Isn't it? I saw where you were going with the astronomy/telescopes analogy, but you lose me on the numbers bit.

[* are == need only be. The mathematician is allowed to be a realist about numbers and think that there are the real numbers etc., and that these are what she is reasoning about.]

But in my view, math is about getting the right numbers, and you don't fully understand a problem until you can get the right numbers out. Here are two examples:

1) Theorem: det A != 0 implies Ax=b has a solution for a square matrix A.

If you actually try this, you'll discover all sorts of matrices A for which the textbook method fails. Thinking carefully about how to find x in the presence of rounding errors leads you to discover condition numbers, numerical range and all that other interesting stuff.

2) Theorem (circa 1870): Fourier series work. (In particular, they converge pointwise.)

Gibbs: I tried, it didn't work for discontinuous functions. I made some graphs, they are terrible. WTF!

Eventually, people paid attention to Gibbs and discovered Gibbs ringing. Trying to figure this out led us to learn about uniform convergence, Hilbert spaces and all that.

Of course, I'm a numerical analyst, so I might be a little biased.