A Math Problem Posed by a Reason Commenter

In the course of a discussion of a Ronal Bailey column on overpopulation hysteria, D Kingsbury (that's the author best known for a novel about eugenicist cannibals trying to breed professional politicians) wrote:

"We are headed for the stars, whether you're ready or not," says Mr Lol. Ha. ha. Lol, you are obviously not a mathematician and probably cannot even add. Learn something about exponential functions. If the human race started to expand TODAY at the velocity of light, and was able to convert ALL mass in our path to serve the human need for bodies, and at the present rate of population increase, we'd run out of mass in less time than it took to get where we are now from the time of the pharaohs. Lol, RIGHT NOW you are in the middle of a violent population explosion and you just don't see it because you are such a transient mayfly. With your limited command of reality you couldn't possibly see as far as a future where man might be involved in interstellar exploration!

In response to that, I wrote:

Okay. If we're expanding close to the speed of light, time dilation will slow the rate of growth.

In turn, D Kingsbury replied:

Hertz, you don't understand time dilation. In the first place, time dilation requires an INCREASE in mass. People may be screwing at a slower rate but they are multiple-thousands of times heavier and multiple-thousands of times more resource (mass) hungry. AND they have to STOP to colonize. Nor do you understand volume to surface ratios -- the bigger you are (volume) the smaller is the "surface to volume ratio." Jeez, take a math course. Sure we can go into space, maybe even interstellar space -- but you can only do that from a stable population base. Note that I said "stable" not "constant." If you look at stability as a mathematical concept, what we have today CANNOT be defined as stable. Don't worry, nature's constraints will FORCE a stable configuration whether it is 9 billion or Trantor's 30 billion or whatever You may not like that world when it comes. If you are a teenager now, in forty years you may be one of the desperately poor multitudes -- unless you're one of the few who will have figured out how to exploit your fellow man and are rich enough to hire guards and own an armored car. Even then you might end up in a spider hole like Saddam after a few years of living off your fellow man. If you seriously think we humans can avoid nature's constraints you have tipped over into insanity. Run as fast as you want, the constraints will catch up. The boogy-man lives under your bed.

Oh boy, it's a calculus problem! (Calling it differential equations might have been a bit pretentious.)

Let's set up the equations according to the model. The rate of increase in total human biomass M is Ṁ = αT_dM, in which α is the growth rate and T_d is the time-dilation factor. Since we are assuming “the human race started to expand TODAY at the velocity of light, and was able to convert ALL mass in our path to serve the human need for bodies” and if we further assume that the human need also includes transportation needs, we must recall that the time dilation T_d = Ec²/M, where E is the total energy available and c is the speed of light. The total energy E = Dc²(ct³), where D is the density of the universe and t is the time. Putting it together, we get: Ṁ = αM²/D(ct³). This can be expressed as: dM/dt = αM²/D(ct³) or d(1/M) = (α/2Dc³)d(1/t²). When we integrate both sides, we get: 1/M = (α/2Dc³)(1/t²) + Constant. To sum up: M = 1/[(α/2Dc³)(1/t²) + Constant].

Long breath now …

The long-term behavior of M depends on whether the constant is positive or negative. If M is small enough, the constant will be positive, which means time dilation will ensure that population levels off. If it is large enough for the constant to be negative, the population will reach ∞ in a finite time. It looks like Kingsbury assumes the constant was negative.

In order to judge whether the constant is positive or negative, we must compare M (the human biomass) with 2Dc³t²/α. If we want to know what t is, we should consider the total mass available to humanity at the time of the singularity, M_h and set it equal to Dc³t³. When we put everything together, we get: D^1/3M_h^2/3c/α. The density of the universe (according to some astronomers) is 5×10^-27 kg/m³. We can assume that M_h is the mass of the Earth: 6×10²⁴ kg. c is 3×10⁸ m/s. The fastest human population growth rate, according to Malthus, is 6%/year, which is α = 2×10^-9 s^-1. This amounts to 8×10²⁴ kg. The current human biomass at 7×10⁹ people and 100 kg per person (a round figure) is 7×10¹¹. In other words, the population is small enough that Kingsbury's scenario is nonsense.

It is, of course, possible to move the goalposts and point out that the above calculations are oversimplified to “the limits of the preposterous and beyond” (as Poul Anderson and Gordon Dickson would have said), but there's no reason to believe that a more thorough analysis would be any more supportive of Malthusian theories.

One last trivial little point: D Kingsbury said in the quotes above: “Jeez, take a math course.” I think I have.