpikā and the Trailokyasāra, two cosmological texts give a list of measuring units of time according to the Digambara tradition. See Kirfel (1920: 338 ̶339). Satinsky (2015: 5) observes that ‘In the Śvetāmbara and Digambara traditions, “the number eighty-four and its multiples” are omnipresent in the category of “calculable” (gaṇiya) time measures. Their function is to designate calculable time periods of great magnitude within the osappiṇī (“down- moving”) and ussappiṇī (“up-moving”) two half-motions of cosmic time’.

The TP gives a number representing the unit of time palya, which is composed of thirty-two digits followed by eighteen zeros. It is there explained that three pits of the extent of one yojana long, one yojana broad, one yojana deep based on the measure of pramāṇāṅgula are dug out. These are packed with the smallest ends of the wool of rams from one to seven days old. Then the small bits of wool are taken out one by one in every one hundred years; the time taken to empty the pits in this manner is vyavahārapalyopama. Each bit is again cut into as many pieces as there are instants in innumerable koṭis of years. The pits are filled up with such bits. Then these bits are taken out one by one every instant; the time taken for emptying the pits is called uddhārapalyopama. Ten koṭis multiplied by one koṭi uddhārapalyas make up one uddhārasāgaropama. The continents and oceans are as numerous as the bits in two and a half uddhārasāgaropamas. The pits are filled with bits got from cutting each bit of uddhārapalya into the number of instants in one hundred years; this denotes the period of time called addhāpalya. Then these bits are taken out one by one every instant; the time taken to empty the bits is called addhāpalyopama.

Matter and Space

In Jainism, the first measuring unit of space is the aṇu, which is constituted by an infinite number of paramāṇus. In his mathematical work, Mahāvīrācārya mentions the basic principles of the Jaina theory of atom and begins the list of space measuring units by observing that a paramāṇu is the imperishable unit of matter (pudgala) which neither fire or water can destroy.47 An infinite number of paramāṇus makes an aṇu. Niyamasāra 20 states that the substance matter pudgala, constituted by the five guṇas colour, smell, taste and touch, is of two kinds: “atom” (aṇu) and molecule or “aggregates of atoms” ().48 This text specifies that the karaṇaparamāṇu is the “cause-atom” of the four root-matters earth, water, fire, and air. The kāryaparamāṇu is the “effect-atom” and

47 See Gaṇitasārasaṃgraha 1.25 ̶29. 48 See also Dravyasaṃgraha 26. These are of two and six kinds respectively: gross-gross, gross, gross-fine, fine-gross, fine, fine-fine. Karma is matter made up of very fine karmic molecules.

smallest part of a molecule. When any molecule is dissolved into the smallest possible atoms, the atoms obtained are called effect-atoms. Niyamasāra 26 specifies that a paramāṇu is that substance which is the beginning, the middle and the end by itself, not apprehensible by the senses, and indivisible. When two or more atoms are combined together, a molecule (skandha) is formed. 49 The text explains that atoms of matter are “numerable” (saṃkheya), “innumerable” (asaṃkheya), and “infinite” ().50 TĀS 5.8–16 states that the every substances is made up of indivisible particles. The space occupied by an atom is called pradeśa, which contains not only atoms of matter, but also particles of other substances. The particles of the five of the six substances are mixed-up so they have many pradeśas. Kāla on the other hand consists of particles which never mixed up and it is said to have only one pradeśa.51The Sarvārthasiddhi, while commenting upon TĀS 8, states that paramāṇu is the primary atom, it occupied a single point of space and is eternal. The material atoms possess the capacity of compressing an infinite number of themselves into one molecule. Fire, air, earth and water have their constitutive cause in the atom. The primary atom is indestructible and eternal and is ekapradeśa, as it occupies only one point. Also, commenting on TĀS 8, this work explains the difference between the categories “innumerable” and “infinite”. “Innumerable” (asaṃkheya) is what can not be counted and it is of three kinds: minimum, maximum and that which is neither minimum nor the maximum. “Infinite” (ananta) represents numbers beyond innumerable. While “innumerable” means that it is not possible to count but it has a limit, “infinite” means “endless” (Sarvārthasiddhi on TĀS 9).52 In Jainism, “space” (ākāśa) is one of the six . TĀS 18 observes that the function of space is to accommodate other substances. The Tattvārtharājavārtika commenting on TĀS 18, which explains the interpenetrability of ākāśa, states that as water allows a swan to enter in itself, so ākāśa allows the other substances to penetrate itself. Another important aspect of the Jaina idea of space (thought more related to cosmology) is the difference between lokākāśa and alokākāśa. According to Jainas, the cosmic space (lokākāśa) does not exhaust all space, for it is a tiny part of the great “non-

