You might come back as a cow...
You might come back as a cow. Or a mouse. Or an insect. Or another human being, depending on how good you were in the previous life. So say some religions, anyway.
It seems that as soon as you die in one place, you can be reborn in quite another. Usually you don’t remember your previous life, but in one famous case a poor boy, born at about the same time a boy from a higher caste died, claimed to remember the wealthy boy’s life. Given the distance between their two villages and the small time involved, none of the matter of the dead boy’s brain or body could have gotten to the newborn’s village. What could the two boys share? At most, a soul.
In that case (if you believe the stories) many of the psychological traits of the wealthy boy were retained. But note, you don’t need to retain skills, or memories, or personality traits, to be reincarnated, according to certain religious traditions. Cases in which people remember previous lives are rare, after all. And as just noted, you do not need any physical matter. You just need one item, the soul, even if it isn’t doing much to connect your prior and later lives together.
But what if you don’t believe in souls? What if there are no such things?
No shirt? No souls? No problem!
You gotta believe in atoms. The adult human body contains about 6*10^27 atoms. After you die, these get scattered about the local environment and incorporated into plants and animals. Even far into the future, even if your atoms mix evenly into the entire biosphere of the earth (less than 4*pi*(4000mi*1.6km/mi)2*2km height*10^12 liter/km3* 1kg/liter*1kmol/10kg*6*10^26 atoms/kmol = approx 10^47 atoms), that still leaves millions of your atoms in a human-sized future organism! So relax! If you thought just one part of you transmigrating was enough to keep you going, now you’ve got millions. So you see, reincarnation for unbelievers is not only lower in metaphysical calories than the leading brand of reincarnation, it gives you more of what you want.
Transmigration of atoms beats souls, millions to one!
What’s that you say – transmigration of atoms gives you too much of what you want? An embarassment of riches? OK, so there are many future people, each of whom will have some of your atoms. And you want to know which of them will be you, reincarnated. No problem!
Each of these people will have many, many of your atoms; some people will have more than others. It’s extremely likely that there will be one person with more than any other. If you insist on being reincarnated only once over, just declare this person the winner of the “contest”.
That’s the easy solution. But it’s not the elegant one.
The elegant one is to drop the insistence on being reincarnated only once over. After all, you were cool with the idea of living many lives in series, right? So why not in parallel?
“But which of these lives should be the one I look forward to?” All of them, of course.
“But these are all different people, separate from each other; they can’t all be me, because then they’d all be one and the same person, instead of many.” And you know that these are many people, not one, because of their separate bodies? Fine: then don’t call them “you”, call them “reincarnations of you,” same as you would call reincarnations on a more traditional, spiritual view. After all, on that view reincarnations have separate bodies from yours, too.
"But wait," you say. "Atoms are made of smaller particles like electrons, and electrons don't seem to have identities. Electrons are crucial to the way atoms behave within molecular reactions. So can the atoms really have identities when crucial parts don't? And what work are the atoms really doing in this reincarnation argument, anyway?"
Ah. Good questions.