Kaba wrote: >email@example.com 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.
(3) The writer is a wise, experienced teacher who has a clear understanding of his likely readership and knows which lemmas would be trivial for his readers to prove, and
* While the author needs the statement as a lemma, it's offered to the reader as an easy exercise.
* To set the expected level of difficulty of the text by omitting trivialities (which would be less trivial if the reader was less experienced or less able).
* (repeating from (2)), doesn't want to detract from the main points by watering it down with trivial details.
>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.
Better metrics are:
* The readability and elegance of the writing.
* Intelligent, natural choices for notation and terminology.
* The right motivation for key concepts.
* A well chosen mix exercises, some easy, some medium, some challenging, chosen so that working through most the exercises builds the students' power over the subject and confirms their mastery of the concepts.
* Clear, well-worded proofs, not cluttered with trivial details, and with the key parts said "just right".
>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;
Why not offer it as an in-place exercise for the reader?
>it should take about the same space as stating it trivial.
Right, but Lang expects the readers to participate.
Just like some math teachers who, when teaching lessons, ask (mostly) easy questions as they go along. This serves a dual purpose. Firstly, it reassures the teacher that the students are "with it". Secondly, it gives the students an opportunity to be part of the development. Most of the questions should be easy ones with quick answers, so as not to slow down the lesson too much.
>If it takes more than a few sentences, then it is not trivial.
Unless those sentence flow in a trivial way. But I agree, most of such omitted proofs should be of the kind for which the student should be able to do it mostly in their heads, with at most a few lines of math written on the side.
In your indictment of Lang's Algebra, let's see some examples. Show us statements from that text for which, assuming the student
* has the assumed prerequisites for the text
* has successfully understood (and done exercises to prove it) for the material up to that point in the text
have proofs left to the reader, but which, in your opinion, would actually not be trivial for the average reader, as qualified above.