- If, as Advaitins say, only Brahman ultimately exists then why do saints like Shankaracharyas and others strive to realise Brahman and for that live an austere life? After all, in Advaita Brahman itself cannot be realised as no one is there to realise ?
The answer is in your question itself. Brahman ultimately exists. But we are not feeling one with that 'ultimate' existence as we are thinking ourselves residing in a relative world of dulaist nature because of Maya and getting both pleasure and pain. As Swami Vivekananda puts it:
Each soul is potentially Divine. The object is to manifest this Divinity within by controlling nature external and internal.
The austere life is to purify the chitta so that the inherent Divinity can be manifested and one is freed from this dualistic world of strife --of pleasure and pain forever and be one with the Absolute bliss which is the True Self.
- As from paramarthik point there is no bondage, liberation, jiva, world and from vyavharic point any idea, experience, realisation, nirvikalpa samadhi is a time bound process, and will turn Brahman into objective thought that involves the use of concepts such as mind, space and time.
What they do to solve this unbridgeable, cognitively closed gap?
Again you are correct. Advaita is true only in the paramarthika stage. We normally are mistaken by the world 'Jagat mithya'. The word 'Jagat' literally means something which is ever-changing.It comes from the root 'gam' which means movement. 'Mithya' here just does NOT mean false. It means :that which is not eternal or is ever-existing'.
As mentionedy ,we get both pleasure and pain because we are deluded by Maya. To cross the ocean of Maya, we need Brahma-Jnana or the knowledge of the never changing eternal truth Which is one with our True Self.To attain that knowledge, we first need 'Chitta-suddhi' according to Sri Adi Sankaracharya.For Chitta-Suddhi, we need 'Nishkama-Karma' and 'Ishwara-Upasana', and only then reach the True Self by 'Neti-Neti' vichara.So long as one has the sense of 'I' and 'My', he has to go on doing good works prescribed by the holy scriptures. So karma is essential to realise 'Advaita' in life.There is no contradiction in it.
So long as we are ignorant of our True Self, we are Jivatmas. We have basically three bodies- Gross or Sthula,Subtle or Sukshma and Karana or Causal. The subtle body comprises of subtle karmendriyas and subtle jnanendriyas and manas. The causal body is caused by the ignorance that identifies the Self with these gross and subtle bodies. It is the causal body and the subtle body which together are together called Jivatma.This Jivatma dies and is born again and again so long as the true knowledge of the real Self is not obtained. The knowledge of the self destroys the mind which is root of all our ignorance. Being one with our True Self ends our endless journey between births and deaths.
'Nirvikalpa Samadhi' is not a time bound process. It breaks the shackles of Maya forever.Time is just one aspect of Maya.So the entire object is to end all suffering by reaching our True Self.
I would request you to read 'Jnana-Yoga' by Swami Vivekananda minutely to clear your doubts.You could also read the sayings of Sri Ramana Maharshi.Both are available online.
Reference :1.Jnana-Yoga by Swami Vivekananda, 2. Jnani-Guru by Paramahansa Nigamananda, Sarasvata Math, Halisahar. 3. Vivekanander Vedanta-Chinta by Dinesh Chandra Bhattacharya Sastri, RMIC, Kolkata.