Many of today’s mathematicians use computers to test cases that are ei

When you see a verb-ing modifier after a clause, the most important thing is to figure out whether that modifier logically describes the preceding clause. There's a pervasive myth that the verb-ing modifier should logically describe the previous subject, and this kind of narrow analysis can get you into a little bit of trouble.

Take a silly example. "The neighbor's dog bit Natalie's pinkie toe, prompting her husband to call an ambulance." This is perfectly fine. (Although let's all agree that Natalie's husband is overreacting here, and this dog has weirdly precise aim. ) Did the neighbor's dog prompt Natalie's husband to call an ambulance? Not exactly. This isn't Lassie magically telling someone to call for help. But it is 100% true that the fact that the dog bit Natalie -- in other words, the dog's act of biting Natalie -- prompted her husband to make the phone call. In other words, "prompting" logically modifies the preceding clause, even though it doesn't logically modify that clause's subject in isolation.

The same thing is true of the question in this thread. "Mathematicians" is the subject of the first clause. Do mathematicians allow the exploration of difficult theoretical issues? Not exactly. But does the fact that "mathematicians use computers to test tough cases" allow the exploration of difficult theoretical issues? You bet it does. So the "verb-ing" usage is just fine here, even though it doesn't modify ONLY the subject.

I hope that helps!