What is the significance of Space?
Does space store everything, the past, present and the future? From a mundane perspective space can store data. Like we see a small electronic chip holding tera bytes of data. However when you talk of human memory this is a function of soul and subtle body rather than physical space. Some people can remember past lives but their gross body that occupies material space is completely changed.
Also, is space associated with any god (like Sun, Fire, Water etc)?
This is the subject matter of sankhya and described in Srimad Bhagavatam in different places. The best place where this is described is by Lord Kapila to His mother Devahuti in Canto 3 Chapter 26.
SB 3.26.32 — When egoism in ignorance is agitated by the sex energy of the Supreme Personality of Godhead, the subtle element sound is manifested, and from sound come the ethereal sky and the sense of hearing.
So the sound is the basis of space and also other subsequent elements like air, fire, water, earth. Now later the form of Virata Purusa is described and for different organs there are different devatas that were described. Fire is associated with the mouth, Vayu(wind-god) with Nostrils, Sun with eyes and so on.
SB 3.26.54 — First of all a mouth appeared in Him, and then came forth the organ of speech, and with it the god of fire, the deity who presides over that organ. Then a pair of nostrils appeared, and in them appeared the olfactory sense, as well as prana, the vital air.
SB 3.26.55 — In the wake of the olfactory sense came the wind-god, who presides over that sense. Thereafter a pair of eyes appeared in the universal form, and in them the sense of sight. In the wake of this sense came the sun-god, who presides over it. Next there appeared in Him a pair of ears, and in them the auditory sense and in its wake the Dig-devatas, or the deities who preside over the directions.
From the above descriptions the Dig-devatas preside over the directions and they are related to the sound (auditory sense) as well. So this gives us one association with space and a devata.
What is the relation between space and the chanting of mantras? From the above Bhagavatam references we can understand that sound is the source of space and hence mantras(specific Vedic sounds) can deliver us from this material world which we perceive in terms of time and space.
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