Are there any attempts to prove the eternal blissful etc. self? The mahaparinirvana sutra seems to claim that there is an eternal blissful and pure self

'Self' appears in the positive very many times in that sutra, e.g. here (Chapter 33) ,,,


  Knowledge ["jnana"] sees the Void and the non-Void,  the  Eternal  and
  the  non-Eternal,  Suffering  and  Bliss,  the  Self  and  the 
  non-Self.  The  Void refers to all births and deaths. The Non-Void
  refers to Great Nirvana. And the non-Self is nothing but birth and
  death. The Self refers to Great Nirvana.


or here (Chapter 37) ...


  ""The Eternal of Great Nirvana is the Self.
  "The Self is the Pure.
  "The Pure is Bliss.
  "The Eternal, Bliss, the Self, and the Pure are the Tathagata.
  
  "O good man! For example, space is not the east,
  nor is it the south, nor the west, nor the north, nor the four directions, nor up or down. It is the same with the Tathagata


etc. e.g. here (Chapter 23) ...


  The Three Jewels of the Buddha, Dharma and Sangha truly exist, and the
  Tathagata always expounds the essentials of all laws [Dharma]


Did any Buddhist philosophers try to prove that nirvana is a self, or that a self exists?
Did any Buddhist philosophers try to prove that nirvana is a self, or that a self exists?

I might start by saying that I don't know how a philosopher can prove anything.


A mathematician proves theories by defining axioms and showing that the theories derive from axioms
A physicist proves laws by defining laws and showing that the laws predict observed physical behaviour


This topic about 'proofs' in philosophy says,


  
  [Except in "formal logic"] the best you get is an analytically-reasoned argument.
  If I can show that a given position is self-contradictory then I have proven the position false. In terms of positive proof for a position, then it does not seem that there is anything available quite equivalent to mathematical proof in either rigour or epistemic certainty.
  


So I don't understand the question as-asked. But (instead of answering the question) perhaps analysing the text and looking at definitions of the words will be a partial answer instead.



I just read Chapter 37 (only).

It's translated from Chinese, which is difficult -- I must read an English translation.

I do't entirely trust that because I think that word-for-word translation are inevitably inaccurate -- e.g. the Pali word dukkha has various translations, like "suffering", "unsatisfactoriness" etc. -- or Pali has three words for different aspects of "mind" -- but anyway I'll try what I can with the English translation.



To start I note it starts with orthodox/standard doctrine ...


  Because of this, know that Sramana Gautama has no father and no mother. If he has, how can he say:
  
  
    "All things are non-eternal, suffering, void; and all have no self, no doing, no feeling?"
  


... i.e. it says that the Buddha's doctrine is as quoted above (e.g. "all things have no self").

I'm not sure what "have no self" would be a translation of, if the original were in Pali -- I think it could be anatta or sunyata -- but the previous clause said "void" which I take it was a translation of "empty" or sunyata so I guess that "have no self" is a reference to anatta.

Incidentally the Pali phrase or doctrine is something like sabbe sankhara anatta (i.e. "all things are non-self") -- but "things" there isn't a great translation of sankhara, a better translation might be "compound things" (or "things put together from other things" or "things which depend on other things"), and perhaps specifically in contrast to nibbana which is called a dhamma and not a sankhara.

Anyway I think the word "self" (in common English and in English translations of buddhist texts) might be ambiguous and depending on context it could refer to different things:


Something eternal (a soul, a god, perhaps something "space" or "light" or "time")
Something unconditioned, which exists independently of other things
Something in the I/me/mine kind of realm, e.g. "my soul" or "my consciousness" etc. -- or conversely that the Self is the possessor not the possessed, etc.




The first mention of "Self" seems to be here:


  That is to say that all things are non-eternal, suffering, non-Self, and non-Pure, and that only the Tathagata is the Eternal, Bliss, the Self, and the Pure.


I have to guess at what "non-self" and "non-pure" mean; but the Wikipedia article you quoted says some things about that.

