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As I discuss in this question, by far the most popular school of Hindu philosophy is the Vedanta school, which bases its tenets on the doctrines laid out in the Brahma Sutras, a work by the sage Vyasa that summarizes and systematizes the philosophical teachings of the Upanishads. (You can read the Brahma Sutras here.) In any case, in Adhyaya 1 Pada 3 of the Brahma Sutras, Vyasa discuss the issue of who is eligible for Jnana. As I discuss here, he says that low-caste people are not eligible for Jnana, since they don't wear the sacred thread. But before that, he says that the gods are eligible for Jnana.

And that leads him into a discussion of the gods in general, in particular regarding how the Vedas can be eternal if mention the names of gods who do not remain in their positions forever. As I discuss in this answer, the resolution is that the same names recur in different ages. But in his commentary on Adhyaya 1 Pada 3 Sutra 29 of his commentary on the Brahma Sutras, Adi Shankaracharya gives a few other arguments for the Vedas being eternal:

[T]he eternity of the Veda is founded on the absence of the remembrance of an agent only.... The eternity of the word of the Veda has to be assumed for this very reason, that the world with its definite (eternal) species, such as gods and so on, originates from it.--A mantra also ('By means of the sacrifice they followed the trace of speech; they found it dwelling in the rishis,' Rig-veda Samh. X, 71, 3) shows that the speech found (by the rishis) was permanent.--On this point Vedavyâsa also speaks as follows: 'Formerly the great rishis, being allowed to do so by Svayambhû, obtained, through their penance, the Vedas together with the itihâsas, which had been hidden at the end of the yuga.'

I discuss the notion that the Vedas are a blueprint for the creation of the world in my answer here. But my question is about the Rig Vedic verse that Adi Shankaracharya cites, which is verse 3 of Book 10 Hymn 71 of the Rig Veda:

yajñena vācaḥ padavīyamāyan tāmanvavindannṛṣiṣupraviṣṭām |
tāmābhṛtyā vyadadhuḥ purutrā tāṃ saptarebhā abhi saṃ navante ||

With sacrifice the trace of Vāk they foIlowed, and found her harbouring within the Ṛṣis.
They brought her, dealt her forth in many places: seven singers make her tones resound in concert.

As you can see in the Rig Veda Anukramani in my answer here, this verse was heard during Tapasya by Brihaspati, the guru of the gods, and the subject of the hymn is listed as "Jnanam". But my question is, why does Adi Shankaracharya think that this verse proves that the Vedas are eternal?

The verse discusses how Vak, which as I discuss here is a name of Saraswati in her capacity as goddess of speech, was found by the wise as "harbouring within the Rishis." But why does that imply "that the speech found (by the rishis) was permanent"? Is Adi Shankaracharya arguing that the fact that it says the speech was "found" within the rishis as opposed to "composed" by them implies that the Vedas don't have a human author?

Are there any commentaries on Adi Shankaracharya's Brahma Sutra Bhashya that shed light on this?

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The Rishis, from various places, gather every once in a while to meditate as is enunciated through the tradition of Kumbha Mela. In such a confluence, in deep group meditation, the Rishis of yore heard sounds or mantras (Sruti). This is before the advent of any language. These sounds were then used to formulate a language which came to be known as Sanskrit. Also, the sounds or the mantras heard were very recently compiled into Vedas, since 1500 BC,. There was no written accounts before this and as is widely known this was an oral tradition. I reproduce my answer from here to support this:

Q: Guruji, why are all mantras in Sanskrit? What is so special about Sanskrit?

Sri Sri Ravi Shankar: No, Sanskrit was derived later. Mantras were already there. The words of the mantra were already there; later on the languages was created. Not only in Sanskrit, but when you go deeper into all languages, you will find all these mantras.

Coming to the Shloka:

यज्ञेन वाचः पदवीयमायन तामन्वविन्दन्न्र्षिषुप्रविष्टाम |

तामाभ्र्त्या वयदधुः पुरुत्रा तां सप्तरेभा अभि सं नवन्ते ||

The word वाचः has multiples meanings. It means voice, word, sound amongst other things. (It does refer to Goddess Saraswati too, the Goddess of all sounds and speech). These meanings are taken from this dictionary.

The first part of the Shloka:

यज्ञेन वाचः पदवीयमायन तामन्वविन्दन्न्र्षिषुप्रविष्टाम |

With sacrifice the trace of Vācha they followed, and found her harbouring within the Ṛiṣhis.

Hence, Adi Shankaracharya says that when the sounds or Vacha were traced to their origin, they were found harboured in the Rishis. In other words, it is the Rishis who heard these sounds first in deep meditation.

Here is another translation by Prof. Shrikant Prasoon to support this:

The learned or the Rishis receive the meaning of the great words through their indulgence into Yagna. The words were revealed to and received by the heart of the essence-knowing Rishis. After receiving they popularised them. In this way they expressed them in the form of the prayers.

Chronologically, these sounds were later used, spread through oral means for a very long time, a language was developed on the same, the sounds were later compiled as texts (Vedas) around 1500 BC, and grammar of the language enunciated by Panini in his Ashtadhyayi which only happened around 4th century BC.

Hence, as Adi Shankaracharya says, the Vedas which are the Srutis as heard by the Rishis in deep meditation are eternal.

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    But what is there in the verse that indicates that the sounds that were found harboured in the Rishis were eternal? – Keshav Srinivasan May 2 '16 at 17:24
  • @KeshavSrinivasan: My understanding on this is that these are the sounds that resonate closest with the consciousness or Brahman. (Thats why they can be only heard in deep meditation). There are so many other sounds too but whose vibration is not as much in sync. Hence, as consciousness is said to be eternal, these sounds are the closest ones to be eternal. – Amit Saxena May 2 '16 at 17:37
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    Well, I for one certainly agree that the Vedas are eternal. I just don't see why, from the perspective of someone who didn't believe that the Vedas were eternal, this particular verse would convince them that the Vedas are eternal. – Keshav Srinivasan May 2 '16 at 17:42
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    @KeshavSrinivasan: It is quite evident to me. I am not sure what point you are missing. I think this Shloka is chronologically very deep as it talks about tracing all speech back to the Rishis which is a very very long process. Once that link is established, it inherently implies that the sounds closest to the Brahman are closest to being eternal. – Amit Saxena May 2 '16 at 17:46
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    Well, the verse doesn't actually say that the sounds found harboured in the Rishis are sounds that are closest to Brahman or anything like that. – Keshav Srinivasan May 2 '16 at 17:49

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