I disagree that you have done arithmetic here. You have constructed an expression that describes arithmetic. Would you be satisfied if you entered 3 + 8 into a

I disagree that you have done arithmetic here. You have constructed an expression that describes arithmetic. Would you be satisfied if you entered 3 + 8 into a calculator it would simply give you back 3 + 8? Would you say this calculator has performed arithmetic? No. You described to it the arithmetic it must perform, and it failed to do so. In mathematics we almost exclusively work with the descriptions (expressions), using the rules for simplifying and manipulating those descriptions (algebra). But the foundation underlying the algebra is (currently) the reals, which contains uncomputable objects. Ultimately we want every description of a number to map to an actual number. Why are we allowing descriptions that are not computable, and then call the result a number anyway?