on what the Buddha taught
We don’t need to consult many buddhist texts to read that according to them death is not the end of it all. Were it so a knife would then already have been a far more easier tool. Rebirth and reincarnation don’t have this problem but share the idea that a certain something is carried over, or remains, from one life to another and that is there the problem.
Eternalism (sassatavādā) and nihilism (ucchedavādā) are two extreme (worldly) views on existence. Eternal is seen as lasting forever and thus must hold time as never ending (eternal) first. The breaking up, disintegration, perishing, of this existing-through-time is nihilism (existing-through-time until no more, is just a delayed version of it). Both rebirth and reincarnation are based on these misconceptions [1].
If rebirth and reincarnation were to be redefined to imply dependent-co-arising (a buddhist version of rebirth or reincarnation? why such horrible construct) it would still solve nothing. Rooted in wrong view they don’t disentangle anything but build further on top what is not understood, it added another concept to the confusion thus requiring more explanation not less [2]. With rebirth and reincarnation beings are seen as reborn or reincarnated, yet beings are born. Both views (which are sakkāyadiṭṭhi, a view of embodiment) must at least acknowledge birth. And it is precisely birth which is not being understood.
The discourses talk about birth, next birth, a following existence, etc.; translations as rebirth and reincarnation are here just translational liberties. And unlike them, birth is not just a view. Birth already includes the possibility for a next birth (birth is birth) for it too is dependently-co-arisen. Thus depending on the context the usage of birth, next birth, future birth, further existence, etc. will do perfectly fine; there is no need to seek shelter in wrong views. Two wrongs don't make a right.
The world fares towards extremes (eternalism, nihilism); it was the exalted one who taught by the middle and a middle simply can not be grouped with either extreme.
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