Ultimate population limits

Given the recent discussion on population that the release of Bryan Caplan’s Selfish Reason to Have Kids has triggered (my posts here, here and here), I have been contemplating some physical limits of what the ultimate human population could be. The current global population is around 6.8 billion and is growing at around 1.2 per cent per year. Let us suppose that the population growth rate persists (as I argued before, there is a case to argue that it might increase). So, how long does it take before the physical limits are reached? (I am sure other people have done this before, but as with most things, I always like to do these exercises myself)

Let’s start with an outer limit. Suppose that each person can be uploaded onto a virtual machine and each person only requires one atom to exist. There are about 10^80 atoms in the universe (most of which are hydrogen). At the current growth rate, it would take less than 14,000 years for there to be more humans than there are atoms.

However, there are a few constraints. The diameter of the observable universe is around 93 billion light years, so assuming we are roughly in the centre, it would take 46 to 47 billion years to reach all points if humans could travel at the speed of light. Yet, there are only 14,000 years before the population needs to access all the matter in the universe. It would be a fair assumption that most of the matter in the universe would be outside the potential travel distance (unless someone wants to argue that we will figure out how to travel faster than light or skip around the universe in some other way). Humans will not even be able to access all of the matter in our galaxy, which is around 100,000 light years in diameter.

How about a closer limit - the number of atoms in the solar system. I could not find a figure for this, but as the sun has more than 99 per cent of the solar system’s mass, I will use the comparative figure for the sun - around 10^57 atoms. Using our one atom per person calculation from above, there would be more people than atoms in a touch over 9,000 years. That’s still a fair bit of time, but of course, does rely on use utilising every atom in the solar system. It is perfectly plausible that humans will have travelled across the breadth of the solar system by that time. Limiting us to earth, there are around 10^50 atoms, which allows around 7,800 years. Adding Mars gets humans less than 10 extra years.

What if we don’t move to this virtual utopia, still need (or want) to be in human form and are limited to earth or the solid planets in the solar system? Again allowing every piece of matter to be used in human form (no need for land to stand on, air to breathe, a biosphere etc), giving each human an average weight of 70 kilograms and with the earth’s weight of 6*10^24 kilograms, current population growth rates can continue for around 2,500 years before hitting the physical limits. Again, Mars allows less than 10 years further population growth.

Breaking the example down further, what if humans are limited to the surface of earth or, again, the solid planets in the solar system? The earth has a surface area of 510 million square kilometres (or 510 trillion square metres), including both ocean and land area. If we assume that each human needs one square metre to provide all of their sustenance and needs, the population scenario hits the wall more quickly - in around 950 years. Mars allows around 20 extra years population growth. If we only need a square centimetre each, the population can grow for a further few hundred years.

Having said all the above, I think it is fair to say that within 10,000 years, which is less than the time since the first use of agriculture, humans will run up against hard physical population constraints, or population growth will be voluntarily constrained. Even allowing a virtual world where each human only needs an atom and there is no need for physical bodies or a biosphere, sooner or later population growth must stop.