> the mathematical objects we generally refer to as "numbers" have no constructive basis.
Eh? I’m confused by your terminology here.
You can construct the set of Real numbers using Dedekind cuts. Less conventionally, you can construct the Surreal numbers, and find the Reals as a subset.
I should say computable basis to be precise. Our fundamental object of computation - the number - generally does not have a computable basis in modern mainstream mathematics (where we generally assume the real numbers by default). The standard Dedekind cut construction does not give computable arithmetic because it involves cartesian products of arbitrary (not necessarily enumerable) infinite sets.
I can understand some of that discomfort. However, ultimately, uncountable infinities and (the axiom of) choice leads to other interesting scenarios that, IMO, it's best to make peace with.
Besides, mathematicians rarely study arbitrary, probabilisticly-common, objects. We focus on objects with “nice” algebraic properties. Similarly, it doesn't matter that almost all numbers are uncomputable when we don't use them in practice.
Eh? I’m confused by your terminology here.
You can construct the set of Real numbers using Dedekind cuts. Less conventionally, you can construct the Surreal numbers, and find the Reals as a subset.