I thank @Jayarava and @cdpatton for bringing this up.
As @cdpatton said in another thread:
I’m not myself aware of early Buddhist sources believing that ordinary people randomly remember past lives; that was a supermundane knowledge. …
As I noted there, the liberating knowledge involving past lives seems very much tied up with seeing through the illusion of self, seeing the taking up of many identities through many lives. It’s a high level of attainment, a precursor to understanding the Noble Truths:
When their mind has become immersed in samādhi like this—purified, bright, flawless, rid of corruptions, pliable, workable, steady, and imperturbable—they project it and extend it toward recollection of past lives.
They recollect many kinds of past lives, that is, one, two, three, four, five, ten, twenty, thirty, forty, fifty, a hundred, a thousand, a hundred thousand rebirths; many eons of the world contracting, many eons of the world expanding, many eons of the world contracting and expanding. They remember: ‘There, I was named this, my clan was that, I looked like this, and that was my food. This was how I felt pleasure and pain, and that was how my life ended. When I passed away from that place I was reborn somewhere else. There, too, I was named this, my clan was that, I looked like this, and that was my food. This was how I felt pleasure and pain, and that was how my life ended. When I passed away from that place I was reborn here.’ And so they recollect their many kinds of past lives, with features and details.Suppose a person was to leave their home village and go to another village. From that village they’d go to yet another village. And from that village they’d return to their home village. They’d think: ‘I went from my home village to another village. There I stood like this, sat like that, spoke like this, or kept silent like that. From that village I went to yet another village. There too I stood like this, sat like that, spoke like this, or kept silent like that. And from that village I returned to my home village.’
In the same way, when their mind has become immersed in samādhi like this—purified, bright, flawless, rid of corruptions, pliable, workable, steady, and imperturbable—they project it and extend it toward recollection of past lives.
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When their mind has become immersed in samādhi like this—purified, bright, flawless, rid of corruptions, pliable, workable, steady, and imperturbable—they project it and extend it toward knowledge of the death and rebirth of sentient beings.
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When their mind has become immersed in samādhi like this—purified, bright, flawless, rid of corruptions, pliable, workable, steady, and imperturbable—they project it and extend it toward knowledge of the ending of defilements.They truly understand: ‘This is suffering’ … ‘This is the origin of suffering’ … ‘This is the cessation of suffering’ … ‘This is the practice that leads to the cessation of suffering’.
They truly understand: ‘These are defilements’ … ‘This is the origin of defilements’ … ‘This is the cessation of defilements’ … ‘This is the practice that leads to the cessation of defilements’.
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Jayarava criticises the research on past lives as not science. However, for me, that’s not the important point. It may or may not be poor science but it is definitely trying to be science. My criticism is that appeals to such research as support for Dhamma (along with speculation on connections between quantum mechanics and Dhamma) falls into the very same materialist view that the proponents say that we should be avoiding. It is an appeal to science to support Dhamma. If the Dhamma is not bound by a materialist world view it should need no such support.