Buddhism vs Advaita Vedanta

Advaita Philosophy was popularized by Acharya Shankara , who succeeded in convincing Indian Buddhists , that Buddhism is a branch of Hinduism.You may refer to the book " Shankara Digvijay--traditional life of Shankaracharya" ,written by Madhava-Vidyaranya ,published by the Ramakrishna Math , Chennai ( Madras) , India and available at Ramakrishna Mission Centres.
Shankaracharya lived in the second half of Seventh Century AD (aroud 650 to 690 AD ).He succeeded in convincing Indian Buddhists , that Buddhism was a branch of Hinduism and succeeded in integrating Buddhism to Hinduism. Shankara also portrayed the Buddha as the Ninth Incarnation of Vishnu . Those Buddhists , who did not integrate with Hinduism ,were later (around Tenth Century AD) , slaughtered or converted to Islam by force , by the Muslim Invaders.

Shankara's Philosophy is known as Advaita Vedanta Philosophy. But over the years ,it has undergone many modifications , by different preachers ,trying to popularise the idea in their own ways. Latest attempt at popularisation of Advaita Vedanta ,was by Swami Vivekananda (1863-1902) ,who also founded the Ramakrishna Mission.

Hinduism has many Books of knowledge ---the Vedas ,the Upanishads ans the Six Schools of Philsophies.
The Vedas popularise worshiping different powers of Nature --like the Earth ,Sun ,Air ,Fire ,Lightening ,Sky etc. symbolised as Gods.
Upanishads are philosophies of Individual Teachers -who tried to popularize the concept of One God --Creator ,Sustainer and Destroyer of the Universe .They also tried to integrate Individual Consciousness to Universal Consciouness and to Space and Time--assuming the existence of a set of Universal Laws.
The Six Philosophical schools ,attempt to integrate the ideas of Upanishads , into a self consistent logical system. They are --(1)Shaankhya (2) Yoga (3) Nyaya (4) Vaisheshika (5) Purva Mimansa (6) Uttara Mimansa (Vedanta).
Advaita Vedanta has evolved out of Vedanta . Literally Advaita means Not Two-ordinarily accepted as One or Zero.
So the Universal God , can be One or Zero( Shunya or Space).In this philosophy , Existence and Non-Existence , are treated as two sides of the same coin.
It assumes three sates of consciousness -waking , sleeping (dreaming) and deep sleep (unconsciousness ,coma or memory less)--all to be TRUE . And tries to INTEGRATE them to a SUPER_CONSCIOUS state--called the Ultimate Reality.

For a beginner, trying to understand Advaita Vedanta and its links to Buddhist philosophies ( or Tantra) , I would suggest going through the works of Swami Vivekananda.

And going further, he reviews bones covered with skin, flesh and blood, and he knows the unbroken stream of consciousness as established in this world and established in the next.

...

And going further, he reviews bones covered with skin, flesh and blood, and he knows the unbroken stream of consciousness as both unestablished in this world and unestablished in the next.

Sampasadanīya Sutta (Quoted from Piya Tan's Translation of DN 28)

It may appear at early stages there is and observer. But later on it becomes apparent that even "the observer" is not unbroken stream or permanent hence not self.

Craving arises at any of the six viññāṇas of the sense doors. As a description of mind, the four aggregates of viññāṇa, saññā, vedanā, and saṅkhāra generally suffice. Deeper Vipassana separates them. Before that stage is reached however, philosophies start because, despite the experience of arising and passing, the observer—which is viññāṇa—seems to remain and it is not divided or dissected. It is viewed as eternal soul: je viññāya te āya ye āya te viññāya (Whatever is viññāṇa is soul and whatever is soul is viññāṇa). However at a deeper level it does become separated: eye viññāṇa cannot hear, ear viññāṇa cannot see, any or all of these viññāṇas can stop, and when mind viññāṇa also stops, nibbāna comes.

Discourses on Satipatthana Sutta - S. N. Goenka

Also see this answer which has further elaboration. Numerous philosophies about viññāṇa is the soul arises since the observers meditation is not developed enough to see the observations / observer is not continuous and impermanent.

