Mathematicians conceived of black holes long before astronomers actually found any?How? I read this statement in Lockhart's - "A Mathematician's Lament". But how could mathematicians  figure out something like black holes even before astronomers noticing any of them?
 A: Without going into the details (which I'm not intimately familiar with), it was possible because physicists and mathematicians (such as Lorentz, Einstein and Minkowski) had developed a good mathematical model of gravitation.  Using this model - which is a system of partial differential equations - physicists and mathematicians were able to predict that gravitational singularities should form under certain conditions.  As a very (very) simplified example of what I mean, suppose that we have modeled some quantity with the simple ODE 
$$
\dot{x}=x^2,\quad x(0)=1
$$  The solution to this equation is the function 
$$
x(t)=\frac{1}{1-t}
$$
This function exhibits finite time blow-up, since when $t=1$, we have a singularity - a place where $x(t)$ becomes infinite.  If the ODE models our system accurately, we should be able to observe this blow-up with measurements.  The same is true of gravitational singularities - the model predicts a blow-up, so astronomers went looking for (and found) those singularities.
