AFAICT mathematical logic has always felt like a curious side show in the greater math community. If you were to ask professional mathematicians to list off all the axioms of ZFC they'd probably shrug if they didn't get all of them. They probably wouldn't know about the more exotic parts of set theory such as large cardinals nor would they would probably know too much about alternate foundations of mathematics. They might not be familiar at all with the specifics of model theory.
The logical foundations of mathematics have always had a much smaller impact on the day to day lives of mathematicians than their foundational nature might suggest.
That's because formal logic is a (relatively) recent attempt to formalize the intuitions behind the math that people were already doing. Doing mathematics doesn't really depend on foundational logic in the same way that, e.g., a web app depends on transistor physics.
As somebody once said, if we ever found a contradiction in the ZFC axioms, we wouldn't throw out math, we'd just throw out ZFC.
> Doing mathematics doesn't really depend on foundational logic in the same way that, e.g., a web app depends on transistor physics.
Web apps don't depend on transistor physics at all, though. An important consequence of Turing universality is that computer science is not a subfield of electrical engineering.
Also, mathematics underwent very dramatic transformations during the late 1800s and early 1900s alongside the early development of formal logic and set theory. To say that logicians were merely "formalizing the intuitions behind the math that people were already doing" strikes me as misguided at best. The mathematics that rose to prominence in that era was very different from what preceded it, often controversially so.
>if we ever found a contradiction in the ZFC axioms, we wouldn't throw out math, we'd just throw out ZFC.
It's like saying: if we ever found a contradiction in the base case of a mathematical induction, we wouldn't throw out mathematical induction, we'd just throw out the base case. The indudction step remains sound.
> what fraction of web programmers understand transistor physics?
I'm not an engineer, but I imagine that even most professional electrical/electronic engineers do not understand transistor physics in full details. In engineering, a transistor is essentially treated as a blackbox, and its behavior is described and approximated by various small-signal and large-signal models (i.e. treated as a lumped component), with their parameters characterized empirically by vendors through experiments. How exactly things work at atomic or quantum level is essentially for a transistor to work, yet, a subject of study unrelated to electrical engineering.
On the other hand, I imagine there is no shortage of EEs who have studied a physics major, or EEs with a background of semiconductor physics - they can understand transistor physics really well.
So we can say the relation between EE and transistor physics sounds a lot similar to mathematicians and logicians, it's a good analogy.
Well maybe not quite that extreme. It's more analogous to the fact that most professional programmers do not fully understand the programming language they use (nor honestly do they need to for the most part).
Nonetheless, it's perhaps surprising for a beginning student of pure mathematics when all they can see is the sky-high demand for rigor that "professional" pure mathematics is seemingly content to hand-wave its foundations. (For the most part this hand-waving is the postrigorous stage described by Terence Tao https://terrytao.wordpress.com/career-advice/theres-more-to-... rather than the pre-rigorous hand-waving of the student).
The logical foundations of mathematics have always had a much smaller impact on the day to day lives of mathematicians than their foundational nature might suggest.