In Buddhism both extremities of there is a self and here is no self is rejected. Better rendering in English is not/non self. For more information see this answer and this answer.

Anatta is non-self, which is neither 'self' (brahmanism) nor 'not-self' (annihalationism). Nothing is annihalationist and whilst the 'sphere of nothingness' is one of the last states the mind experiences before cessation, it is not the goal and is 'wrong view'.

The Buddha observed that empirical experience lacks these extremes like 'nothing' and 'everything`, 'always', 'never'. In my mind, I see it as whenever you draw a border or concrete dividing line between two concepts, you create a construct that does not appear in experience - every 'thing' is correlated to every other 'thing',. There is no 'solid' base to stand on within experience.

No permanent soul or substance transmigrates within Buddhism. The Buddha simply saw the consequence that Anatta has fo 'the all' and all within it - regularity (for a modern scientist, on average). And even then this is still an essentialist definition. Because rebirth is a construct that concerns 'the all' (there is never a concrete barrier to block where a person, 'soul', goes within the Buddhist universe, so it can't be though of with regards to only a fraction of experience), it can't be based on relational constructs.. But these are the only constructs we can know conceptually, as they are the only ones we experience. If Existence is 'infinite' and effectively has all states present at all times, then somewhere or other, a being dying with X Kamma, will always be succeeded by a being born into conditions fitting X Kamma. Rebirth is to be understood or not understood fully by a mind that has had long and persistent training in Jhana imo..

When one annihilates all delusion of self, all defilments are overthrown, and one rests in Liberation, there is nothing to be reborn. There is no "eternal" - which is only a notion of time unending. Advaita is the idea that an eternal self reincarnates throughout the universe. However in Buddhism the "ultimate" mystery is Emptiness which although the mind powerfully moved by it, ultimately is inconceivable, unknowable. This Emptiness is the essential nature of everything, but it lacks a "self" nature. Emptiness is beyond any concept of time and space, birth and death, being and non-being. So, it isn't a self that reincarnates in Buddhist conjectures, but something else. Normally this is where the concept of Karma is introduced. Karma operates something like Fate, it too is unknowable, except to the few with special insight into paranormal insights into reality. Why does it matter at all? Because one lives dealing with karma arising all the time, we don't know where it comes from, and that which is our karma becomes our life. One cannot merely step away from this.

It is said that buddha upon enlightenment could see past lives and all other knowledge. Rishis also attained this level. Therefore buddha merely wanted people to be able to attain such wonderful state, that was kept hidden from them and him or lost. Krishna that vedas was lost and he came to teach the knowledge back to humanity. During Buddhas time this knowledge was lost as well, again. This is why it is so easy for hindu to see buddha as reincarnation of vishnu. Before regressing into sophistry, I would like to point out that logic and discernment is very important, as well technique to obtain enlightenment. The technique buddha propounded are in the vedas. It is not different. Just like sankhya and yoga are not different, so is buddhas teaching and many rishis and acharyas are not different.

1. Advaita practitioners believe that we have an eternal self (consciousness) that is one... we can realise the truth.

"one with everything", do you mean one eternal self for every one of us, i.e., we are all one eternal self; if everyone enlightened, i.e., there will only be one thing left, the eternal self? Since not familiar with Advaita concept thus ask for clarification.

2. Theravadin Buddhists on the other hand believe that there is no 'self'...

Purely a personal opinion, the Theravada School is incomplete, from a Buddhist practitioner's point of view. To remark is, especially if one reading articles on the internet in English about the Theravada School's teaching based on many Pali translated English (modern) Sutras formed the view believed that's what Buddha taught, they could be inaccurate; upto this moment these views and the analogies behind them have a lot to be examined and verified, or even updated.

