The Buddhist Philosophy of the Middle

1. Mathematical and Linguistic Models in Indian Thought: The Case of Zero and Śūnyatā

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1. Mathematical and Linguistic Models in Indian Thought: The Case of Zero and Śūnyatā1


BEGINNING WITH the ancient Greek thinkers, the exact sciences have played what may be called a paradigmatic role in the formation and development of Western thought. And as the exact science par excellence, mathematics has indubitably exercised there a fundamental influence not only in the natural sciences but also by extension in the sciences of man inclusive of the “humanities.” Especially in linguistics and philosophical logic mathematical models, formalizations, and calculi occupy a prominent place. And despite the fact that Saussurean structuralism was not derived from mathematics, the mathematical mode of thought has clearly not been without impact in the development of some forms of structuralism. At the same time linguistics and the linguistically based form of structuralism have also exercised very considerable influence, so that a linguistically inspired paradigm has taken an important place in the main stream of modern Western thought.2

In this respect a certain convergence has arisen, consciously or unconsciously, 3 between modern Western thought and classical Indian thought. For when the Indologist looks for ideas and methods that have played a modelling and paradigmatic role in the history of Indian thought, it is probably 2 above all to the grammatical śāstra that he would turn. The Indian thinkers have indeed themselves pointed to the paradigmatic significance of grammar in their civilization, and one of them has referred to it as a universal science (sarvapārṣada-śāstra).4 Mathematics appears on the contrary to have generally occupied a relatively less prominent place in the history of Indian thought taken as a whole.


At least one interesting exception to this general tendency may, however, be recalled here. As is well known, the Abhidharma schools of Buddhism developed a theory of elements or factors (dharma), each of which is considered to bear its own specific characteristic that determines it.5 But the precise nature of the dharmas was the subject of much discussion in the Buddhist schools, which differed appreciably as to their ontological status; and one of the major problems that arose concerned the nature of a dharma with respect to its efficiency (kāritra) in the three times, namely present, past, and future. The Sarvāstivāda or Vaibhāṣika school maintained that the dharmas exist in all three times, which are in fact determined by just this efficiency. Thus, when a dharma exercises its kāritra, it is said to be present; when it does not yet exercise it, it is said to be future; and when having exercised its efficiency it has ceased, it is said to be past.6 But there was disagreement among the Ābhidharmikas as to how such a dharma goes from one point in the time-scale to another.

Four distinct solutions to the problem have been recorded by Vasubandhu in his Abhidharmakośabhāṣya (v.26). One solution is ascribed to Dharmatrāta, who maintained that there takes place in time a change of condition (bhāvānyathātva) rather than of substance (dravya), like in the Sāṃkhya system; this change is then compared to the change (pariṇāma) of 3 milk to curd, which involves a change of one property—e.g., taste—but not of another—e.g., colour. The solution connected with the name of Ghoṣaka maintained that the change in question is one of characteristic (lakṣaṇa), which does not involve a change of condition; the process is then compared to the case where a man becomes enamoured of one woman without being free of passion for others. Another solution is ascribed to Buddhadeva, who spoke of relational otherness, explaining that a dharma that evolves in time may be said to be other (anya) with respect to the preceding and the subsequent without this involving a change of substance, just as one and the same woman is said to be a mother in relation to her child and a daughter in relation to her own mother.7 In each of the three solutions thus offered, the example adduced to illustrate the process of change in time remains of an everyday, non-scientific kind.

The case is somewhat different in the fourth solution connected with a certain Bhadanta Vasumitra, who is probably to be identified as one of the leading figures at the time of Kaniṣka’s Great Council.8 Vasumitra explained the process in question by saying: “A dharma evolving in the [three] times is stated to be other according to the different states it enters, [the change in question being then] due to otherness of state (avasthāntarataḥ) but not of substance.” The example adduced to illustrate this is that of a marker or counter (vartikā) in reckoning which in the unit position has the value of a unit, in the hundred’s position that of a hundred, and in the thousand’s position that of a thousand.9 It is this explanation by Vasumitra that the Kośakāra has described (v.26c)—provisionally—as correct, although as a Sautrāntika Vasubandhu would criticize all these solutions (v.27).

