A flag is a type of global variable raised as a side-effect of exceptional floating-point operations. Also it can be sensed, saved, restored and lowered by a program. When raised it may, in some systems, serve an extra-linguistic diagnostic function by pointing to the first or last operation that raised it.
Any modern programmer knows that side effects are, while inevitable, hard to tame and some discipline is needed. Pure functional programming, mutable XOR shared, software transactional memory, you name it. This part of talk completely handwaves a difficulty of side effects and forces every language to be handcuffed with those global states and side effects. No good.
[1] Joe Darcy, one of the students of William Kahan, later went to Sun to improve fp support in Java; see https://web.archive.org/web/20090402234711/http://blogs.sun.... and https://web.archive.org/web/20100224062224/http://blogs.sun.... for example. But his work was more about library supports AFAIK.