Mathematics in the large can be considered robust because

• the most important results get checked by sufficiently many people,
• frequently using a wrong result would lead to a contradiction,
• there are enough proofs that lead to the same result

In other words, redundancy and replication is key. In that regard, mathematics is not fundamentally different from empirical research or, say, airplane safety measures.

Whereas, for example, medical research has experimental replication and reproduction to ensure robustness of the results, replication and reproduction takes place by mathematicians who read and understand the papers.

That is why what you find in standard textbooks on central and long-established topics can be considered safe at this point.

However, the situation is different the more specialized and arcane the field becomes. I would not unconditionally trust research

• in fields that only few understand
• that is not widely used and hence insufficiently "tested"
• that has not intellectually reproduced by other researchers

For example, I know of incorrect proofs in my area of research that do not get addressed for various reasons: the results simply don't matter enough to be addressed by anybody, they are too difficult for the small community to actually read and check, or they are ignored because people know it can be fixed. Last but not least, some errors are not addressed because people are scared of the original author. Would my small subfield suddenly become central and important to the larger mathematical community, then the scrutiny would be much harsher.