There is actually a scientific way to calculate how many civilizations are out there. It's called the Drake Equation. We need to look at how many stars there are in the galaxy, how many habitable planets there are, etc. The problem, of course, is that we really don't know what numbers to use to solve it... yet.
You can play with the equation itself at BBC Future.
You may ask yourself, then, could we ever see other alien civilizations? The answer is, unfortunately, no. Or at least, it would be very, very unlikely. The universe is simply too large.
How large? Well, light, which is the fastest thing that exists, needs to travel more than 5 hours just to get to Pluto. It takes only 8 minutes to reach Earth. (Again, you can check out another great visualization by Josh Worth to see just how large, and, well, empty, the Solar System really is).
And that's just the Solar System. If you wanted to travel to the nearest star, Proxima Centauri, it would take you 4.22 years at the speed of light. It's 39,900,000,000,000 kilometres away!