APPENDIX A: MITOCHONDRIAL EVE Dennett, p. 97.

Definitions of terms:

A = CURRPOP Current population
B = M(CURRPOP) Mothers of current population
C = M(M(CURRPOP)) Mothers of mothers of current population
D = M(M(M(CURRPOP))) Mothers of mothers of mothers...
etc.

Remarks:

1. CURRPOP is the set of all human beings, male and female, alive today. "A" is the number of individuals in that set.

2. M(CURRPOP) is the set of all mothers of those human beings. "B" is the number of those mothers. Some are alive; some are dead. The two sets intersect over live mothers.

3. M(M(CURPOP)) is the set of all mothers of the mothers of all human beings alive today. Certain of the M(CURRPOP) mothers have the same M(M(CURRPOP)) mother. These are the M(M(CURRPOP)) mothers who gave birth to two or more daughters who subsequently became mothers themselves.

4. Dennett points out that B (the number of all mothers of human beings alive today) is of necessity less than A (the number of all human beings alive today) "since no one has more than one mother, and some mothers have more than one child."1

5. In the normal course of events, holocausts aside, B will be very much less than A, reflecting the fact that the current population consists roughly 50% of males and of many juvenile and barren females.

6. Next, Dennett represents, again as a matter of "necessity," that C is less than B; i.e., that the set of mothers of mothers is smaller than the set of mothers. Also, that D is less than C; i.e., that the set of mothers of mothers of mothers is smaller than the set of mothers of mothers, and so forth, etc. However, there is no reason given for this necessary decrease in the number of mothers of mothers of.... etc., as we go backwards in time.

7. If there is a reason for this decrease, it is certainly not the reason given by Dennett appearing in (4), above; namely, that "no one has more than one mother, and some mothers have more than one child." (That reason supports only the claim that the number of any one group of mothers is smaller than the number of all of their children — which number contains non-mothers.)

8. Nor is it because "no one has more than one mother, and some mothers have more than one female child." (That reason supports only the uninteresting claim that there are some sisters.)

9. If the mother sets do in fact get smaller as we go back in time, it is because "no one has more than one mother, and some mothers have substantially more than one child-bearing female child." (See Par. (3), above, and DISCUSSION, below.)

10. However, it is not necessarily the case that some mothers have substantially more than one child-bearing female child. If some do, they do only as a matter of empirical fact. A society in which only one female (and not necessarily fecund) child per mother is allowed is imaginable, and I think actual. Infant girls are frequently put to death.

11. It is also not necessarily the case that the number of "extra" daughters produced, even if they become mothers, will be greater than the number of "regular" daughters who remain barren.

12. It is thus not necessarily the case that the sets C, D, E, etc. get smaller and smaller as we go back through time. They may; they may not.

13. Even if the regression be allowed, as a matter of fact, we will not necessarily have at the end of the road one "Eve" on our hands — the most recent female ancestor of all human beings alive today.

14. What will happen is that, as we go back in time, starting with any mother of a human being alive today and examining her mother, and then the mother of that mother, etc., we will eventually come to the last of these mothers who had two daughters who themselves became fecund, and whose own daughters (at least one of them per generation) remained fecund down over the generations to the present day. That's Eve1 — the most recent female ancestor shared by some human beings alive today. Eve1 of course had a mother herself, but the mother of Eve1 had only one daughter, not two or more, and so does not stand as the one most recent female ancestor shared by some human beings alive today.

15. Starting with any other mother of human beings alive today might lead to the same Eve1, but it might also lead to Eve2, or to Eve9870. These Eves may have lived in different places at different times, constituting in that case a sort of trans-spatial-temporal sorority which, by its nature, could never meet as a group.

16. That sorority set would be one in number only if we knew on other grounds that the bloodlines all converged as we moved backwards in time.

17. However, in that case we would have already known that there was one stand-alone Eve, and would never have been concerned to prove the case all over again.

18. In sum, a regression in the number of mothers, etc. would be shown to have been necesssary if we knew there was an Eve, but does not itself, even if established as a matter of empirical fact, prove that there was only one such lady.

19. The empirical facts which will show it to be one way or another will probably involve the comparison of the mitochondrial DNA found in contemporary human beings with the DNA recovered from bones of the most ancient of our ancestors that we are able to recover.

DISCUSSION: A society in which all females replaced themselves by giving birth to one and only one daughter who becomes a mother herself is in a "steady state" in regard to mothers. In such a society, B would be less than A (in respect of the birth of males), but there would be no backwards convergence of the sets of mothers towards ever-smaller sets. C would equal B, and D would equal C, etc.

It is only if and when mothers start having "extra" daughters who themselves become fecund that the number of mothers of mothers may possibly increase as we go forward in time. If, for example, out of 100 mothers, 10% of these give birth not just to one daughter, but to two daughters, there will be ten (10) extra daughters. If all of the sisters become mothers, the number of mothers of mothers will increase by ten, minus the fraction of the hundred first-born daughters who do not become mothers, whatever that fraction might be. If it is 10%, the society will be in mother-steady-state once again — with the number of fecund extra daughters just making up for the physiologically barren women and the career girls.

What just "turns out to be the case" carries with it no necessity. Either the set of mothers of mothers will increase, or it will decrease. One can easily imagine a wide variety of contingent factors (Ms. magazine, or the abolition of birth control devices) which might make it turn out one way, or the other.

Looking not forwards, but backwards in time, we are in a position to investigate whether or not the number of extra daughters who turned out to be fecund in fact "made up for" the daughters who were to remain barren. I think that the answer is "Yes," that they did, if only reasoning from tabloid coverage of a world-wide population explosion during recent times.2

I do not know about earlier times, but I think the evidence points towards there being fewer people around. And so I think it's safe to say that the very early sets of mothers of mothers of mothers, etc, were in fact smaller than the later ones, and that these sets do "trend downwards" in size as we go further and further back in time.

However, I do not see them becoming smaller with logical rigor, with one Eve under the cross-hairs. Not unless we are first prepared to demonstrate independently that every example of the species Homo sapiens can be traced back to one physical location and to one woman — to show that human speciation did not occur in different places at different times.

Absent that demonstration, we should be prepared to live with the notion that there were a great number of Eves, each living in different places at probably very different times. We cannot look to the regression of probably ever-smaller mother sets, which is itself an empirical matter — not a matter of necessity — and to the bare logical possibility of one stand-alone Eve, as a substitute for that demonstration.

1 Ordinarily so, but not necessarily so. During the course of a partial or mass extermination of the human species, there might come a point in time when the number of mothers of all human beings remaining alive would be greater than the number of these live human beings.

2 Interesting actuarial statistics would include (a) the percentage of recent mothers who gave birth to more than one child-bearing daughter, and, in those cases, (b) the average number of such daughters, and (c) the average age of these daughters when they gave birth to their first child. This would allow us to calculate how fast the Eve engine has been running, at least over the time periods that we are familiar with, and when Eve lived (if she happened to!)

Editor's note: Professor Dennett has come to agree that there is no logical necessity attaching to the notion of one, stand-alone Eve, and has so stated in his Errors memorandum clickable here.