The more I correct copies, the more I realize something I intuitively knew for a while (for observing other fellow students last year) but that I now have the confirmation of: the way we teach undergraduate students (and in particular first year students) to write mathematical proofs is very bad. This, because we actually don't teach it to them.
There are two ways to write mathematical proofs: the correct way, which is the way they are written in most books and journal articles, and the bad way, which is the way they are written on the board during lectures (or seminars).
During a lecture (or a seminar), the lecturer is running out of time (pretty much from the beginning of the hour), so he will shorten the presentation (a little bit, if not a lot). He will use symbols instead of sentence fragments, for instance using "=>" instead of "implies", if not missing entire complete sentences (mentally hoping, or knowing, that the audience will "fill the blanks"), but moreover will say out loud many details that do not go onto the board but that are actually part of the complete proof (details that never make it to the students written notes).
While being on this subject I really dislike lecturers habit to mix English sentence fragments and expressions using quantifiers, for instance using the (TeX) symbols "\forall" or "\exists" in the middle of a sentence that was meant to be a proper English sentence. It's not that it really bothers me (I do that as well --but in general really try and avoid that--, because I am all too aware of how much more convenient it is to use them instead of the complete English sentence fragments, above all when it's already quarter-to and you realize that you haven't done half of what you wanted to do during the hour), but this style should only be used with audiences that have already been properly introduced to the art of the correct writing of mathematical expressions, so that they know that the shorthands used on the board are just that, shorthands. Instead, we maintain a sort of insane misunderstanding in our students mind about what exactly constitutes a mathematical expression.
For instance, here are fragments from an hypothetical proof
- (...) and therefore we know that P(x) is true for all x. Now, let \epsilon>0, we can write that ... [correct]
- (...) and therefore we know that P(x) is true for all x. Then, this implies that the expression (\forall x \in [0,1]) P(x) is true, ... [correct]
- (...) and therefore we know that P(x) is true for all x. Then, \forall x, we see that ... [incorrect!], because "\forall x" is neither English nor a mathematical expression, the correct form is "(...) and therefore we know that P(x) is true for all x. Then, for all x, we see that ..."
By making the rules of mathematical writing clear to the students (from the very start), they will stop mystifying the proofs that are given to them. I can clearly see that they "try and imitate" the form that is showed to them during lectures, while unfortunately not really understanding (maybe because nobody told them) that mathematical writing/thinking is nothing else than (regular) writing/thinking, but about mathematical objects (albeit more precise and rigorous thinking than what we can usually get away with when reasoning about mobile phones, food or Paris Hilton). We should highlight to them the fact that if they would not be satisfied with what they have written (logical flow and style) under replacing mathematical objects of the text by real life objects, then they are doing it wrong. They think that mathematical writing (or even mathematical thinking) is something new and unique, which has no other equivalent in real life. This is very sad.
Along those lines it feels absolutely wrong to me that first year university students in mathematics do not have a course called "Writing mathematics" or "How to write mathematics", even Computer Science students, under Donald Knuth, had something like that, and a (draft) copy to the actual course can be found here (I blogged about this document in 2010]. Otherwise, I cannot overstate how much I like Queen's Mary introduction course: Mathematical Writing for undergraduate students. Not offering this kind of teaching to first year students in mathematics is like teaching Chess's Theory of Openings to people without having taken time to first explain the rules of the game to them. It's the most important (!) thing we can teach them during their first year. They might not know a lot of mathematics when they start, but at least they will be able write the little they know correctly.
... avoiding me to look at some copies and wonder "WTF was he/she trying to say here, this doesn't make *any* sense". Or more likely I will be wondering: "Why do they seem to think that English sentences are not welcome in a mathematical proof ? Oh, yes, I know! For having been brainwashed in high school into thinking that mathematics is about calculations". Not all of my first year students are like that though, some of them are really perfect, both in writing and reasoning, and I am very happy when I write "Absolutely Perfect!" in red on their copies.
The students are given lectures notes, but they take them as stuff to look at if one misses one lecture, not as what the lectures was really meant to be in the first place and what they should be learning from (not only mathematically, but also in terms of style and rigor). When I was undergraduate, I rarely showed up to lectures as soon as the lecture notes where made available.
The resulting effect is that students, who often during their first university years are too intimidated by books to learn from them, are under the false impression that proof writing is what they have seen on the black board all those years. In fact, and this is what really bothers me, they will graduate without having ever seen or never been told how to *properly* write a proof. And no, I am not overreacting, most Masters students I saw last year, cannot correctly write a simple proof. Actually, one of them told me, after having seeing copies of my assignments (which from the point of view of the lecturers where perfectly written), that my writing was too verbose, and that I should use shorthands, like in the lectures, that I should not use complete sentences, because (I quote) "mathematics is about going to the solutions as fast as possible". Do I need to mention that this girl failed her year and is redoing it now...
A mathematical proof is a document (piece of text) that one write to establish the fact that a mathematical statement is true (within a given axiomatic system). This is fine, but then how do you actually write them ? Some advice:
- As far as I am concerned, a mathematical proof is a piece of text in a given natural language (English, for instance), and should be written with complete sentences. With a starting capital letter and an ending point.
- The sentence must be grammatically correct.
- One is invited to split the text into paragraphs, each highlighting a sub-argument of the complete logical flow that makes the entire proof.
- Mathematical expressions, formulas (equalities, inequalities etc.) or quantifiers statements can be written either inline or isolated in their own lines (centered), but the meaning must be fluid.
The rule that I follow is this. Mathematical proofs are like weblog entries (or emails), but rather than talking about the news or other everyday life things or activities, they talk about mathematical objects (with aid of the special notations we need to describe them). In both cases the rules of prose, style and clarity are the same. (This is why I often say to people: you want to learn how to write correct proofs ?, start by learning how to write correct emails.)
Terry Tao's weblog, is a very perfect example of how to write mathematics. He uses a style which is perfectly clear, grammatically and stylistically correct and rigorous, but also (cherry on the cake) balanced and engaging.