is that after death the spirit gets fragmented into shards. Each shard becomes part of a new person but is scrambled and the new person does not have the memories of the person the shard came from.
Let's say for example, I die, and my spirit is fragmented into seven shards. Each shard forms with new materials to form a new spirit and a new person. All seven people have no memory of being me, save many some cases of deja-vu or something, and all seven are spread out throughout the globe.
Yet the question is does any of the seven people have any memories or ideas that they used to be me?
In this way the population can grow and the spirits just keep splitting up after death.
There are no new spirits being made, just shards of the old spirits being split up into several or maybe hundreds of spirits and people with no memory of past lives.
I seem to recall in the Bible that Jesus heard musicians playing music and said they could have only gotten that way by thousands of years of playing. I forget the chapter and verse, but it might be possible that one time Christianity had a reincarnation idea like the Hindus have. If true, something survives the reincarnation or else the musicians being reborn wouldn't remember the music they played in their former life. There is also that being reborn thing that Jesus said and Christians take it figuratively as being reborn spiritually, and others thought it meant being reborn literally like physically. It depends on the interpretations though.