49 These are: gross-gross, gross, gross-fine, fine-gross, fine, and fine-fine aggregates of atoms. 50 See also TĀS 5.10. 51 According to Dravyasaṃgraha 22, substances have innumerable pradeśas. 52 Jaina theorization of “numerable”, “innumerable”, and “infinite” is investigated in Petrocchi 2016.

world space” called alokākāśa. 53 This concept id unique to Jainism, no parallel to this is found in other Indian cosmologies; it appears for the first time in the Bhagavatīsūtra. In few words, that part of the space (ākāśa) that is pervasive and in which substances exist is known as lokākāśa,54 that which is beyond is alokākāśa.55 With the above in mind, what can be said about the way Jaina thinkers often expound metaphysics and answer questions related to the nature of reality, karma and, hence, salvation ? Here the most important matter is to establish that quantification offers to Jaina thought an important method of categorising and hence understanding the complexities of the Universe and of the conditions of the human predicament. I shall provide some examples which demonstrate that in the Jaina tradition the atomistic characteristic of time and its derivative conceptualization of space produce, as I earlier mentioned, a distinct philosophical enquiry based on qualitative methods of classification and quantitative means of calculation. The Aṇuogaddārāiṃ (ca. fourth century CE) is a Prakrit text belonging to the Śvetāmbara Āgamas56 and deals with investigations on hermeneutics.57 This work explains the methods for the investigation of a scriptural text and deals with everything a Jaina is required to know. The Aṇuogaddārāiṃ offers a significant example of the way in which Jainas thinkers have produced a vast vocabulary to describe and understand units of time and space, sets of realities, and matters related to karma and liberation. Petrocchi 2016 demonstrates that the Aṇuogaddārāiṃ expounds a Jaina theory of numbers where calculations and concepts such as innumerable and infinite are mentioned in relation to philosophical/religious issues.58 The argument of this author is that the Aṇuogaddārāiṃ exemplifies the way mathematics was a perfect, functional instrument for representing Jaina cosmological and soteriological model.

53 Lokākāśa is measured in rajjus.A rajju or “rope” was defined by medieval Jaina cosmographers as ‘the distance covered by a god flying non-stop for six months at a speed of 2,057,152 yojanas (say 10,000,000 ) a second’ (Caillat and Kumar 1981: 20). 54 Pañcāstikāyasāra 90. 55 Dravyasaṃgraha 19. The alokākāśa is eternal, formless, and without activity. 56 The word Āgama denotes a group of sacred texts considered to be authoritative. An early study on Jaina Āgamas is by Weber, see the English translation by Smyth 1999. Texts belonging to the Jaina canon are also found in the study of the Jaina doctrine by Schubring 1935. 57 Published in 1968 by Puṇyavijayo, Mālavaṇiyā, and Bhojaka. English translation of the text by Hanaki 1970. 58 See Petrocchi 2016.

In the GSJK, conceptualizations of space-time and their relations to karma and salvation are significantly expounded by means of numbers and calculations. For instance, GSJK 113–117 gives the exact numbers (all very large numbers, with a minimum of 13 digits) of the various kinds of bodily materials (kinds of matter) assimilated by earth-bodied, water-bodied, fire-bodied, air-bodied, and vegetable-bodied beings; by two-sensed, three-sensed, and four-sensed beings; and by five-sensed water- inhabiting, air-flying, quadrupeds, serpents, human, celestials, and hellish beings.59 The total number of the various kinds of bodily materials assimilated by such beings is 197500000000000 (gāthā 117). Furthermore, the number of the class of celestial beings called Vyantaras and the Jyotiṣkas is calculated by dividing the universal surface area60 by 300 hundred square yojanas and 56 square , while in the first and second the number of Bhavanavāsi devas are obtained multiplying the universal line by the square and third root of one cubic (ghanāngula), and so on in relation also for the number of other heavenly beings. Adjectives such “infinite” and “innumerable”61 are used to denote t molecule of atoms forming a unit of bondage (samayaprabandha), which represents the number of karmic or quasi-karmic molecules that the soul attracts in one instant.62 The text gives the method to ascertain the amount of molecules of the units of bondage of the five bodies: physical, fluid, assimilative, electric, and karmic.63 In chapter 10, one finds the method to ascertain the number of humans and sub-humans as well of each class of hellish and celestial beings affected by four passions (pride, anger, deceit. and greed). The last chapter mentions units and conceptions of time and space, twenty-three kinds of molecules, the six kinds of skandas (aggregates of atoms), the various degrees of smoothness and roughness in matter (these are said to be the causes of union of atoms into molecules), and the number of saints between the sixth and the fourteen stages of the guṇasthāna– their sum is 89999997, which is the number of saints

