Someone who says that is, ironically, not concerned with the details. All of published math is wrong? No, a lot of is is understandable and verifiable by advanced undergraduates.

Most of the fundamental theorems in each field are understandable by humans. As you get more and more abstract, and get into more and more obscure areas, you start to see non-understandable proofs built on top of non-understandable proofs. There's a danger that those are wrong, but when they collapse it just means that one professor has wasted their life, not that all of mathematics will suddenly collapse.