To quote Humpty Dumpty:

    "When I use a word, it means just what
     I choose it to mean — neither more nor
     less."

If the author chooses to define proof in a different way, and chooses not to use "proof" in the same sense that mathematicians use the term, and then goes on to tell them they're not producing proofs at all, is he surprised that they get annoyed at him?

As always, XKCD gets there:

    Communicating badly then acting smug when
    you're misunderstood is not cleverness.

-- http://xkcd.com/169/