I understand the typical answer involves advancements in technology/farming, but I've always wondered this from a more mathematical standpoint. I asked my history professor this once, but he didn't really seem to understand my question.
We have reason to believe that population growth should follow an exponential model (because the rate of growth is proportional to how many people there are). Comparing a typical population graph to a typical exponential graph this hypothesis is supported.
However, if you zoom in on "flat" part of the exponential graph, after scaling your axes, you get something that looks familiar: it looks like the recent years of our population graph. In fact, with a little bit of math, it's clear that after scaling your axes, an exponential graph will look like it has this sharp spike in recent times whereas before that it was flat. So, how do we know that the growth in recent years was due to technological advancements and not just because a population growth rate is dependent on the population itself?
I think this is more of a math misunderstanding than a historical question. It is easy to see the rate of growth is higher by looking at the graphs on a logarithmic scale, in which case an exponential function is a straight line while the population growth is curved.
This is something generally understood in terms of the demographic transition model. Initially, both birth and death rates are high, but as societies change, death rates fall, followed by a delayed decline in birth rates. It just happens that much of the world is in the early stages of the transition, so birth rates are high but death rates are low. This explains why the growth right now is faster than exponential.
To interpret the previous plot (and piggybacking on /u/aka_someone's comment), consider what would happen if the population growth rate r were constant. Then the rate at which the population size N changes would depend only on its current size, multiplied by a constant:
dN/dt = rN.
A little bit of calculus yields the time dependence of N, or N(t):
N(t) = N(0)e^rt
where N(0) is the population size at time zero. Taking the (natural) log of both sides yields
log N(t) = log N(0) + rt,
which has the form y = b + mx, i.e., a straight line.
So the size of a population growing at a constant rate should, on a semilog plot (i.e., log-scaled y-axis, linear x-axis), look like a straight line, and yet it doesn't.
