India might be considered the original homeland of generative linguistics on at least two counts.
Firstly, it was in India (and on the basis of the structure and lexicon of Sanskrit) where the idea of an Indo-European language family was developed, a concept which was to form the basis for the study of historical and comparative linguistics from which the modern linguistic traditions ultimately derive. Generative linguistics, as pioneered by Noam Chomsky, grew out of the earlier structuralist approach to linguistics as developed in America by Leonard Bloomfield. Bloomfield himself was greatly influenced by Ferdinand de Saussure's Course in General Linguistics (1916), which suggests that language should be understood as a system of interconnected parts. Saussure's structuralism, though it focusses on the synchronic study of language as a static system, has its root in the earlier study of historical (diachronic) linguistics. In the 19th century, linguistics meant historical linguistics, and Saussure himself had previously worked on Indo-European from a diachronic perspective (famously anticipating the later discovery of Hittite 'laryngeals' in his proposal of Proto-Indo-European 'sonant coefficients').
The starting point for the modern study of diachronic linguistics can reasonably be argued to be 1786, when Sir William Jones delivered his paper on the history and culture of the Hindus before the Asiatick Society in Calcutta (later published in 1788 as "The Third Anniversary Discourse, On the Hindus"). In this paper Jones stated that:
The Sanscrit language, whatever be its antiquity, is of a wonderful structure; more perfect than the Greek, more copious than the Latin, and more exquisitely refined than either, yet bearing to both of them a stronger affinity, both in the roots of verbs and the forms of grammar, than could possibly have been produced by accident; so strong indeed, that no philologer could examine them all three, without believing them to have sprung from some common source, which, perhaps, no longer exists; there is a similar reason, though not quite so forcible, for supposing that both the Gothick and the Celtick, though blended with a very different idiom, had the same origin with the Sanscrit; and the old Persian might be added to the same family, if this were the place for discussing any question concerning the antiquities of Persia.
It was with Jones' discovery of the genetic relation of Sanskrit (the classical language of India) to Latin and Greek (as well as Celtic and Germanic) that the study of historical linguistics came into existence.
Secondly, there is an indigenous Indian tradition of linguistic analysis, dating back to at least the 4th century BC, which has helped to shape modern generative linguistics: Chomsky's development of a theory of language as involving formal generative rules clearly owes a large debt to Pāṇini's Aṣṭādhyāyī (6th-4th c. BC?), a grammar of Sanskrit morphology stated in terms of transformation rules.
When I was in India (2003-5), one day when I went to post a letter I was pleasantly surprised to find that the Indian Department of Posts had issued a Pāṇini stamp (see the press release here).
Interestingly, Pāṇini is depicted on this stamp writing his grammar. Since one of the characteristics of Pāṇini's grammar is its conciseness (3,959 sutras), it would seem more likely that it was orally-composed and transmitted. If it had been a written text, why would Pāṇini have favoured brevity over clarity? (No traces of true writing systems have been found in India predating the 3rd c. BC Brāhmī Ashokan inscriptions.)
One of the characteristics of Indian writing systems like devanāgarī is that they are--unlike for instance Greek and Roman alphabets--arranged according to phonetic principles: the vowels come first, then the velar stops, palatal stops, retroflex stops, labial stops etc. This seems to be a result of having invented linguistics before writing.
The Sanskrit grammatical tradition is known as vyākaraṇa, and is considered one of the six vedāṅgas (auxiliary disciplines for understanding the Vedas [the oldest sacred texts of India, consisting largely of ritualistic formulae and hymns]). Vyākaraṇa (= vi + ā + √kṛ 'to do') has a general (and compositional) meaning "separation, discrimination, distinction", but also possesses a narrower sense "explanation, description", as well as the specific sense "grammar, grammatical analysis" (as in the title of this blog).