How important is the concept of uncomputability in mathematics?

I apologize if this question sounds too philosophical. I am reading about Turing's paper on computability and it got me thinking:

Why do we bother defining the mysterious "undefined things" in-between on the real number line? How important are they?

I agree that it is ok to define anything and work with any structures. But undoubtedly, some structures are more interesting than others.

Does distinguishing between computable and uncomputable real numbers generate any important results in mathematics?

(Edit: Thank you for the answers so far, I understand that we do not have to distinguish computability in the set of real numbers. I am curious what happens if we do (in any way). Has anyone investigated?)

• We defined real numbers before computers or the theory of computability even existed. We didn't "make up" uncomputable numbers, we already had the reals, and we made a new interesting set of computable numbers, and noticed that the two are not the same. So the numbers missing from the second set get called uncomputable.
– 5xum
Sep 28, 2018 at 13:35
• Seems related: math.stackexchange.com/questions/948171/… Sep 28, 2018 at 13:35

The "uncomputable" comes in computability theory often in the form of undecidability. A subset $$A$$ of $${\Bbb N}_0^k$$ is decidable iff its characteristic function is recursive (computable and total). One prominent example is the halting problem. As far as I known it has no real impact on programming these days. But in terms of the theory of computation the situation gets even more frustrating since by the theorem of Rice, we can say that each nontrivial program property is undecidable.