Eliezer S. Yudkowsky (sentience@pobox.com) Thu, 16 Sep 1999 22:19:51 -0500 wrote:
>Damien Broderick wrote:
>>
>> More tricky ... is her claim that if we
>> double the human life span it will double the peak size of the global
>> population. This strikes me as an obvious non sequitur, but I'd like some
>> clever mathematics to prove it.
>
Um, er, actually, ...I think she's right.
But that doesn't mean Eliezer is wrong really, ...well,...let me explain.
Eliezer says:
>It's not a matter of math.
Well, this is wrong. Population is numbers and numbers is math.
> If population is determined by reproduction, it keeps expanding and
>expanding with no end in sight.
This is right historically, but conditionally. Historically, human population has been continually expanding. But the condition for expansion is that the average *per person rate* of reproduction be more than 1. Now this has up till now been the case, but lately, in the more affluent countries the *rate* has declined, so the possibility exists that population size could--I'm talking about without coercion here--stabilize or even decline. If someone were to correlate the rate of decrease of the per person rate of reproduction to the rate of increase in wealth we might project that population stabilizes when the world reaches a certain state of global affluence. All of this would be a projection based on *current trends*, subject to all sorts of uncertainties and possible errors. Absent extropian factors then, a conventional, but wealthier world might have a population stabilized at 10 or 15 or 20 billion people.
>If population is determined by carrying capacity, it stops at the
>same number of long-lived humans as short-lived ones.
Um, yes, this is true, but it's a very odd statement. First, it pushes the entire context to a theoretical extreme which (I would say unfairly) radically alters the conditions on which the original contention is based. Also, while it's mathematically true, it represents a situation which, in human terms, seems (to me) unlikely to be reached (and if ever approached, likely to be nasty and unstable). And finally, I would ask, what is the maximum carrying capacity of the planet? I suspect the answer is highly variable depending on lifestyle, technology, and several other things. 100 billion? 200 billion? With energy imported from off the planet, and nanotech to *make* food, and three dimensional use of the surface of the planet--water as well as land--you could fit a lot of folks on this orb. So...
For the purpose of discussion--for the purpose of showing how Margaret Wertheim's contention that doubling the lifespan doubles the peak population is plainly reasonable--let's just say that a doubled lifespan is achieved in 50 years--from max age 100 to max age 200--and that in the ensuing 200 years the human population does not reach the maximum carrying capacity of the planet. After 100 years the entire population will be long-lived folks, and 100 years after that the long-lived folk will start dying off and the situation will have reached a steady-state condition. At the same time I postulate a steady population growth RATE. Either steady at zero, ie. one offspring per person, or steady at some constant exponential rate, say, 1.2 offspring per person.
Under these circumstances we can compare what the populations would have been with and without the 100 to 200 year extension. Now feel free to criticize, but I'm going to use the method I suspect Ms. Wertheim may have used. It's quite straightforward. If people have their children at (average) age twenty, then in a population where people live to be (average) 100 years old, five generations will be alive at any given moment. In a population where people live to be 200 years old, ten generations will be alive at any given moment. In a zero population growth situation--one offspring per person-- this implies that, in the doubled-lifespan population, twice as many people are alive at any given time. I suspect this is what Ms. Wertheimer did.
In the constant exponential rate situation, however, the situation is *worse* than 2:1. The ratio of long-lived to short lived populations depends both on the offspring per person and the average age of child-bearing (which effects the total number of generations alive at any one time in each of the populations), according to the relation:
long-lived population/short-lived population =
P(0)*[r^0+r^1+r^2+...+r^(2n-1)]/P(0)*[r^0+r^1+...+r^(n-1)]
=1 + r^n
where P(0) equals identical starting populations at time zero, r equals the offspring per person, and n and 2n are the number of generations of the short and long-lived populations respectively, alive at one time .
So for example, at an offspring per person rate of 1.2, with 5 and 10 generations alive at one time for the short and long-lived populations respectively, the ratio of long-lived to short-lived populations is 3.49 to one.
Hmmmm. Looks like one child per person for the biologicals, and the rest of us upload and/or go explore the stars. I can live with that.
Best, Jeff Davis
"Everything's hard till you know how to do it." Ray Charles