Vasumitra’s explanation is of interest to us both because it adduces a more “scientific” example, and because it provides us with one of the earliest available references to arithmetical place-value and hence to one of the functions of zero.10


This example is indeed quite well known in Indian thought and it turns up in a number of later texts. An equivalent term—nikṣepavartikā—is to be found in the Vibhāṣāprabhāvṛtti.11 Yaśomitra in his Vyākhyā uses the term gulikā “ball, bead”; and in his Pañjikā on the Tattvasaṃgraha (1786), Kamalaśīla specifies that the counter was of clay (mṛdguḍikā). On the other hand the Vyāsabhāṣya on the Yogasūtra has the term rekhā “line, stroke”;12 and Vijñānabhikṣu glosses rekhā as the aṅkaviśeṣa, which denotes 100 when joined with two zeros (bindu), but 10 when joined with only one zero. The Vyāsabhāṣya differs from Vasubandhu by connecting the example of the rekhā in the different places and the example of a woman who may be either a mother or daughter according to the particular relationship in question (which in the Abhidharmakośabhāṣya was connected with Buddhadeva’s solution). The term rekhā in the same context is also found in Śaṃkara’s commentary on the Brahmasūtras (ii.2.17).

The terms vartikā and gulikā/guḍikā seem to presuppose the abacus system of reckoning, whereas the term rekhā evidently refers to a written zero (cf. Vijñānabhikṣu’s explanation).


In view of the fact that in the Sanskrit mathematical and astronomical literature the word śūnya has the meaning of zero, there exists another point in the history of Indian thought where, at least in principle, it might be supposed that mathematics has had some influence on philosophy: the Buddhist theory of the emptiness (śūnyatā) of dharmas. And some scholars have in fact suggested a connexion between the Madhyamaka theory of śūnyatā developed by Nāgārjuna and the mathematical zero.13


It has however to be noted in the first place that śūnya as a term for zero appears much later in our sources than the canonical Buddhist concept, and also after Nāgārjuna (no later than ca. 200 C.E.). The earliest datable attestations of śūnya in the meaning of zero seem to be in the Paitāmahasiddhānta of the Viṣṇudharmottarapurāṇa, which apparently goes back to the early fifth century, in the Bṛhatsaṃhitā (viii.20) of Varāhamihira (sixth century),14 and in another work by the same author based on five astronomical siddhāntas—the Pauliśa, Romaka, Vāsiṣṭha, Saura (Sūrya), and Paitāmaha—entitled Pañcasiddhāntikā.15 In Sphujidhvaja’s Yavanajātaka—a versified version, made in 269/270 C.E., of another Sanskrit work (the source of which was an Alexandrine manuscript in Greek) going back to 149/150 C.E.—the words used for zero are kha and bindu, terms that were to remain in regular use later. 16 In the new Sūryasiddhānta (i.29) kha is used for zero.17 Amongst literary works, the Vāsavadattā of Subandhu (ca. sixth century) employs the expression śūnyabindu for the symbol for zero.18

As already observed above, however, place-value (and hence a function of zero) was known to Vasumitra. And if this Vasumitra is the well-known contemporary of Kaniṣka, as may be the case, he could also be the contemporary of Nāgārjuna (if not older), and also of Sphujidhvaja’s Sanskrit source of 149/150.19


A few words need now to be said about śūnya and some cognate or semantically equivalent words.

Already in the Rgveda the cognate word śū́na appears, as a neuter noun, with the meaning of “lack, absence, emptiness” (in sentences introduced by the negative particle mā́). Later it is found as an adjective meaning “swollen, increased.”20 As for the word śūnyá, it appears as an adjective in the Brāhmaṇa literature with the meaning “hollow, deserted,” and then in the Epic with the meaning “empty, vacant (of a look); devoid,” etc. In the Epic śūnya is in addition attested as a noun meaning “desert; empty place.”

