> True, but with such numbers you will normally not do anything else except computing an approximate value of them.

That's what I think people do with other numbers like "pi" at the end of the day, no? :)

> That is what I meant by "interesting", i.e. the necessity of using symbols of such numbers, obviously for use in symbolic computations, since in numeric computations you would never use the actual numbers, but only some approximations of them.

It's very much an encoding problem, I think. Though we probably, on aggregate, use "unnamed computable numbers" implicitly on the order of as much as we use "named computable numbers" the former just has way more of a "tail" of uses where the "encoding of the symbol" is, e.g., "here's the PDE you use to compute this number"!

(It gets a little weird since we're kind of not distinguishing between the approximation that can be used to construct said numbers to arbitrary precision vs the specific program instance that constructs one specific approximation, but the idea is mostly there.)