Ah, so you're suggesting that the similarities between Mu'tazilism and Shi'i thought are more a result of convergent evolution than evidence of a similar logical process? That's interesting, I haven't seen that before, though it makes sense. The Mu'tazila certainly don't rely on a belief in some conspiracy of Abu Bakr, Umar, and Uthman to "edit" the Qur'an (an idea, of course, which has basically collapsed in the face of
modern-day archeological discoveries of Qur'an manuscripts identical to the modern Uthmani Quran and scientfically dated to the lifetimes of Muhammad and the Rashidun Caliphs) to make their points, relying on arguments that start from given first principles and trying to reason their way out from there.
Could it be possible, though, that considering the similarities of these positions that later Usuli theologians have borrowed from the arguments of the (nearly-extinct by then) Mu'tazila to defend their positions, even if their initial logic for taking those positions was quite different?
I feel as if you simplify or make weaker the argument of the Shi'i who claim the Tahreef or distortion of the Quran. These are also complex ideologies and differing levels to which they advocate this. These following types go in such a way:
1. Omission: Instead of a distortion, the words of Allah contained in the Quran are filled with various omissions once given to Muhammad. These omissions exist for the reason that Allah had foreknowledge that if he revealed the total Quran or his total commands to Muhammad, these commands would be distorted or misused by the apostate sahaba, especially, Abu Bakr, Umar, Uthman, Mu'awiyah, Ayesha, Hafs, Abu Huraira, etc... This thus, allows the holders of this believe to say that the Quran is free from distortion in terms of nothing has been added or taken away after it was revealed to Muhammad or how it is passed down to the Sahaba, but instead, there is more to it which is held by Ali ibn Abi Talib and his Imamiyah and his Wilaya (authority).
2. 'We' never held the Qu'ran: Ali ibn abi Talib was given the Quran and with his wilaya alone, can he reveal it with the ascendance of the final Imam, Imam Aj-Aj. Further, he was the one who compiled the Quran in its entirety, with 17,000 surahs and these are compiled in a correct order unlike the current Quran. Thus, the Quran in which we have today, is incomplete. Scholars in this though, usually hold that Ali presented this completed and perfect Quran to the Sahaba and they rejected it, claiming that what they compiled was preferred. Ali thus, placed this Quran in what can be described occultation or hidden from the senses yet ever present as is the position on the final Imam of both Twelver and Nizari faqih.
"Imam Amir ul-Mu'minin (Ali ibn abi Talib) collected the Qu'ran and this is of the truths. He came to the 'Muslims' with this Qu'ran and said to them, this is the Qu'ran. After Imam Amir ul-Mu'minin read the Qu'ran to them, they said, 'we do not need this.' So, Imam Ali closed the Qu'ran and returned it back (~to where, I do not know, perhaps the recitor is referring to occultation~). After this, they (Sahaba) discussed among themselves, and decided that they chose wrong. They said, 'let us go and tell Ali to give us this Qu'ran so that we may destroy it.' Thus, they came to him and said, 'O Abdul-Husayn (Ali) give to us the Qu'ran which you posses, so that we may examine it.' Amir ul-Mu'minin said, 'Nay! By Allah, you will never see the Qu'ran until my son, Imam of Time (Aj-Aj) appears before this earth." -Shaykh Abdul Hamid al-Muhajir referencing a common ahadith from on the Shi'i collections supposedly from Sahi' Imam Ja'far.
3. Distortion: The Qu'ran is actively distorted by the Sahaba, especially by removing words that refer to Ahl ul-Bayt (family of Ali) and removing phrases which curse the Sahaba. Thus the Qu'ran was completely distorted, both in meaning and to whom each section is addressed as well as the way in which it is organized.
"All of the prominent Sahaba left the religion and became apostates. They delivered to us, a distorted Qu'ran, the Qu'ran which we have in our hands today is distorted and changed. Today, we require and must use this distorted Qu'ran, until our Imam returns with the True Qu'ran." -Shaykh Ali al-Hashimi
"Allah said that he had perfected your religion now, yet after this he mentions the prohibition upon certain meats? We have doubts regarding to the placement of this verse. Why is it that we have doubts regarding this placement and other placements? It is due to them (Sahaba), who interfered with the context of the Qu'ran, inserting and removing sections." -Ayatollah Ali al-Qorani
"Imam Amir ul-Mu'minin is mentioned by name within the Qu'ran. Ali is within the Qu'ran. Why then would we of the Shi'a then oppose such a well known issue (~the Qu'ran has no mention of Ali and is the opinion of the Sunni~)? It is because we hold to what was revealed to us by Ahl ul-Bayt, which confirm to us that indeed Ali and the Ahl ul-Bayt is mentioned. Ali mentioned to us, that 1/3 of the Qu'ran is regarding himself, Amir ul-Mu'minin." -Shaykh Yasr al-Habib
4. Revealed to incorrect person: A more antiquated view, is that the Angel Gabriel mistakenly revealed the Qu'ran to Muhammed instead of to the preferred individual, Ali ibn abi Talib. This was held by some scholars in the Shi'a past, however less so currently. The idea however, is still featured in some Shi'a texts and ahadith from various collectors of the middle ages.
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Yes it is possible they adopted some ideas from the Mu'Tazila, at least far more than current Sunni have borrowed from the Mu'Tazila.