the paper lays this out: it’s just really weird that essentially all real numbers cannot be named, described, or known to us. (or computed by any Turing machine, if you want to avoid paradoxes of natural language.) if you randomly sample a real number, with 100% probability it will be one of these strange, unknowable numbers.
this is quite weird! i’ve only ever seen nameable, computable numbers in my whole life, yet apparently drawing one of these from a uniform random sample has probability 0?
fortunately IIRC the subset of computable real numbers still forms a field and behaves how we want, although you can only test equality up to some epsilon.
The negative numbers and the number zero have similar problems with being fully manifested in our physical experience. Numbers are a logical tool and what the full range of Real Numbers gives us is the logical foundation for "smoothness".
Also, of the discrete Natural numbers, you are unlikely to witness in your entire lifetime any number larger than 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10, i.e. virtually all of the numbers are inaccessible to you except for in your imagination.
The real numbers are perfectly accessable in our imaginations. They just have different properties than some other numbers and are used for different things. It would be similarly absurd to say that we can't "access" negative numbers in our heads. Of course we can. We can construct the Real numbers and consider them in their entirety. In fact, we have a conception of the Reals as we use them in other mathematics.
this is quite weird! i’ve only ever seen nameable, computable numbers in my whole life, yet apparently drawing one of these from a uniform random sample has probability 0?
fortunately IIRC the subset of computable real numbers still forms a field and behaves how we want, although you can only test equality up to some epsilon.