59 All the sub-classes constitute the yoni or physical centre. 60 Jainism considers the universal line (jagaśreṇī) to be 7 rajjus in breadth. The jagaśreṇī is mentioned as a unit of measure in TP 1.93 61 Gupta 1932b is a study on the category “innumerable”. 62 See GSKK 4. 63 Audārika, vaikriyika, āhāraka, , and kārmana are the five śarīras or “bodies”. These have sub-categories (GSJK 230–244). To the five śarīras correspond five kinds of molecular bondage and five kinds of molecular vibration. Karmic matter is bound to the soul by the vibrations () of speech, mind, and body. The five bodies are also listed in TĀS 2.36.

existing at every instant.64 Gāthā 634–637 gives a rule for calculating the number of celestial beings in vowless, mixed, and downfall stages at every instant. GSJK 577–580 specifies that practical time consists of short or long durations such as hours, minutes, and seconds, and it is of three kinds: past, present, and future.65 The number of the time- atoms in the past is said to be equal to the number of the liberated souls multiplied by saṃkheyāvalis or “numerable winks”. 66 Explaining GSJK 578, J.L. Jaini observes that:

‘in every period of six months and 8 instants, 608 souls leave the primitive common, or Nitya , condition; and the same number of souls enter the abode of liberation from 2 ½ continents. The number of or liberated souls is infinite part of the total of all the souls (mundane and liberated). 6 months and 8 samayas being reduced to āvalis and divided by 608 would be the numerable āvali mentioned in the gāthā. It would be noticed that numerable āvalis in the gāthā are constant. They do not vary. They represent the average time for one soul to attain liberation. The number of liberated souls increases every 6 months and 8 instants by 608. This explains the ever-increasing-length of past time by the constant number of āvalis.’ J.L. Jaini (1927: 288 ̶289)

Concluding Remarks

It has been demonstrated that within Jainism, quantification represents the attempt to reproduce universal, eternal intelligible structures which determine a system of laws of ideal constructions. The main argument of this paper is that numbers and mathematical practices served to construct and communicate ideas about the Universe, karma, and the

64 The “scale of salvation” (guṇasthāna) represents the path to liberation, which in Jainism proceeds through fourteen stages of purification. 65 GSJK 579 states the future practical time is infinite times the total number of all souls and all substances. 66 One āvali is the twinkling of an eye.

human condition, and hence were instrumental to Jaina religious and philosophical enquiry. A thorough investigation of philosophy of space-time highlights that, within early Jaina doctrine, numbers represent means able to reproduce the metaphysical structure of the Universe. Jaina cosmology reflects a structured and ordered Universe; the Jaina cosmos is perfectly predictable. Text expounding early Jaina doctrine offer elaborate conjectures about matter, the whole of reality and its spatial and temporal aspects, the ideas of motion, and the geometrical structures characterising the cosmos. As Plofker (2009: 65) observes, ‘Jaina doctrine stands out as exceptionally bold and profound in its approach to quantification’. Textual evidence demonstrates that Jainas specify all minute descriptions with the help of numerical data and detailed calculations, and were profoundly committed to construct units of measures and quantities able to reflect their metaphysical theories. Cort (2001: 18) has drawn attention to the fact that, differently from other cosmologies, in the Jaina definition of the Universe both the spheres of cognitive, sensory soul and non-cognitive, non-sensory matter exist. In Jainism, bondage is physically real and a deep, correct knowledge of the schemes in which the Universe operate, on all the kinds of beings, their conditions and places in which they live, represent an object of meditation that eventually, together with the performance of strict ascetic practices, may lead to liberation.

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