We don’t even get to “the reals” before this issue crops up though. Irrational numbers were rigorously studied before infinity and the reals, as far back as 500

We don’t even get to “the reals” before this issue crops up though. Irrational numbers were rigorously studied before infinity and the reals, as far back as 500 BC. I’m not a computer scientist so I may have gap here, but demonstrating two quantities are incommensurable (showing no unit makes up two quantities m and n times, m and n being integers), does not seem like something possible to approximate empirically or computationally in many cases. The precision required may be one step beyond your current capacity. Constructivism may be right. But I don’t have a good argument for why finished computation or empirical approximation (there’s always limits to measurement) is the be all end all. Unless we take them to be the final adjudicators, why shouldn’t there be incommensurable quantities? We need very strong arguments they provide the final say, but we know they have limits, their capacity/memory.