Book review of Eric Olson

Eric Olson:
THE HUMAN ANIMAL (Oxford Univ Press, 1997)

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The British philosopher Eric Olson has written an odd treatise on personal identity.
What makes you you? What would make you somebody else? If we transplant a brain from one body to another body, who is who? Is the person its brain or its body? Olson is one of the few who answers: the body. Olson believes that identity comes from biology, not from psychology. You can be brain dead, but still be "you". For as long as some biological functions continue, you are you. If someone transplants your brain to another body, you are still the same "you", and someone has received "your" brain. This, of course, flies in the face of the definition of "I". Olson thinks that psychological continuity is one thing, and identity is another thing. Psychological continuity occurs between two "people" (he doesn't quite define "people", so we are not sure if he means "bodies" or "minds") at two different times: one at a certain instant is psychologically continuous with the other one at a later instant.
I take this to mean that minds change all the time, and therefore I am no longer the mind I was a second ago. Therefore it is improper to claim that "I" was. In a sense, "I" can only "be" now. "I am" is correct, whereas "I was" is a contradiction (the "I" that was is another one).
But I don't quite understand why the same argument does not hold for the body: biological continuity occurs between two "people" at two different times, because the body changes too. I, as a body, am not what "I" was a few minutes ago, because millions of cells have died and have been replaced by new cells.
Olson repeatedly justifies his faith in the body based on his view that we are, first and foremost, animals, therefore bodies. But where did he get the notion that animals are only bodies, and not minds? The asymmetry that he thinks he has found is a wild speculation. The fact that we are animals does not, per se, determine whether bodies or minds are more important. His emphasis is on the "life-sustaining functions" (they are the ones that define my identity, who "I" am), but he does not quite explain why these functions are more important than, say, the "mental-sustaining functions". He writes that "many animals... manage to persist without having any psychological capacities at all". One wonders how he found out.
If the body is the only thing that matters, one wonders what happens when exactly half of the body is transplanted into another body. Which half is the original person? Who is who?