[redvenom]

There are two separate statements here:

1. Mathematicians often use relatively powerful systems, like reasoning in ZFC about Peano arithmetic. The consistency and completeness of axiomatic systems is what this article is about

2. Mathematicians pretty much exclusively use informal systems (compared to formal logic) for reasoning. The reason why that is is because it is just infeasible to reason typical mathematics using formal logic for most cases.

However, it's important to realize that these two are orthogonal.