Not quite. Mathematicians realised that "counting" is just defining relations between sets of things and sets of the form {1,2,3,...}, in such a way that every thing gets assigned 1 and only 1 number.

Usually, the relation is defined by pointing to each of the things we want to count and saying the number we're assigning to it. However, using this whimsical definition of counting allows us to define the relation in equally whimsical ways and "count" stuff we normally wouldn't be able to, however impractical (or even infinite). For example, we know we can "count" the natural numbers, even though they're infinite, because we obviously can assign a number to each one of them, namely themselves. But did you know we can also "count" the rational numbers? The thing about the reals, though, is that not only we haven't been able to find this relation, but we actually proved that it's impossible to find. The proof isn't actually hard to follow so I recommend you check it out.