All of us follow the information that Ai Shankaracharya established four Maths, one each for direction.
East-Padmapara-Govardhana Peetha-Rig Veda-Pragaynam Brahma is the Mahavakya-Bhogavala Sampradaya.
South-Suresvara-Sringeri Sarada Peetham-Yajur Veda-Aham Brahmasmi-Bhurivala Sampradaya.
West-Hasthamalakacharya-Dwaraka Peetham-Sama Veda-Thattvamasi-Kitavala Sampradaya.
North-Totakacharya-Jyothir Matha-Atharvan Veda-Ayamatama Brahma-Nanada Vala Sampradaya.
Adi Sankaracharyas’ date varies from BC to 3, 8 Century AD.
While the BC date is nor proven by records, I am speaking of Indian records for the western records might post date him or might even deny his existence
However,.the AD dates seem to be reasonable.
Thirugnana Sambhandar , a Tamil Saint, is quoted by Adi Shankaracharya in his Soundarya Lahari, as Dravida Sisu’
So Shankaracharya should have been before the close of the 15 Century AD.
The earliest recorded visit to Madurai Adheenam , established by Gnana Sambhandar was by Robert De Nobili in 1605.
“F. Robert was pleased and visited the temple as well as our Aadheenam during the rule of our 230th Guru Maha Sannidhanam Srila Sri Jnana Bhotaha Sri Jnana Sambanda Desika Paramacharya Swamigal and had discussions on common interests. Fr. Robert D. Nobily was in love with Tamil and learned it well.”
The Commander Masikapoor of the Emperor Aurangzeb visited our 237th Guru Maha Sannidhanam Srila Sri Mithiyeshwara Sri Jnana Sambanda Desika Paramacharya Swamigaland they had a long sitting of sharing many avenues of their interests.
Commander had presented an “Ivory Paduka” to our 237th Guru Maha Sannidhanam. Together with big silver places for Naivedyam (sacred food offered to divine), full set of Sodasa Upachara Pooja/Deeparadhandi, silver glasses, jugs etc and two Rajasthan horses for Rishi and palaquin for relaxing after long travels.”
So if we take one Adheenam was place for 30 years at the minimum, then the first mutt by Sambhandar should have been 6900-1600.
So this places Sambandar 4300 BC!
Then Shankaracharya dates back to between 4300-4000 BC!
But the Bhakti Literature mentions Gnana Sambhanda around 7 Century AD when Tirunavukkasar is mentioned as a contemporary of Sambanda.
This is supported by other evidence of the same period in Tamil.
So Shankaracharya’s date may be around 7 Century AD.
Now to the view that Shankaracharya did not establish the four Mutts.
Shankaracharya,according to the Legends, engaged in Arguments with the Heads of established systems at Mutts and won them over.
This means that he did not etablish these these Mutts in the strictest sense , inherited them and molded them .
Then he inherited these Mutts.
“According to Pandey, these Mathas were not established by Shankara himself, but were originally ashrams established by Vibhāņdaka and his son Ŗșyaśŗnga. Shankara inherited the ashrams at Dvārakā and Sringeri, and shifted the ashram at Śŗngaverapura to Badarikāśrama, and the ashram at Angadeśa to Jagannātha Purī”
* I shall be posting on Kanchi Mutt from the Historical perspective