John M. Lee is a famous mathematician, who bears the reputation of writing the classical book "Introduction to Smooth Manifolds". In his article, "Some Remarks on Writing Mathematical Proofs", he gives us concrete and complete suggestions about how to write mathematical proofs in a concise and unambiguous way. In my opinion, most of them are quite pertinent and enlightening. In the following, I'll list some key points from this article and some comments are also appended.
- Identify your audience: know your audience is and what they already know.
Comment: this is very important! For audience at the beginner level, explanations should include minute details, which are similar to the annotations in a traditional Chinese book and usually their text length largely exceeds the original text. Don't be afraid that the explanations may look naive and trivial in the eyes of expert. Actual examples are also recommended to be provided. Vivid illustrations are quite helpful for the ease of understanding. For professional mathematicians, the writing should presented in a rather formal and abstract way for the purpose of clarity and brevity. We should use well-defined and unambiguous mathematical symbols to describe facts by starting from definitions, then lemmas, theorems etc. and gradually unfolding the complete logical network.
- Write in paragraph form
- Avoid most abbreviations like "s.t.", "w.r.t", etc. However, "e.g." and "i.e." are still acceptable.
They are suitable for handwritten in a notebook or on a blackboard, but not suitable for a formal mathematical writing.
- State what you're proving, i.e. restate the theorem to be proved in a formal way before the proof starts.
- Label your theorems by using the following keywords (generally speaking, they all mean the same thing, that is a mathematical statement to be proved from assumptions and previously proved results):
- Theorem: an important proposition.
- Proposition: a result that is interesting in its own right, but not as important as a theorem.
- Lemma: a result that might not be interesting in itself, but is useful for proving another theorem.
- Corollary: a result that follows easily from some theorem, usually the immediately preceding one.
For handwritten mathematics, underline these keywords with an emphasizing effect.
- Show where your proofs begin and end
The proofs start with Proof and end with (Box). In (LaTeX), this is done automatically by various predefined mathematical environments. Handwritten mathematics should also follow the same convention.
- Why is it true?
Every mathematical statement in a proof must be justified in one or more of the following six ways:
- by an axiom;
- by a previously proved theorem;
- by a definition;
- by a hypothesis (including an inductive hypothesis or an assumption for the sake of contradiction);
- by a previous step in the current proof;
- by the rules or logic.
Comment: we can see the logical rigorousness in mathematical proofs.
- Include more than just the logic
Mathematical proofs are not simply stacking formulas. The formulas should be concatenated by meaningful and logical descriptions.
- Include the right amount of detail
Knowing the audience is a precondition.
- Writing mathematical formulas
- Every mathematical symbol or formula should have a definite grammatical function as part of a sentence. Therefore, if they end a sentence or a clause, a punctuation mark must be followed.
Read aloud each sentence is a good way to check.
- Do not begin a sentence in a paragraph with a mathematical symbol. For example, "(l) and (m) are parallel lines" is not good, write "The lines (l) and (m) are parallel" instead.
- Avoid writing two in-line formulas separated only by a comma or other punctuation mark. For example, "If (x eq 0), (x^2>0)" is not good, write "If (x eq 0), then (x^2>0) instead.
- Do not connect words with symbols. For example, "If (x) is an function (in X)" is not good.
- Symbols for logical terms like (exists) (there exists), (forall) (for all), (wedge) (and), (vee) (or), (
eg) (not), (Rightarrow) (implies), (Leftrightarrow) (if and only if), (
i) (such that), (ecause) (because) and ( herefore) (therefore) should only be used in handwritten mathematics. In formal mathematical writing, they should be replaced with English words.
Exception: (Rightarrow) and (Leftrightarrow) can be used to connect complete symbolic statements. For example:
We will prove that ((a) Leftrightarrow (b)).
- Every mathematical symbol or formula should have a definite grammatical function as part of a sentence. Therefore, if they end a sentence or a clause, a punctuation mark must be followed.