Here's a possible good faith interpretation: Irrational numbers exist as a limiting process, not as a static atomic unit. That is, irrational numbers are a verb, not a noun.

We use irrational numbers as nouns, when convenient, but this is an abuse, in some sense. When we want some digit of sqrt(2), say, we need to interrogate an algorithm to get it. We talk about how much time it takes to extract the amount of precision we want. At best sqrt(2) can be thought of as an abstract symbol that, when we multiply it by itself, is 2. That is, an algebraic manipulation that we can reduce to an integer under certain circumstances, but it doesn't "exist" the same way that an integer or a rational exists.