Well, if you want debate the physicality of numbers, it's pretty clear rationals exist; that's just a matter of definition. The rest, whatever you like.
Irrationals, maybe, but it's not like we can construct nor measure a perfect right triangle. Even if one made a crystallographically perfect shape and counted the atoms along each side, one would find whole numbers, a mere approximation to the irrational ratio. How should you even count the corners? They aren't perfect right angles, there's at least one ionic radius around the edge. Should you project the lines/planes towards their ideal intersections, or should you measure the real distances, taking account of the corner somehow?
One likely cannot even construct a 3-4-5 right triangle, as the hypotenuse is not on a crystal plane -- unless there happens to be a symmetry group with exactly such a plane, in which case merely pick any other of the infinite Pythagorean triples you can choose from and check again.
I'm fond of a little rant about "real" numbers, as there are so very few indeed that can be constructed or computed in any meaningful way -- pi and e happen to be two of the most powerful ones, showing up in a great many places. The whole space of real numbers -- let alone just those that can merely be computed, through any arbitrary (and likely impossibly slow) process -- is truly, unsettlingly vast. It's not just an uncountable infinity, but most -- indeed, "nearly all", well and truly defy
any possible description, by any finite means, in infinite time, in the real universe-as-we-know-it.
For example, we can construct the number which is the decimal concatenation of the maximal number of steps of execution (with halting) of an N-state Turing machine. That is, the "Busy Beaver" function. The first few values of which are known, and which grow incredibly quickly. The currently known bounds on BB(6) are unimaginably large, and no computer can exist in the known universe to compute it directly. The problem in general, is a statement about the decidability problem, a problem which is provably unprovable. Therefore, the exact value of this constant cannot be known. And it doesn't have any handy identities to allow a shortcut; this isn't your average pi we're talking about.
See:
https://math.stackexchange.com/a/462835They give a slightly less "dense" form, using powers of two rather than concatenation. Obviously, there are infinite ways we could encode this function/sequence as a real number, so there's already a small infinity of real numbers constructed with this uncomputable function, in part.
So, I would humbly submit that, you can give or take algebraic (irrational) or computable numbers, but "real" Real numbers, most definitely aren't!
Tim