In the meaning of “nought, cipher, zero,” śūnya has as a synonym the word kha as well as such words as ākāśa, ambara, and viyat, all denoting empty space. The word kha is attested from the Ṛgveda in the sense of “hollow, cavity,” and from the Brāhmaṇa literature in the meaning of “empty space, sky, air.”21

It is to be remarked that, as a word denoting the unlimited (apeiron), ākāśa served in ancient Indian thought so to speak as a model for the concept of bráhman. And ākāśa and kha are in fact attested as names for bráhman.22

The word śūnya has been usually derived from the root śvi- (śū-) “to swell, increase, grow.” Hence, both etymologically and in its actual usage, śūnya is linked with the ideas of swelling and blowing up and hence of hollowing out.23

Finally, just as according to the Bṛhadāraṇyakopaniṣad (v.1.1) the full or “plenum” (pūrṇa, i.e., bráhman) is subject to neither increase nor decrease, in Buddhist thought the absolute element (dhātu)—which is empty (śūnya) only of all adventitious factors but aśūnya of certain constitutive informing factors—is subject to neither diminishing nor increase (Ratnagotravibhāga i.154-55). In this connexion it is interesting that the word pūrṇa has been used for zero by Bhāskara (twelfth century) in the Gaṇitādhyāya of his Siddhāntaśiromaṇi.24

But concerning the question whether the theory of śūnyatā was inspired by or modelled on a mathematical concept, in the present state of our knowledge we can only say that neither the history of the Buddhist doctrine 7 starting with the canonical scriptures, nor the semantic history in Sanskrit of the words śūnya and śūnyatā, nor the history of mathematics and astronomy in India appears to establish that the philosophical concept of śūnyatā in the Madhyamaka or elsewhere in Buddhism has anything to do directly with the mathematical zero. This is consistent with our initial observation that in ancient India mathematical concepts did not often serve as models or paradigms in philosophy. At the most one could perhaps speak only of a subjacent and remote resemblance or cognitive homology between independently developed concepts.


It may now be asked whether the theory of śūnyatā has connexions with the other model and paradigmatic type of thinking mentioned above: ancient Indian linguistic theory.

It is to be observed in the first place that the Chandaḥsūtra, a treatise on metrics attributed to Piṅgala, employs the word śūnya (viii.28-31). There the expression is used when a unit is not to be halved (since this would produce a fraction), so that it is there the sign of absence of a metrical operation.25 But although the Chandaḥsūtra has sometimes been dated as early as 200 B.C.,26 its date is uncertain, and the relevance of its testimony for the early history of the philosophical notion of śūnyatā is accordingly doubtful. Moreover, while śūnya in this text does clearly enough relate to the voidness of a particular operation so that its use might be compared with the philosophical śūnya which concerns a certain kind of inoperativeness—namely that of all entities since they are empty of own-being (i.e., aseitic existence, svabhāvaśūnya) this usage has no bearing on the meaning of the term in the older Buddhist literature, where the allusion is in the first place to emptiness.

In grammatical theory going back to Pāṇini too we meet with certain concepts which are of interest in the present context, even though their historical links with the śūnyatā theory are anything but perspicuous.

In the Aṣṭādhyāyī we find the term lopa “elision, syncope” defined as non-appearance of a linguistic element (adarśana, i.1.60). The term refers to the non-realization in the actual speech (bhāṣā) of ordinary usage (laukikaprayoga) of an element posited in grammar as existing on the level 8 of the sthānin, i.e., the theoretically postulated form. In other words, what is postulated on the level of the sthānin may, according to grammatical theory, not in fact be represented on the level of the ādeśa “substitute,” i.e., the linguistic element which is in actual speech substituted for the sthānin.27 And lopa as a zero substitute is then opposed to śravaṇa “hearing,” that is, realization of the sthānin by means of a full substitute-morph in actual speech.

Lopa = adarśana affects the postulated grammatical element that is termed prasakta “applicable”—or prasaṅgavat “having application”—on the level of the sthānin.28 And the sthānaśabda is then termed prasaṅgavācin,29 sthānin and prasakta being thus largely equivalent expressions.30

The concept of lopa thus corresponds, at least in part, to that of zero in modern linguistics.31 Furthermore, through lopa the sthānin might be regarded as in some sense void or empty on the level of real speech-usage. And perhaps to this extent a parallel (though a very limited one) presents itself with the dharmas which serve as theoretical factors of philosophical analysis, although in fact they are empty of own-being (svabhāvaśūnya) according to the Mahāyāna.

