How does mathematics work?

If I am given a parking lot with ten thousand cars and I want to determine whether one of the cars is orange, the only way I can do this is go through the parking lot examining each car until I find one that is orange or I examine each car and conclude that there are no orange cars. However, if I want to the determine whether there are any nontrivial integer solutions to the equation of Fermat's Last Theorem ( x n + y n = z n, n > 2), I do not have to examine every possible solution, in which there are an infinite number of them. If I am clever I can mathematically prove that there are no such solutions, just as Wiles and Taylor did in the 1990s. So my question is what is it about mathematical problems like FLT which allow a person to bypass the brute force search of the parking lot problem? I am really asking, "How does mathematics work?"