3. I just dont see any evidence of anything living on beyond the death of my brain.

One do not die after dead, this the Buddha has used an analogy to make King Prasenajit and others realized the fact:

...佛言。大王。汝見變化遷改不停。悟知汝滅。亦於滅時。知汝身中有不滅耶。波斯匿王 合掌白佛。我實不知。佛言。我今示汝不生滅性。大王。汝年幾時見恒河水。王言。我 生三歲慈母携我。謁耆婆天經過此流。爾時即知是恒河水。佛言。大王。如汝所說。二 十之時衰於十歲。乃至六十日月歲時念念遷變。則汝三歲見此河時。至年十三其水云 何。王言如三歲時宛然無異。乃至于今年六十二亦無有異。佛言汝今自傷髮白面皺。其 面必定皺於童年。則汝今時觀此恒河。與昔童時觀河之見有童耄不。王言不也世尊。佛 言。大王。汝面雖皺而此見精性未曾皺。皺者為變不皺非變。變者受滅彼不變者元無生 滅。云何於中受汝生死。而猶引彼末伽梨等。都言此身死後全滅。王聞是言信知身後捨 生趣生。與諸大眾踊躍歡喜。得未曾有。──《楞嚴經》

It reads as below in English:

...Buddha said, "Maharaja, you see that all are transforming and changing incessantly, thus realized that you would decease. Also during the decease, do you know that in your body there has the not obliterated?"

King Prasenajit holding his hands in prayer replied the Buddha, "I indeed don't know."

Buddha said, "I now show you the no originated-obliterated quality. Maharaja, how old were you when seen the water of River Ganges?"

The King said, "When I was three years old my loving mother took me to visit Deva Jivaka passing by this flow, at that time I have known it was the water of River Ganges."

Buddha said, "Now you are lamenting your gray hair wrinkled face, the face must be more wrinkled than at childhood. That by now you observe this River Ganges, as in the past as a child observing the river this seeing has the difference of young or aged not?"

The King said "No, World-honoured One."

Buddha said "Maharaja, although your face wrinkled yet this essence of seeing has never wrinkled. The wrinkled is change, the not wrinkled not change; the change is to be obliterated, that the unchanged essentially is not originated and obliterated, how so in it do render to your living or dead? Whilst still quoted those such as Makkhali [Gosāla], and said the same that after this body died all annihilated?"

The King heard this, truly knew that deceased yet living after the body. With the rest of all rejoicing in jubilation, obtained the never have had before.

The above is quoted from the Surangama Sutra translated into English, by me :]. Surangama Sutra has been disparaged by many, both in the Chinese language source and others. Here I will bypass to tangle in examining its authenticity. IMO it's the Buddha's Sutra as solid as the rest. It's said that this is the 1st Sutra to disappear at the ending of Dharma time; the last is the mantra "Namo Amitabha". If you read this conversation between King Prasenajit and Buddha, you will get that there is the essence (of you) that will never die, as King Prasenajit. I got it when reading this part. Since I'd known the non-dying in me in other experience, I was not as exhilarated as the King and others, only slightly gladdened. Hopefully it works for you, and brings about the same effect ;). Else try to contemplate few more times to get it.

4. ...the concept of Annatta... what is it that is reborn? ...Nothing is observing something?...

Wow, I'll leave this to later maybe, since my answer is so long now, and not sure if I could produce an accurate answer... I'm reading the Mahāparinirvāṇa Sūtra and Laṅkāvatāra Sūtra, hopefully I could come up with some more "prajna" to answer your question ;].

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*Below is my own exercise, for the convenient of writing, just disregard.*
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1. ...佛言。大王。汝見變化遷改不停。悟知汝滅。亦於滅時。知汝身中有不滅耶。

   • ...Buddha said, Maharaja, you see that all are transforming and changing incessantly, thus realized that you would decease. Also during the decease, do you know that in your body there has the not obliterated?

2. 波斯匿王合掌白佛。我實不知。

   • King Prasenajit holding his hands in prayer replied the Buddha, "I indeed don't know."

3. 佛言。我今示汝不生滅性。大王。汝年幾時見恒河水。

   • Buddha said, "I now show you the no originated-obliterated quality. Maharaja, how old were you when seen the water of River Ganges?"

