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Are these sentences semantically correct?
I don't see anything through these binoculars.
I see nothing through these binoculars.
A teacher says the second one is wrong because you can't say "see nothing with something" when you use a tool because nothing refers to "non-existence" It doesn't really make much sense to me. Don't native speakers normally use it that way?
Your teacher is being hypercorrect. Native speakers regularly use nothing to mean 'no thing' or 'not anything;' this is the OED sense 1:
Not any (material or immaterial) thing; nought.
For example, unless we're playing Socratic word tricks, when we say "I know nothing," we mean "I do not know anything," not "I know non-existence."
Both sentences you give are correct and very common, and both can mean either "I literally cannot see through these binoculars" or "I don't see anything of interest through these binoculars."