The "six masters" then say something like, "Well if you have Self then so have we. For example maybe Seeing is Self (and we see)", and the Buddha replies that seeing isn't Self. Here is the Buddha's argument:


  The Buddha said to the six masters: "If seeing is the Self, you are wrong. Why? You take up the analogy of an object and say that we see by it. Now, man uses together the six sense-organs to one object. If there surely is the Self and we see unfailingly by (means of) the eye, why is it that we do not cognise all objects with that one sense-organ? If one does not meet with the six sense-fields, know that there is no Self to talk of. If things are thus with the sense-organ of sight, there will be no change even if years pass and the sense-organs become ripened. As "man" and "object" are different, one sees one's own self and the other. If it is thus with the sense-organ of sight, there must be the seeing of one's own self and the other at (one and the same) time. If not seen, how can we say that there is the Self?


I think that's saying:


Seeing isn't Self because we cognise sense-objects with several senses (not just seeing)
Seeing is dependent on contact (with the sense-fields), if there's no contact then is there no "Self"?
Sense organs (e.g. the eye) change with time
Another argument I don't understand


I wanted to analyse the argument in order to understand (or reverse-engineer) the definition of "Self" that's being used.

I think the first bit of the argument is that if something (e.g. seeing) is Self then there can't be something else (e.g. hearing) that's also self.

And so on. If you analyze subsequent paragraphs in a similar way you derive different characteristics or definition of Self, for eample:


  
  If one cannot do as one wills, know that this indicates that assuredly no Self is there.
  The body of the Tathagata is not based on causal relations. Because there are no causal relations, we say that there is the Self. 
  If the Self does (i.e. performs actions), how can we say (it is) Eternal? If it is Eternal, how can a person do good at one time and evil at another?
  


I think this is more-or-less the conclusion:


  The Buddha said: "If the Self does (i.e. performs actions), how can we say (it is) Eternal? If it is Eternal, how can a person do good at one time and evil at another? If a person does good or evil at (different) times, how can we say that the Self is boundless? If the Self does, why would one practise evil things? If the Self is the doer and if it is the wise, how can one doubt about the selflessness of the being? So, we can say that there can definitely be no Self in the doctrine of the tirthikas. The Self is none other than the Tathagata. Why? Because his body is boundless and there exists no doubt. On account of non-doing and non-receiving (of karmic consequences), we say Eternal. On account of birthlessness and deathlessness, we say Bliss. As there exists no defilement of illusion, we say Pure. As he does not have the ten aspects of existence, we say Void (i.e. void of all that causes suffering). Hence, the Tathagata is none other than the Eternal, Bliss, the Self, and the Pure, and the Void, and there is no other aspect to speak of.


I think that's not a clear (or it's a difficult) explanation of why "the Self is the Tathagata" -- I think the reasoning is that it's consistent with the definitions or properties of "Self" -- e.g. (I paraphrase) "seeing can't be self because it's limited, light can't be Self because it's caused by (dependently originated from) a lamp, but the Tathagata is unbounded".

You might like to see also Why is the Buddha described as trackless? -- because I think that's an example of the Tathagata's being called unbounded in the Pali suttas.



I'm not sure what the point is, what the benefit is, for having this kind of doctrine abut Self. It's unusual, the Wikipedia article Ātman (Buddhism) which you quoted says,


  Most scholars consider the Tathagatagarbha doctrine in Mahāparinirvāṇa Sūtra asserting an 'essential nature' in every living being is equivalent to 'Self', and it contradicts the Anatta doctrines in a vast majority of Buddhist texts, leading scholars to posit that the Tathagatagarbha Sutras were written to promote Buddhism to non-Buddhists.


So if the Brahmins are teaching about the Self, then the Buddha has two ways to contradict them:


"There is no Self"
"The Self is the Tathagata"


Another theory is that it might be intended to contradict the Three marks of existence -- i.e. if things are "negative" (unsatisfactory and non-self) maybe there's something else that's "positive" ... even attractive.



Finally I note there's something else in that same Wikipedia article:


  According to Paul Wiliams, the Mahaparinirvana Sutra uses the term "Self" in order to win over non-Buddhist ascetics. He quotes from the sutra:
  
  
    The Buddha-nature is in fact not the self. For the sake of [guiding] sentient beings, I describe it as the self.
  
  
  In the later Lankāvatāra Sūtra it is said that the tathāgatagarbha might be mistaken for a self, which it is not.


I haven't tried to find the location in the sutra where it says that, you might find it interesting to do that.