17.3.2013 14:25, firstname.lastname@example.org wrote: > Maths texts and lectures often refer to observations as being "easy to check", > "trivial" or "obvious."
I find two extreme situations for why someone uses such weasel words.
1) The writer is an expert, and is bored of going around the same argument for himself for the thousandth time. The claim is probably correct.
2) The writer is a novice, and does not have the energy to go into details which detract him from the main point he is trying to make. There is a high risk of the claim being incorrect, or of that the claim is correct, but has a tedious proof.
I'll concentrate on the type 1 writers; the type 2 writers hopefully improve on their writing as time passes.
Speaking of books in particular, whose main purpose is to teach, one quality metric for me is to count the density of weasel words in the text. An unfortunate example is Lang's Algebra, where everything is obvious, easy and trivial. This is almost always contradictory. If it really is trivial, then why not write it down; it should take about the same space as stating it trivial. If it takes more than a few sentences, then it is not trivial. I find the advice in Strunk & White (Elements of Style) relevant: "Do no inject opinion."
A contrasting example is to take any book from John Lee (Introduction to Topological Manifolds, Introduction to Smooth Manifolds, Riemannian Manifolds). These are masterpieces to learn from. No weasel-words, precise, and minimum amount of errors of any kind. It shows that the author is interested on transmitting knowledge as efficiently as possible, and also knows how to do that.
To me, the use of weasel words reflect a lack of effort; that the writer isn't interested on giving the reader the best learning experience he can. They make a book confusing to read, and indeed, I have sometimes missed important points this way. You can afford to be careless when writing to experts, but not when you are writing to students (readers of the book).
In my opinion, weasel words do not undermine the readers confidence. To the contrary: they contaminate the reader with a false sense of security, opinions of what is easy and what is not. What actually happens to me is that, if the claim is not immediately obvious, I skip checking that claim to get back to the flow of the text.
Related, there is this effect which I call the Stockholm Syndrome for Mathematicians :) This happens when people read a book which leave large gaps in their proofs, and force the reader to fill them. From the helplessness of the start of not understanding, because information is missing, the reader works through the proofs, and increasingly builds confidence in himself. After having mastered the book this way, his emotions have gone through a rollercoaster of frustration to a feeling of control. And suddenly those positive feelings are projected to the book; since I know this well, the book must be great. But it's not the book; it's the massive work that was done to recover the details and essential techniques. I hope future writers avoid writing their books this way; it's abuse in disguise.