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 Universe 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 Jainism, 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 nature 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 India, 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 Vedas 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 karma 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, cosmology 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 length 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 Sanskrit 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