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‘If it is true that mantras and ritual have no meaning …’ : even Alper, Harvey. P. 1989. Understanding Mantras. Albany: State University of New York (SUNY) Press, an impressive collection of articles and the most authoritative bibliography until that date, does not arrive at a clear conclusion of what mantras are. He does pay reasoned attention to a variety of views and opinions.

‘Language is a system in which everything hangs together’ : De Saussure, Ferdinand. 1915, etc. Cours de Linguistique Générale. Publié par Charles Bally et Albert Sechehaye. Paris: Payot. I have not found the phrase où tout se tient in this book but it expresses his basic insight which is tantamount to saying that language is a synchronistic system—a reaction against the diachronistic philology of his predecessors. Pāṇini also presents a synchronistic system but for different and erroneous reasons: he believed that Sanskrit did not change—a belief that de Saussure did not share. Beliefs however do not matter: what counts in science are the results, not what scientists say or believe about it. (‘Newton’s Lesson’: Staal. 1993, 94. Concepts of Science in Europe and Asia, Chapter 1. Leiden: International Institute for Asian Studies.)—It would not be surprising for tout se tient to apply to language if it applied to the universe. Evidence is mounting; but it has not been demonstrated.

‘Pautimāṣya from Gaupavana, Gaupavana from Kauśika, Kauśika from Kauṇḍinya,…’ : BĀU 2.6.1.

‘Chomsky replaced them by a simpler but much more abstract system’ : Chomsky, Noam. 1995. The Minimalist Program. Cambridge and London: The M.I.T. Press.

‘Our problems are different. We are interested in infinite recursive structures’ : the theory behind what can and cannot be solved by computing machines. Recursion theory spans the period between Gödel, Church and Turing. Expertly and intelligibly discussed in Feferman and Feferman 2004, who take their reader by the hand, beginning with the ‘Unity of Science’ in Vienna before WWII up to the date of publication of their book, and full of telling details (on pp. 145–6, they depict Church as a solid American citizen, just as stubborn as Tarski who was an immigrant from Poland).—Here is an example of what infinite recursive structures are about: recursiveness leads to infinity but infinity need not be based upon recursiveness. Readers who are not used to logic, recursion theory or computer science, but familiar with numerals, will be in a position to appreciate the difference between recursive infinite and non-recursive infinite numerals. 1/3 or 0,33333 … is a recursive infinite numeral because the ‘0’ is followed by infinitely many ‘3’s. But pi or 3,141592920 … is a non-recursive infinite numeral: there are infinitely many numerals after the ‘3’ but there is no pattern: they do not repeat. We shall see (p. 273 and #) that Vedic mantras and ritual incorporate recursive structures, and that the Vedas loved the infinite which was abhorred by the ancient Greeks. See Stall, forthcoming and b.

‘That is better although Proust himself is miles ahead’ : Proust, Marcel. 1981. Remembrance of Things Past, Vol. II: The Guermantes Way. Translated by C.K. Scott Moncrieff and Terence Kilmartin, New York: Random House, p. 802. There are other masters of the long sentence. The Kādambarī and Thomas Mann come to mind. German and Sanskrit have other means at their disposal that the French language lacks: nominal composition. English can do it to some extent by simply putting words together as in car engine. The eighth century critic DaṇḌin comments on Sanskrit: ‘the frequent use of compounds gives power to its prose, it is its life force’ (oja‘samāsabhūyastvam etad gadasya jīvitam): Hock, Hans Heinrich and Pandharipande, Rajeshwari (1978), ‘Sanskrit in the Pre-Islamic Sociolinguistic Context of South Asia.’ International Journal of the Sociology of Language 16.11–25: 22.

‘Psychologists are interested in these recursions.’ : Miller, George. 1964. ‘The Psycholinguists,’ Encounter, 23 (July): 29–37 discusses three cases of recursiveness: right-recursive, left-recursiveness and self-embedding. The text does not mention the third which is exemplified by: ‘the rapidity that the hummingbird has (is remarkable)’. It is common and easy for every speaker of English. Few can manage the next step: ‘the rapidity that the motion that the hummingbird has has’. Everyone has trouble with: ‘the rapidity that the motion that the wing that the hummingbird has has has’. We return to self-embedding in ritual in the next chapter: p. 228 with note.

