Note that "computable" just means "computable by a Turing machine or equivalent system", at this time. We don't know if we are missing some other form of computation that goes beyond the abilities of a Turing machine, that could upend the whole concept. And given that we haven't proven that general human thinking is a computable process, there is at least one significant candidate for possible computation beyond Turing machines.
Note that I don't personally think it's likely that our thinking goes beyond the capabilities of a Turing machine, not at all. But I also think it would be premature to throw out all mathematics that doesn't conform to Turing machine computability before we are more sure that this is the best possible model.
And even if it were, there is also the question of whether physical processes are computable or not. Right now there are plenty of physical processes where our best and only reliable models for predicting their behavior require assumptions from calculus, like the existence of all real numbers. For example, there is no successful formulation of quantum mechanics where the distance between two particles moving relative to each other can be constrained to any subset of the [minDistance, maxDistance] interval of the real number line. Which means that, as the particles move away from each other, at some times the distance between them will have to be an uncomputable number (and given their density in the real number line, this will be approximately all the time).
Note that I don't personally think it's likely that our thinking goes beyond the capabilities of a Turing machine, not at all. But I also think it would be premature to throw out all mathematics that doesn't conform to Turing machine computability before we are more sure that this is the best possible model.
And even if it were, there is also the question of whether physical processes are computable or not. Right now there are plenty of physical processes where our best and only reliable models for predicting their behavior require assumptions from calculus, like the existence of all real numbers. For example, there is no successful formulation of quantum mechanics where the distance between two particles moving relative to each other can be constrained to any subset of the [minDistance, maxDistance] interval of the real number line. Which means that, as the particles move away from each other, at some times the distance between them will have to be an uncomputable number (and given their density in the real number line, this will be approximately all the time).