Despite the fact that the word śūnya does not form a part of the grammarian’s technical vocabulary, it is remarkable that the notion of a linguistic zero appears in close association with the idea of the prasakta/prasaṅgavat. Now in the Madhyamaka the theory of śūnyatā is quite closely linked with concepts expressed by terms derived from pra-saj-. Thus the Mādhyamika speaks of the prasakta (that which arises or applies within the frame of discursive development and dichotomizing conceptualization);32 of a prasaṅga (the unde 9 sired consequence resulting within this frame);33 of prasaṅgāpādana (a form of reasoning based on a reduction to absurdity or logical impossibility of a conceptual position postulating some entity);34 and of the prasajyapratiṣedha (absolute non-presuppositional negation which, contrary to relative or choice negation, does not commit one to asserting the contrary of what one has denied within the frame of discursive development and dichotomizing conceptualization, since it is shown by philosophical analysis that the subject of the negated proposition is null and empty and that no predicate or property whatsoever can therefore be ascribed to it).35

This terminological convergence is at least worthy of note even if it is difficult to pursue it further by showing that the two systems of thought in question—the Vaiyākaraṇa’s and the Mādhyamika’s—are directly related historically or systematically comparable.

At the same time there are undeniably very considerable differences between the grammarian’s zeroing of a postulated form and the philosopher’s “zeroing” of conceptual entities and factors of analysis (niḥsvabhāvatā, dharmanairātmya).

As has been noted above, the prasakta/prasaṅgavat form belongs in grammatical theory to the postulated level of the sthānin, which in the event of lopa is not actually realized by means of a full ādeśa in speech-usage. In the Śūnyavāda of the Madhyamaka school it is the saṃvrti or vyavahāra level of transactional usage that is comprised of the conceptual entities and the analytical factors which are postulated and applicable (“prasakta” so to say) in the frame either of conceptual thinking (prapañca and vikalpa) with its associated speculative views (drṣṭi), or of philosophical analysis. But in reality these entities and factors are empty of own-being; and on the paramārtha level they are simply not “realized.”36


Hence the relation between the two levels in question—the subjacent one of the postulated ideal sthānin and the actual speech-usage one of the ādeśa (with the prasakta on the sthānin level in grammar), and the absolute paramārtha level and the relative transactional one (with the so-to-say applicable prasakta factors on the saṃvrti level in the Madhyamaka)—is not congruent in the two systems.

Most importantly, according to the Mādhyamika, śūnyatā does not refer to any kind of entity, ultimate or conventional. Nor is it another factor in philosophical analysis on the same level as the dharmas, and it transcends the duality of conditioned (saṃskrta) and unconditioned (asaṃskrta). It is then comparable with a higher-level principle that directs the correct analysis of the (lower-level) entities and factors, and that describes the ultimately real state of affairs, viz. the non-substantiality and emptiness of all dharmas. In the grammatical theory of lopa, there does not seem to be a place for anything corresponding to śūnyatā, which refers metalinguistically to the fact, or truth, that all conceptual entities and analytic factors are empty “ciphers” in philosophical and psychological description.

Furthermore, the conceptual process of construction or (abhi)saṃskāra in Buddhist philosophy and the linguistic process of word formation or śabdasaṃskāra in Indian grammar are hardly isomorphic or directly comparable.


While it has been possible to adduce an interesting link between mathematical and philosophical thought in the case of Vasumitra’s theory of the phases of a dharma, it has not so far been possible to establish any clear and direct influence of the arithmetical zero on the theory of śūnyatā. It has been noted, however, that in the Chandaḥsūtra the term śūnya is evidently used in connexion with the absence of an operation. And it has also been possible to point to a certain parallel, albeit a very limited one, between the lopa of a prasakta linguistic form in grammar and some important features of the Madhyamaka; but this parallel rests largely on a terminological convergence that scarcely suffices to demonstrate the dependence of the philosophical notion on a grammatical concept.

