Re: Date for Last Common Ancestor?
Stephen Barnard (steve@megafauna.com)
Tue, 13 Aug 1996 02:34:59 -0800
Stephen Barnard wrote:
>
> CHESSONP wrote:
> >
> > "Let S_1 be the set of all people whom are alive today. Let S_2 be the
> > set of mothers of the members of S_1. In general, let S_k be the set
> > of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
> > S_k-1), because everyone has only one mother, but some mothers have
> > more than one child. When the size of the set reaches 1 then we have
> > arrived at the mitochondrial Eve.
> >
> > Steve Barnard"
> >
> > This does not prove that the cardinality of the sequence of sets {S_k}
> > coverges to one. To establish the existence of a "Last Common Ancestor"by
> > logical means only will require quite a lot more than you have supplied,
> > starting with a clear definition of what is the "Last Common Ancestor" for
> > a species.
>
> I suppose that in strict mathematical terms it doesn't. That would
> require a strictly decreasing cardinality of sets.
>
> From a practical point of view I'm satisfied with it. The group that
> included the putative mitochondrial Eve would very probably have been
> pretty small. But you never know. :-) I've seen speculation in this
> newsgroup that relies on *much* flimsier assumptions.
>
> Steve Barnard
I've been thinking about this some more, and I'm becoming convinced that the
argument for a single mitochondrial Eve, while not airtight, is about as close to
airtight as anything gets in paleoanthropology. Bear with me.
If the cardinality (i.e., the size) of the sets were strictly decreasing there
would be no problem. We would eventually arrive at the singleton set. That's
just simple mathematical induction.
There is no question that the sizes of the sets does not increase. That would
require at least one person to have more than one mother. Obviously, that's not
possible. I think I'm on pretty safe ground there.
The only problem arises when the sizes of the sets remains unchanged. Suppose
S_k = S_k-1 = N (where I mean the *sizes* of the sets). This would mean that
every member (i.e., each of the N mothers) in S_k had *exactly* N daughters who
were in S_k-1, who in turn must all bear female children (because they are
mothers of mother in subsequent sets).
Clearly, for large N this is extremely unlikely. But it's worse than that. Even
for small N (say N=2), for N never to reach one it would be required that both
mothers in set S_k-1 be the *only reproducing daughter* of the two mothers in set
S_k, and this state of affairs would have to be maintained backward in time *in
perpetuity*!
So it appears to me that the case for a single mitochondrial Eve is made. If
anyone can point out a flaw in this argument (assuming anyone has waded through
all this), I'd be very interested to hear it.
Steve Barnard
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