4. 王言。我生三歲慈母携我。謁耆婆天經過此流。爾時即知是恒河水。。

   • The King said, "When I was three years old my loving mother took me to visit Deva Jivaka passing by this flow, at that time I have known it was the water of River Ganges."

5. 佛言。大王。如汝所說。二十之時衰於十歲。乃至六十日月歲時念念遷變。則汝三歲見此河時。至年十三其水云何。

   • Buddha said, "Maharaja, as you said, during twenty was weaker than at ten years old, nonetheless at sixty; days, months, hours and moment by moment that's changing. That when you were three-year-old seeing this river, up unto at thirteen how so the water?"

6. 王言如三歲時宛然無異。乃至于今年六十二亦無有異。

   • The King said, "Clearly has no difference as at three years old. Even by now at sixty-two there has no difference."

7. 佛言汝今自傷髮白面皺。其面必定皺於童年。則汝今時觀此恒河。與昔童時觀河之見有童耄不。

   • Buddha said, "Now you are lamenting your gray hair wrinkled face, the face must be more wrinkled than at childhood. That by now you observe this River Ganges, as in the past as a child observing the river this seeing has the difference of young or aged not?"

8. 王言不也世尊。

   • The King said "No, World-honoured One."

9. 佛言。大王。汝面雖皺而此見精性未曾皺。皺者為變不皺非變。變者受滅彼不變者元無生滅。云何於中受汝生死。而猶引彼末伽梨等。都言此身死後全滅。

   • Buddha said "Maharaja, although your face wrinkled yet this essence of seeing has never wrinkled. The wrinkled is change, the not wrinkled not change; the change is to be obliterated, that the unchanged essentially is not originated and obliterated, how so in it do render to your living or dead? Whilst still quoted those such as Makkhali [Gosāla], and said the same that after this body died all annihilated?"

10. 王聞是言信知身後捨生趣生。與諸大眾踊躍歡喜。得未曾有。

    • The King heard this, truly knew that deceased yet living after the body. With the rest of all rejoicing in jubilation, obtained the never have had before."

My person does not know how it would be possible to draw a lasting border between this and that ...-ists, as views are not possible to freeze, but slippery they decay. How ever...

Well instructed disciples of the Buddha do not waste much of their time with their identity, self, whether there is a self or not, but focus simply on suffering and the end of it. To best do so, they usually cut off whatever might be grasped as own, unuseful refuge, whatever might bind, seeing it as not real and secure. Doing that way systematical they overcome suffering, birth, oldage and decay.

(Note: this is not given for trade, exchange, stacks or to entertain for sufferings sake but as a tiny door to escape from this wheel)

The Surangama sutta is proof that Buddha was positing a concept which is veritably Advaita!

In his conversation, with King Prasenajit, and in his previous conversations with Ananda and group, he does prove the existence of an unchanging presence of awareness -- an awareness that knows all the changes, but is itself free from changes of any kind!

in Advaita Vedanta, this unchanging awareness at the background of all our experience is known as "Brahman"! It's not a "self" in the sense of the term that it's not a thinking, deciding entity -- thinking and deciding are mental processes that take place in the presence of awareness!

Even consciousness, unconsciousness, all take place in the presence of awareness -- that is why Buddha gave so much importance to The Cessation state or Niroddha Samapatti -- because in this state, when all perceptions and thoughts have ceased, one still is aware -- the presence of awareness is still there, otherwise "who" or "what" experiences Nirvana -- how is the experience of Nirvana possible without an awareness being present there to acknowledge it?!

This is what the Buddha meant when he said that he teaches "the path to the deathless" -- "This is the unborn, undead, unfabricated awareness" -- and Nirvana is the proof of it!

Think about it, Folks -- how can cessation state be permanent, or deathless? It's just a temporary state which the meditator experiences, and then one again returns back to the ordinary consciousness -- it is not cessation that is permanent, but rather it is the awareness, that is present in cessation when all contents have disappeared, which is referred to as Deathless here. Because, irrespective of the contents of the mind, whatever state of jhana mind may be in, there is always a presence of awareness that knows all of these states -- even unawareness is registered as a discontinuity between experiences, in the presence of same awareness!

