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If the Buddha really taught that there is no eternal soul (atta) then Buddhism is an absurd nihilistic religion. If the only self is Samsaric and there is no transcendent self (soul, atta in the Vedic sense), then there is no possibility of enlightenment, rebirth or liberation from Samsara because by definition there is nothing to take rebirth, gain enlightenment or be liberated. Just a phantom and an illusion that ceases to exist at death when it's causes, the 5 aggregates, perish. It cannot be "experience" either, because experience is impermanent and thus dukkha and Buddha said experience is without self (soul - atta). Without atta (eternal soul), the entire project of Buddhism falls apart. You can try to spin it, but whatever notion of "self" you cobble together to explain how something that you have defined as not existing can take rebirth and be enlightened, from a no soul perspective, will be Samsaric by definition, impermanent and thus dukkha. This is the central problem with Buddhism. It has gotten the lesson of anatta wrong. Anatta is an adj. that describes that which is without soul (atta). Therefore the use of atta in suttas does not mean "oneself" (no reflexive term for self in ancient pali) but logically means "Soul" or "True Self" in the Vedic sense since Anatta means an absence of the soul, true self in the Vedic sense. So annatta is being used by the Buddha to point out what atta is not, namely everything within Samsara. If you could identify atta within Samsara, then there would be no possibility of enlightenment because your essential nature will be Samsaric. But because atta cannot be found in Samsara, that means escape from Samsara is possible because your essential nature is Nirvana (the unborn, unmade etc. that makes liberation from the born, made etc. possible). There is zero basis in the suttas to support the claim that the Buddha taught there is no eternal soul and thus, the entire foundation of Buddhism for the last 2,000 years crumbles and falls apart. Understand?
“Whatever form, feelings, perceptions, experiences, or consciousness there is (the five aggregates), these he sees to be without permanence, as suffering, as ill, as a plague, a boil, a sting, a pain, an affliction, as foreign, as otherness, as empty (suññato), as Selfless (anattato). So he turns his mind/will/spirit (citta, Non-aggregate) away from these; therein he gathers his citta (nous/spirit/mind) within the realm of Immortality (amataya dhatuya). This is tranquility; this is that which is the most excellent!” [MN 1.436]
“Whenever we deny something unreal, is it in reference to something real”[Br. Sutra III.2.22].
"The purification of one's own mind/will (citta); this is the Doctrine
of the Buddha" [DN 2.49]
"How is it that one is called a 'Buddha'?...gnosis that the mind/will
(citta) is purified (visuddham)...such is how one is deemed a
'Buddha'." [MN 2.144]
'Etam amatam yadidam anupada cittassa vimokkho' - "This is
immortality, that being the liberated mind/will (citta) which does not
cling (after objectivity)" [MN 2.265]
You may find this video interesting of the Arahant, Ajahn Maha Boowa, describe his enlightenment. Powerful stuff. The money quote: "The basis of death exists right there in the citta. Birth and death are both present in the citta. The citta itself IS NEVER BORN AND NEVER DIES, rather, the defiling things that infiltrate the citta lead us to repeated birth and death."