Well, it is certainly true that introducing unambiguous, very carefully defined, agreed terminology and having a perspicuous notation can make reasoning easier and make us less prone to common reasoning errors. To take the obvious example, mathematicians aren't just being awkward when they use a lot of symbolism and make very careful distinctions wrapped up into technical terms (and borrow from the languages of formal logic to make clear, for example, the 'scope' of their quantifiers). If proofs all had to be written out in unaugmented English, then we'd get lost following them, even in elementary high school algebra: and proof-discovery would be orders of difficulty harder.

I suppose we might say "mathematicians' English" -- meaning English augmented with their new definitions and notational devices -- is a new, better, language, more suited to (mathematical) reasoning than street English. But equally, we might say that it is just one part of a single inclusive language, modern English: it is just a part that is only learnt by those with certain specialist interests. But for present purposes I can't see that it really matters which of those descriptions you prefer. The key point remains that, yes, appropriate linguistic devices, e.g. sharply defined terms and a perspicuous symbolic notation, certainly can expedite reasoning and help us avoid error.