Now, dependant origination applies to everything else in the Conditioned world -- every Conditioned entity arises and passes away, and its arising and passing away are based on causes and conditions! Emptiness in Buddhism refers to the lack of any absolute self-nature of dhammas. Now think about it -- how can the existence of any object be confirmed without the presence of an awareness field in which its perception takes place? How the existence of an object can be inferred, other than the perception of it? And what perceives it, if not the presence of awareness?!

All confusions that have taken place between Vedanta and Buddhism is because of the difference in language and dialect -- Buddhism defines terminology differently from that of Vedanta -- e.g in Buddhism, "Consciousness" means Mind. Vijnana (consciousness) actually is the reaction in the chitta field, which is caused by any of the six senses. Now in Advaita, consciousness means the ultimate subjective presence of awareness that watches the activities of the mind -- according to Vedanta, senses are observed by mind, and mind is observed by consciousness!

Apart from different words, Buddhism relies on a different causation model called pratityasammutpada or dependant origination, but Advaita uses the idea of Vivartavada or principle of superimposition!

One last thing,here some people mistake the "observer self" that is encouraged in Buddhist practises of Vipassana, for the "Witnessing consciousness" in Advaita! They are different things -- the "observer mind" in Buddhism, which is used for mindfulness practises, is actually the trained mental faculty of observation: it's not the consciousness! Similar practises exist in Vedanta, by the name of Sakshi bhava, where the aspirant is asked to develop the attitude of a detached witness to his/her thoughts, emotions, actions and senses, so as to gain greater insight of how these processes operate!

'Re-birth' is merely the re-arising of the 'self-belief' or 'self-concept' (in this life). To quote:

There is the case where an uninstructed, run-of-the-mill person — assumes form, feeling, perception, thought fabrication &/or consciousness to be a 'self'. That assumption is a thought fabrication. Now what is the cause, what is the origination, what is the birth, what is the coming-into-existence of that thought fabrication? To an uninstructed, run-of-the-mill person, touched by that which is felt born of contact with ignorance, craving arises. That thought fabrication (of 'self') is born of that. SN 22.81

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The craving that makes for re-becoming — accompanied by passion & delight, relishing now here & now there — i.e., craving for sensual pleasure, craving-to-be, craving not-to-be: This, friend Visakha, is the origination of self-identification described by the Blessed One. MN 44

This 'self-belief' or 'concept' is a thought that is known or observed by 'consciousness'. Just as consciousness is not the thought of a 'dog' or 'tree', consciousness (which observes) is not the thought of 'self', 'I' or 'me'. The thought is a separate thing to the consciousness.

There is no evidence of anything living on beyond the ending of the activity of 'the' brain (not 'my' brain). The totally lame analogy about a candle flame going from one candle to the next which explains nothing was not an analogy spoken by the Buddha. Buddhism teaches the five aggregates are impermanent. When the activity of life (the aggregates) ends, that's it. If the mind can fully realise & accept all life ends, it will be at peace. That is all that needs to be done. Please read the Anatta-Lakkhana Sutta and the Maggavagga and be free from doubt.

This is a question that will live on in your mind unless you douse it yourself by some belief or another.

Why I say this is that, Buddhists are going to read the texts and say "this is what Lord Buddha said", according to some translation by someone. Advaita people will say something else according to some interpretation. And you will sit here and kill yourself by thinking about this over and over without a proper answer. And this will be never ending.

Unless you are doing this for a particular philosophy course or something where you need an answer to a question, I'd suggest you be okay with not knowing. Because the frame of reference from which you are asking this question about "self" or "no-self" is yourself, and this is a loop. Your mind cannot know what is beyond itself. The mind is a mere accumulation of thoughts and memories from the past. There is nothing original or new about it.

Don't jump to an answer yet because that will be a belief. It is not in your experience.