How much is the use of formal logic useful or even necessary to establish mathematical and scientific theories today?
I understand that computers, i.e. hardware but more importantly software, somehow contain logic and that computers are not only useful but really necessary to mathematicians and scientists. However, that seems to be essentially a practical matter, somewhat like most people don’t and couldn’t keep going to work by foot nowadays.
So, I believe mathematicians and scientists could at least in principle do all their research without computers.
Assuming they did it all by hand, so to speak, would questions of logic, or the logical aspects of their work, be just too much for them to do it relying just on their intuitive sense of logic, and would they have instead, at least in the most complex cases, to go through logical proofs using formal logic? If so, do you have any example of the logic involved?