A banal observation about world population in history

In 1970, a very clever writer wrote once :

A banal observation has left me stupefied, so much, that I guess that it should have inside an unforgiveness fallacy. I started thinking about the principle that I have two parents, four grandparents, eight great-grandparents and sixteen great-great-grandparents. Why not keep going? Using a pencil and a piece of paper I calculated that progression. In year 1780 I had 64 ancestors (estimating 30 years by generation). In year 1480 I had 65,536 ancestors, in year 1240 I had 16,713,216 ancestors, in year 1060 I had 1,069,645,824. And I didn't go on because it went absurd, in the most historic falsehood. Simply because in year 1060 world population wasn't even up to two thousand million of people. What kind of explanation could have my calculations?
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some of those 1,069,645,824 people are the same
miyahiraAuthor Commented:
He just continued:

Incest and polygamia can reduce somehow this numbers, but not to the extreme of destroying its unacceptable quantity.
Dave BaldwinFixer of ProblemsCommented:
I don't believe that "Incest and polygamia" have anything to do with it.  Your apparent assumption that there should be 1,069,645,824 unique people is wrong.  1000 or more years ago, it would have been common for neighboring tribes to inter-marry.  And then inter-marry again.  After a while, everyone is related to everyone.  Not thru incest but second or third cousins.
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You could also start with Adam and Eve and go forward.  Assume every couple has two male and two female offspring, so the size of each generation doubles every generation (callit 25 to 30 years).  Doesn't take that long to get to the current population.
Cornelia YoderArtistCommented:
The answer is something called "pedigree collapse."  This occurs when relatives, usually cousins, marry, in effect narrowing the family tree.  When this happens, many of the "slots" in a given generation of your family tree are filled by duplicates.

Pedigree collapse explains why it's so easy to trace your ancestors back to royalty — go far enough back and you're related to everybody.   In fact, you're probably related to everybody alive today, somewhere around 50th cousin maximum.
" in year 1060 I had 1,069,645,824"
"Simply because in year 1060 world population wasn't even up to two thousand million of people. What kind of explanation could have my calculations? "

You included all the born people,
At a certain moment of time, on Earth there were less people, because some obviously died in mean time. Not all were alive.
The population in 1060 has nothing to do with the number of people/ancestors who were born and died between now and 1060.

A simple math.
1st generation – 1x30 years – 1 person
2nd generation – 2x30 years - 2 persons
3rd generation – 4x30 years – 4 persons
4th generation - 8x30 years – 8 persons
N generation – N x 30 years – 2 power (n-1) persons
>>  A banal observation has left me stupefied . . .

I would say:   A naive observation has left me confused . . .

A simple (and self-consistent) model:   Population grows exponentially forward from a source.  It is larger at the bottom like a pyramid.
    Gen 1 ==> Adam and Eve                            2 people
    Gen 2 ==> Abel Betty Cain and Diane       4 people
    Gen 3 ==> each couple has 2B and 2G     8 people
    Gen n ==>                                                      2^n people    

A simple (but flawed) model:  Your number of ancestors grows exponentially backwards.  It is larger at the top like an inverted pyramid.      
    Ancst 1 ==> Mom and Dad                                       2 people
    Ancst 2 ==> Granny Gramps Nana and Papa       4 people
    Ancst 3 ==> each person has 2 parents                8 people
    Ancst m ==>                                                              2^m people  

Your ancestor pyramid must fit within the human population pyramid.
In this world:
You can have the 65,536  [14 x Great]-Grandparents that you expect.
But further back, the numbers decrease instead of increase.
So, instead of having  4 Billion  [30 x Great]-Grandparents you have to settle for two famous ones.
Think maybe in a binary fashion. Males as '0' and females as '1'. Consider a set of eight couples:

10 10 10 10 10 10 10 10

Now picture those couples each having two offspring, one male one female. The next generation looks like:

01 01 01 01 01 01 01 01

Okay, so mix up the pairings from that generation for a new set of parents that looks like:

10 10 10 10 10 10 10 10

They create a new generation that looks like:

01 01 01 01 01 01 01 01

Again, mix up the pairings to create a third set of parents:

10 10 10 10 10 10 10 10

Repeat as needed.

Okay, so that eventually will probably cause trouble as genetics reinforces any flaws; but there are probably more than 16 members in the 'tribe' at the top of this discussion. Also, some of the 0s and 1s will mix over to tribes in N/S/E/W directions rather than staying within the tribe.

Some/many pairings will produce more than two offspring, though mortality takes a high toll for infants and other youths. Few ever get "old". The overall population doesn't grow very fast. Natural controls keep things in check.

Regardless, it should only take a moment's thought to see that the '1+0' pairs can be extended to many tribes scattered (and scattering) over wide areas. There is zero requirement to avoid incest after the first couple intermixings between tribes. It's unlikely that anyone would have the faintest idea that a blood relationship existed after the fourth (or maybe even third) generation when multiple tribes are involved. And in an isolated tribe, it's unlikely that "incest" would be a meaningful concept.

For as long as basic natural forces are involved, the common ancestry is just the natural result. There is only negligible growth in overall population. At times, disasters will probably even reduce total population. From time to time, genes will come in from widely separated tribes to help refresh the local gene pools.

You simply can't calculate and imagine that the number of unique ancestors grows over any significant span of time. It just doesn't happen. Far too many slots are filled by duplicates. The more generations, the more duplicates will show up.


Let me ask you something: do you know who was your mother's mother's father' father's mother's father's mother? Very few people would even have a vague idea. Most would reply something like " Who cares, what relation is she to me?"

By the same token, you would never know if your mother's mother's father' father's mother's father's mother married your mother's mother's father' mother's mother's father's father (spot the difference?). And, no one in his right mind would consider that marriage anywhere close to incest.

Once you understand this, you may be able to understand how it is that many of your billion people are in fact the same person. In other words, your mother's mother's father' father's mother's father's mother was the same person as your mother's mother's father' mother's mother's father's mother.

Let's assume that all billion stayed in England. The English population at the year 1000 is estimated (with a large margin of error!) to be around  2,500,000. Hence, on average, every person living then is your ancestor in 400 different ways. Of course, that's an average; some of them will connect to you in thousands of ways and others may not be ancestral to you at all.

It also implies that you and I are probably related in many different ways, if you go back only 1000 years. For other pairs of people you may need to go back further in time to find common ancestors, but only about 70,000 years ago there were as few as 1000 people. Hence all people living today will be related to each other in many, many ways via those 1000. More extraordinary still, we all descend from "Mitochondrial Eve" and "Y-chromosome Adam". "Eve" lived around 150,000 years ago, and "Adam" about 250,000 years - again with large error margins.

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