Some people eat, sleep and chew gum, I do genealogy and write...

Saturday, March 26, 2011

Missing Ancestors or the Ancestor Paradox

It starts with a simple concept, in each generation the number of your ancestors doubles. You have 2 parents, 4 grandparents, 8 great-grandparents and etc. Some of my family lines go back 18 generations. At that level, I would have over 262,000 direct ancestors. By 20 generations the number would increase to 1,048,576. Continuing backward, I would ultimately have more ancestors than the total number of people who have lived on the earth. How does this mathematical concept correspond to reality? At some level, it would appear that everyone in the entire world is somehow related. This is called the Ancestor Paradox.

However, in these ancient times, it is supposed that there were far fewer people than at the present time.  (By the way, if you ever want to see every type of way-out theories from UFOs to asteroid impact, try looking up ancient population figures). So how do the calculated figures of my ancestry correspond to the supposition that there were fewer people living in the past? See Wikipedia. Where are all these missing ancestors?

The easy answer is to acknowledge that some of your ancestors married relatives. Would the impact of intermarriage reduce the actual number of ancestors enough to explain the missing grandparents? It would if everyone married their brother or sister, then there would be no increase in the number of ancestors. But, I have yet to meet anyone who can document an ancestral line of only marriages between brothers and sisters. Another alternative would be to marry only first cousins, then the numbers would increase only by two in each generation instead of doubling. Following this line of reasoning soon gets you to another paradox called Zeno's paradox. I'll let you look that one up.

The seemingly simple concept of the doubling of ancestors in each generation quickly becomes a morass of math and supposition. There are three very citing articles on the subject by Brian Pears beginning in 1985 in the Northumberland & Durham Family History Society Journal.  The three articles are: Our Ancestors, Conceptions, Misconceptions and a Paradox, The Ancestor Paradox Revisited and the Ancestor Paradox Yet Again. Let's just say that I don't find all of the logic in Pears' articles convincing. Before you get going on this question, you can begin by noting the fact that the Ancestor Paradox is not a Heap Paradox question. Neither is it a Bald Man Paradox because those paradoxes rely on the problem of fuzzy sets. (You can look those up also).

In some ways the problem is related to the ancient Theseus's ship paradox. (I know, you can look it up). The real question is whether or not the Ancestor Paradox is a problem in logic or math or is really a issue that can be solved. As was stated by Giangiacomo Gerla in his treatise called Why I have an extra-terrestrial ancestor, "The paradox lies in the possibility of confuting a theory on factual reality by a purely logical argumentation.

From a genealogist's point of view, every ancestor of a given person can be identified, just as we accept the proposition that every living human being had a mother and a father. However, we also jointly assume the opposite of this fact to be true, that it is practically impossible to identify every ancestor of any given person. So it may be logical from examining the simple mathematical progression, to assume that the number of ancestors increases in every generation, when in fact that progression may not be true. The number of ancestors would only increase if each ancestor is considered to be unique. So the problem is not so much one of a paradox but a fallacy. Although I said I did not agree entirely with Pears, I do agree that the flaw in the exposition of the Ancestor Paradox lies in a fallacious assumption, that is, the uniqueness of our ancestral lines.

Any thoughts on this subject? Comments would be appreciated.


  1. The first fallacy is quite simple - by the time you have gone back 20 generations you will discover that not all 1,04,576 ancestors were simultaneously alive. This could only be the case if each new generation was born precisely after the same number of years as all of its predecessors. Hence to talk of there being 'too many' ancestors after a certain number of generations is simply incorrect.
    Secondly, allowing for all cousin relationships to be recorded (imagine how difficult that would be to accurately ascertain!) would still leave us with 1,048,576 spaces in our tree but many individuals would appear multiple times, and not always at the same 'level' due to the factor alluded to above.
    Combining these two effects will significantly reduce the number of discrete individuals from whom one is descended AND who were alive at any given point in time and so the ancestor paradox simply disappears...
    PS - I am writing this in 'real time' so it's quite probably wrong!

  2. Anyone who has done French Canadian genealogy understands what is happening. There are fewers than two thousand couples, who sired 7 million French Canadians.
    My wife's mother's ancestors are a case in point. I have traced her grandparents completely through the fourth great grandparents. I have traced about half the lines some generations further, and have traced 50 lines back to the first immigrant. (This is not bragging. The quality of information in French parish registers is incredible, and they're almost all on Ancestry.)
    Of the first immigrants, only one came in the 18th century (and he was banished from France.) All the rest arrived in the 17th Century. 7 million French Canadians descend from 3500 original immigrants.
    What happens is that all the lines start to share ancestors. It's not a question of cousins. The first overlapping line is at the third great grand parents. Most of the immigrants were ten to twelve generations removed from my mother in law. There are 2048 ninth great grandparents. The probability is that some ancestors will repeat.
    And they're all some kind of cousin to Celine Dion. My wife is connected to the Guyon/Dion on both her mother's and father's side.

  3. I find the Ancestor Paradox interesting & have heard about it before. I do not have to go too far back in my tree to find that it is true. My 4th great grandparents, Jeremiah Smith (1773 - 1843) & Joanna Dillon (1778 - 1821) are on my tree twice. They are my double 4th great grandparents. Two of their children married into the family branches. Therefore the number of my ancestors is not as high as it should be from that generation and going back from them.

    I'm sure if we could trace back dozens of generations it would be more & more commonplace. People did not move locations. They paired up with people who were close at hand.

    I enjoyed your post. Colleen

  4. This is a great post and great comments.

    Lemonegg, I am a Dion and have found the French Canadian research to be some of the most challenging and thrilling of all my lines. My grandfather's family were Dion/Guyon and, you are correct, we are all related to Celine Dion. i have always told people she is a cousin since all French Canadians came from such a limited set of original ancestors.