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Mīmāṃsā and Vedanta consider vedas to be eternal and authoress. But if a book exists it must have an author. And what is the source of vedas according to these schools. They don't even believe vedas are authored by the God. Source

The author of the Vedas should either be God or humans.

My questions : How can vedas be authorless?

And what is the source (if they are authorless) of the Vedas according to these schools?

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    Authorless maybe a wrong translation for apaurusheya. Apaurusheya as clarified by many of the recent answers means the Rishis perceive mantras (a set of syllables to access certain energies) in a state of meditation. Meaning they’re not created by humans (apaurusheya) – Archit Mar 5 at 15:47
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    Have a look at this excellent answer for Purva Mimamsa point of view – Pandya Mar 5 at 16:40
  • @Archit Nyaya believes God is the author of the Vedas but in Vedanta it was authorless – Dark Knight Mar 6 at 9:10
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    @DarkKnight okay I have no knowledge. Anyway see Pandya’s mimansa answer which he linked it’s quite nice – Archit Mar 6 at 11:09
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Kindly note that Mimamsa school does not admit of a single supreme God (Ishvara) and, therefore, the question of the Vedas being authored by Ishvara does not arise. Thus, the question is now limited to whether humans authored the Vedas.

Sabara's bhashya on Jaimini's Mimansa Sutras give long explanation on why the Vedas are eternal and authorless.

Shabdas of two varieties: apaurusheya (Impersonal) and paurusheya (personal). The Vedas are the former. Mimamsakas argue that that the Vedas are svata-pramanya (intrinsically valid) and because the sabda is eternal, the vedas are eternal.

Their argument relies on an epistemological theory by which all utterances are valid. All utterances are assumed to be true unless falsified by evidence. The validity of a human utterance; however, is questionable because it could be falsified later.

From Sabara Bhashya 1.1.4

The cognition by a person is only by sense organs and it is not a means of dharma, as these organs are incapable of knowing dharma [in the matters of beyond time and perception].

The previous set of text in the bhashya 1.1.2 explains it in more detail.

It is the Vedic injunction that is capable of knowing what is past, present, future and also what is subtle, hidden or remote. This cannot be from sensory perception….If the words emanate from a trustworthy person, it must be true; if it emanates from an untrustworthy person, it is unreliable. However, in matters that cannot be perceived directly by the senses, all words from men are unreliable as they emanate from the mind of a human being. Dharma is not perceptible by senses and it is only by means of words. For these words to be entirely reliable, they should not proceed from a human source…If a man has learnt it from some other man, he must have learned it from some other person and so on..but the original source itself may be unreliable. Therefore, they cannot be from a person…. Just as the words of persons born blind cannot be a reliable source of knowledge regarding shades of colour.

For example, doing a yagna X begets heaven. How can this be verified by ordinary humans or senses? Therefore, one has to rely on apaurusheya shabda pramana to realize this.

There can be two epistemological approaches. One is to say that all utterances are false unless proved otherwise. The other approach is to say that all utterances are true unless proved otherwise. The Mimamsas take the latter approach. Most scientific laws operate this way. It is your job to disprove that yagna X begets swarga. As you can not do that, it is true till falsified. As swarga etc are matters that cannot be perceived by senses, they should have emanated from impersonal source and uncorrupted by the human mind.

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