Your argument is similar in flavour to Intuitionistic Mathematics:

https://plato.stanford.edu/entries/intuitionism/

"the vast majority of the real numbers are things that nobody will ever be able to write down ,.. they’re less well-defined."

This reflacts the relationship in intuitionistic mathematics between the intuition and the language.

Note the following:

"Brouwer was not alone in his doubts concerning certain classical forms of reasoning. This is particularly visible in descriptive set theory, which emerged as a reaction to the highly nonconstructive notions occurring in Cantorian set theory."