Yeah, this was a VERY controversial subject for a long time. Maybe even still today!

David Foster Wallace's Everything and More was an enjoyable history of the topic.

I think the rebuttal to the author's "write all the primes down, you can't" would be something like "tell me how many numbers there are between 0 and 1." I'd die before I'd finish writing out all the primes, but the author or that professor referenced in the article could just never even have a way to give me an answer to my request. (EDIT: If they say that's not "real-world" enough, how many possible pieces of wood of distinct length could they cut that would be between 0 and 1 meters long? Unless someone wanted to try to brute-force it based on like molecule size... maybe it's not truly infinite? But could you even measure that difference? Which brings you back to infinity from a precision aspect...)

I think this depends on if you're a constructivist, no?