‘This picking up of language … is similar to the way scientists learn about the world’ : Gopnik, Alison (1999), ‘Small Wonders,’ The New York Review of Books, May 6: 41–5.—In Tantrism it is said that 70 million of them exist in superior worlds (‘higher levels of the hierarchy of RV 1.164.45’: p. 296 below): Brunner, Héléne. 1986. ‘Les membres de Śiva’, Asiatische Studien/Études Asiatiques, 40:91 and note 9.

Kautsa : pp. 144ff.

‘It holds not only for mantras like OM,… but for mantras such as the Gāyatrī mantra’ : last section of this chapter, p. 213ff.

‘That mantras are untranslatable, like proper names, was recognized by Chinese pilgrims such as Xuanzang or Hiuan-Tsang’ : # 27.—‘Though often kept secret and guarded jealously, some Vedic mantras were inherited by Buddhists and Tantrics’: Tantric mantras play an important role in several of the essays published in Goudriaan, Teun, (ed.), 1992. Ritual and Speculation in Early Tantrism. Studies in Honor of André Padoux. Albany: State University of New York and White, David Gordon (ed.), 2000. Tantra in Practice. Princeton: Princeton University Press. In Buddhism, the Mantrayāṇa is called after them. Tantrism is not unique to India, Buddhism or Hinduism: Strickmann, Michel, (ed.), 1981–83. Tantric and Taoist Studies in Honour of R.A. Stein.—I-III, Bruxelles: Institut Belge des Hautes Études Chinoises.

‘Maintaining the original forms of a ritual language is advocated in a similar spirit’ : Jucker 2006 for the Catholic Mass which is also recursive in structure as the author shows.—‘Arabic in the Qur’ān’: Qur’ān 12.1, 20.112, etc. ‘this is an Arabic Qur’ān.’ The Qur’ān, therefore, may not be translated into another language.

‘Bertrand Russell called them “egocentric particulars”’ : Russell, Bertrand. 1940. An Inquiry into Meaning and Truth. London: George Allen and Unwin, Chapter VII and elsewhere.

‘Many students of mantras have argued that mantras are speech acts’ : McDermott, A. C. S.. 1975. ‘Toward a Pragmatics of Mantra Recitation’, The Journal of Indian Philosophy 3, 3: 238–98; Wade T. Wheelock and John Taber in Alper 1989: #193. More recently, the study of pragmatics in language has developed in various new directions. In 1993, the Journal of Historical Pragmatics devoted a special issue to ritual language behaviour. Gabriela Nik. Ilieva (‘The Rgvedic Hymn as a Ritual Speech Event. About some grammatical-rhetorical features of 10.39 from a pragmatic perspective’: pp. 171–94) discusses many of the complexities that a pragmatic study of a single hymn entails and that involve speaker- and hearer-oriented terms of address, plots, ellipses and ideological background. I do not know whether ‘ritual speech event’ as a technical term adds to conceptual clarity. What is important in the present context is to distinguish between a particular Ṛgvedic verse and a mantra. The former may become a mantra but is not or not yet a mantra. The two basic principles or meta-rules of the Śrauta Sūtras (below, pp. 263–5) that govern the relation between mantras and acts in Vedic ritual are: ekamantrāṇi karmāṇi, ‘each act is accompanied by one mantra’ and mantrāntaiḥ karmādīn saṃnipatayet, ‘one should let the beginning of the acts coincide with the end of the mantras’ (Āpastamba Śrauta Sūtra 24.1.38). There are of course exceptions, but these two principles are, on the whole, sufficient for our purpose.

‘The Chāndogya tells us about dogs that are gathered’ : CU 1.12.5.—‘The syntactic study of bird song was initiated by a composer and musicologist’: Mâche 1983 inspired Staal. 1985. ‘Mantras and Bird Songs,’ Journal of the American Oriental Society 105: 549–58.

‘The syntactic structures of language, mantras and bird song’ : note syntactic, not semantic. The distinction is not always made when the press reports on ‘talking’ birds like the famous African Grey Parrot ‘Alex’ who passed away on 6 September 2007. Such birds, when trained by humans, do not only repeat, but construct new patterns that are syntactically more complex. Whether they understand and, if so what, is controversial.

‘As for recursiveness, there are different kinds and natural language, birds, mantras and ritual’ : #198–201.

‘Similar in formation to upaniṣad (‘sitting near’)’ : p. 160.

‘The boy is then given a staff, a grass girdle and several instructions’ : Kane. 1941. Vol.II, Part I, Chapter VII: 268–415 for a full description.

‘P.V. Kane refers to its “Grand simplicity”’ : Kane (publication details #60), II. I, p. 303.