> I have to disagree. A trained mathematician understands why a proof is or it not valid, based on a combination of axioms and a logical sequence predicated on axioms, but with no gaps or overlooked assumptions.
First off, to dispel any misunderstanding, I was talking about the process of becoming a mathematician, which you necessarily go through before you can call yourself one.
But even for full-fledged mathematicians, what I'm saying is true to an extent. For instance, although ZFC set theory is usually accepted as the basis for all mathematics, most mathematicians (precisely: those who do not study formal logics or other areas of metamathematics) do not state their results in terms of set theory, but instead write informal proofs that appeal to other established results in their field of study.
That is absolutely not the same as saying that their thinking is fuzzy or sloppy. The fact that I personally do not state (or, even, understand) all the details behind an established theorem X has no bearing on the validity of my proof for a theorem Y that hinges on X being true.
> The Incompleteness Theorems show the degree to which mathematicians expect to know why something is true, and if they cannot, why not.
This is a separate matter. Whether in theory it is possible for you to ascertain whether X is true or not, and whether you fully understand (down to first principles) why X is true before you use it as a stepping stone for other results are different issues.