The main point of contact between the Śūnyavāda and the departments of Indian thought examined above appears, then, to lie with the grammarians’ (and the ritualists’) prasajyapratiṣedha and the theory of absolute negation as developed by the Mādhyamikas, especially from the time of Bhā(va)viveka 11 (sixth century) and Candrakīrti (seventh century), who both elaborate on it. The (onto-)logical use to which the Mādhyamika puts the prasajyapratiṣedha is of course distinct from the more limited application it had elsewhere.

If our search for either a mathematical or a linguistic background or epistemic homology for the philosophical theory of śūnyatā has thus resulted largely in a non liquet, the outcome of the investigation has at least the usefulness of underscoring the specificity—indeed the uniqueness—of the philosophical concept. The theory of śūnyatā evidently represents a new threshold in the history of Indian thought. And this uniqueness may help to account for the fact that, in the course of the development of Indian philosophy, the theory has raised many a problem—and perhaps also for the fact that, so often, it has been the object of misunderstanding on the part of philosophers, Buddhist as well as non-Buddhist, belonging to other tendencies who were unfamiliar with its background, significance, and implications.

Postscript (2008)

In the decades intervening since the first publication of the preceding article, the theme of Indian grammar in relation to geometry and mathematics has been taken up by J. Bronkhorst, “Pāṇini and Euclid: Reflections on Indian Geometry” in Journal of Indian Philosophy 29 (2001), 43-80 (containing an extensive bibliography of secondary literature). J. F. Staal, who had raised the question in his Euclides en Pāṇini (Amsterdam, 1963, cited in note 2 above), again addressed these topics in the frame of the idea of mathematics and Pāṇinian grammar as what he terms artificial languages; see his “Artificial Languages across Sciences and Civilisations” in Journal of Indian Philosophy 34 (2006), 89-141 (including a critique of Bronkhorst, and in note 10 a rejection of Joseph Needham’s curious suggestion that zero originated in Southeast Asia), and “Artificial Languages between Innate Faculties,” Journal of Indian Philosophy 35 (2007), 577-96. None of these articles considers the matter of śūnyatā in Buddhist thought, however. Beginning in 1980 Kamaleswar Bhattacharya published a series of articles relating to the place of grammar in Nāgārjuna’s and Candrakīrti’s thought: “Nāgārjuna’s Arguments against Motion: Their Grammatical Basis,” A Corpus of Indian Studies (Essays in honour of Professor Gaurinath Sastri, Calcutta, 1980), pp. 85-95, and “The Grammatical Basis of Nāgārjuna’s Arguments: Some Further Considerations,” Indologica Taurinensia 8-9 (1980-81, Dr. Ludwig Sternbach Commemoration Volume), 35-43; the relevant articles are listed in his “A Note on Nāgārjuna’s Sanskrit,” Śemuṣī (Baldev Upādhyāya 12 Janmaśatī Volume, Varanasi, 2004), p. 659, including his “Nāgārjuna’s Arguments against Motion,” Journal of the International Association of Sanskrit Studies 8 (1985), 7-15, and (the last of the series) “Back to Nāgārjuna and Grammar,” Adyar Library Bulletin 59 (1995, C. Kunhan Raja Birth Centenary Volume), 178-89.

Concerning zero—which marks place value (known already to the Babylonians)—and numerals in general, see K. Menninger, Zahlwort und Ziffer, eine Kulturgeschichte der Zahl (Göttingen, 1958), G. Ifrah, The Universal History of Numbers (London, 1998), and R. Kaplan, The Nothing That Is: A Natural History of Zero (London 1999). For the transmission of the Indo-Arabic numerals, see recently C. Burnett, “Indian Numerals in Arabic, Greek and Latin,” in Journal of Indian Philosophy 34 (2006), pp. 25-26. According to M. Yano, Journal of Indian Philosophy 34 (2006), p. 154, Brahmagupta (sixth century) was the first Indian mathematician to have defined zero as one of the numbers.

Regarding grammar as a universal science and model for other Indian knowledge systems, to the references in note 3 above may be added Ānandavardhana’s observation in his Dhvanyāloka i.13 (KSS, pp. 132-33): prathame hi vidvāṃso vaiyākaraṇāḥ, vyākaraṇamūlatvāt sarvavidyānām.

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