World Popluation

At some point can there be more people alive then have ever lived?
Nope.

I plotted the world population and the total world population versus time (negative values are B.C.), and the cumulative population was always larger.

If we approximate the growth by a simple exponential, \begin{aligned} P(t)=P_0 e^{mt}\end{aligned}, the problem is equivalent to solving:

\begin{aligned} \int_{0}^{t} P_0 e^{m t'}dt'= P_0 e^{m t} \\ P_0 \frac{e^{m t} -1 }{m} = P_0 e^{m t} \\ t = \frac{ln{\frac{1}{1-m}}+2 \pi i n}{m} \end{aligned}
Only at t=0 is time non-imaginary (here n is just an integer), which agrees with the plotted results.