The Self cannot be ‘known’ in any objective sense because it is the ultimate subject – there is no other subject that could know it. This is why science can never tell us anything about the Self. Science works by collecting data and analyzing it; formulating theories and then using them to predict what will happen when data are gathered in a different situation. This can never be applied to Self/brahman, because brahman has no data.
Strictly speaking, vichAra refers to investigation into ‘things’ so that Atma vichAra is effectively a contradiction in terms; the Self is not a thing. Spiritual investigation has to be done rather differently. The correct term is shAstra mImAMsA and it is really scriptural ‘investigation’ that we must conduct in order to find out about the Self. Monier-Williams translates mImAMsA as “profound thought or reflection or consideration; investigation, examination, discussion”. The philosophical branch that studies the Upanishads etc at the end of the Vedas (Vedanta) is called uttara mImAMsA. (uttara means “later, following, subsequent, concluding” but also “superior, chief, excellent, dominant”.)
We ‘discover’ the Self by removing ignorance. If someone holds up a screen in front of our face and then brings an object to show us, but keeps it behind the screen, we can say nothing at all about the object. However, as soon as the screen is taken away, the object is revealed to our senses and the perception takes place automatically. Similarly, knowledge of the Self is obscured by ignorance but as soon as that ignorance is removed, the Self is immediately self-evident; we do not have to do anything to ‘investigate’ it.
Scripture functions like a mirror. When we look into a mirror, we do not literally see our face and body, we only see an image of it. Yet this enables us directly to perform whatever actions are required on the body itself – combing the hair, shaving and so on. We do not shave the image but the actual hair on the face. Similarly, the scriptures do not directly represent the Self but the information therein, when explained by a qualified teacher, directly enables the ignorance in our mind to be removed, revealing the Self-knowledge which is as though hidden beneath.
Actions will never bring about Self-knowledge, since action is not opposed to ignorance. Nor will practices such as meditation or prayer. As Swami Paramarthananda puts it, meditation will only bring about quiet ignorance.
As Shankara puts it (if he was the author of vivekachUDAmaNi v.13): “It is through reflection over the words of a truly benevolent soul that one comes to a knowledge of reality, and not through bathing at sacred places, charity or hundreds of breathing practices”.  I.e. it is through shravaNa, manana and nididhyAsana and not through asking ‘Who am I?’ that one gains Self-knowledge.
 The Crest Jewel of Wisdom; viveka-chUDAmaNi, commentary by Hari Prasad Shastri, Shanti Sadan, 1997. ISBN 0-85421-047-0.
Two different Ways to choose from (¿) – or being attracted to.
VENKAT (via Ramana, Nisargadatta, JK) mostly YOGA – (Through “Ceaseless investigation into oneself / one’s ego is the means to ‘go beyond’ and results in ‘no-mind’ or dissolution of the ego”). MEDITATION, NIRVIKALPA SAMADHI have a prominent place. ANNIHILATION OF THE MIND the goal.
DENNIS (Via Upanishads and Shankara mostly) ATMA VICHARA – (Through real understanding by the mind, REMOVAL OF IGNORANCE… ORDINARY MEDITATION IS A FORM OF ‘DOING’. ANNIHILATION OF THE MIND NOT SOUGHT. EXPERIENCES (Samadhi) NOT SOUGHT.
SHRAVANA – MANANA – NIDIDHYASANA as method (“and not through asking ‘Who am I?’ that one gains Self-knowledge”.
What is self-enquiry if not effectively nididhyasana?
Ramana did not prescribe the simple “who am I” question as is widely misunderstood. He prescribed self-investigation – i.e. to question what is the ‘I’ that I think I am.
Sri Candrasekhara Bharati, a Sankaracharya of Sringeri, in his commentary on your v.13 of Vivekachudamani had this to say:
“The sloka is intended to convey that the means to obtain conviction of truth can arise only by listening to the words of a beneficent person and then by self-inquiry based on that, and not by action.
Vicara is a mental operation which conduces to the conviction of the actual nature of an object . . . Having inquired about the atman following the words of the sruti and the upadesa of the guru, getting direction perception of its true nature, one should disentangle the atman from samsara which is the non-atman which has been superimposed on it”
In other words inquire intently into the nature of the super-imposed ego, and see through it to THAT which THOU art.
V.S.Iyer, the teacher of the early Ramakrishna monks, and a staunch advocate of a philosophical rather than mystical Vedanta, had this to say about realisation, which concurs with Ramana’s recommendation to self-investigate whenever an “I”-thought arises:
“When you know the difference between drik and drsyam, and include the ego with the latter, there is nothing objective you need run after henceforth”
“The practical test whether Vedanta has been grasped is to ask yourself whether the ego is behind your thoughts, actions or not. Ask this a hundred times daily.”
“Whatever you say about Brahman, it is only an idea, ie drsyam. Reason when applied to drsyam you can grasp it; but it can never grasp the drg. Reason can tell you the drg is there, but it cannot grasp it. This is the limit of reason. But the drg is always there; it cannot be known or understood because knowing implies a second thing. But in all acts of knowledge, the drg is there when you are thinking. Hence Brahman is known only in idealessness. It is impossible to be free from Brahman for it is impossible for any thought to arise without it. You must analyse yourself and see that whenever there is drsyam, there must be that (the drik) which is aware of it. When you see this, you know the Atman. Hence Drg-Drsyam analysis is so fundamental and so difficult.”
“In advanced Vedanta, we use neti, neti, which is the negating of every thought or idea that can possibly arise. “Neti” means “don’t think”. It is not a new thought to be added. All thoughts are useless in truth. Find something uncontradictable in Absolute silence where ideas there are none”
Thank you, Venkat, for the excellent selections (item: “When you know the difference between drik and drsyam, and include the ego with the latter, there is nothing objective you need run after henceforth” – V.S. Iyer).
Can we not say then, ‘Ego is not Atman, but Atman is ego’… ‘Ishvara is not Brahman, but Brahman is Ishvara’?. This is tadatmya – non-reciprocal relationship.
Evidently there is not only overlap between the two Via/s I mentioned, but also convergence in their aim and final result (hopefully). It could not be otherwise, for Truth is One. However, differences there are. Some others of our friends might comment on this, but perhaps it is unnecessary.