Given that proofs/demonstrations are a recurring topic on this blog, I think it would be nice to talk about something that is rarely talked about openly in math books and classes… how to write in math. This seems to be a somewhat neglected subject from a didactic point of view, because as we delve deeper into readings and exercises, we “absorb” these sociomathematical norms (socially accepted by the mathematical community) until we use them without realizing it.
But what motivates me to talk about this is that when we start in this life of demonstrations, sometimes having the idea for a demonstration already demands a lot of mental effort, but then, with organized thoughts we find an additional obstacle… writing. We use symbols in place of words, words in place of symbols, we try to put what we have in mind on paper so that someone else can read and understand, but this is not trivial at first. Thinking about the demonstration is no longer something easy, why make life even more difficult for those who are starting to walk this path?
Thus, the tips I present here were found on pages 107-109 of the book “Book of Proof” (2nd edition) by author Richard Hammack. Although they won’t help as much in the process of thinking/elaborating the arguments that make up the proof, they can help other people to understand more easily what you wrote. Curious that when I took this to students of the 1st semester of the Degree in Mathematics, they found it strange/different/peculiar that we discussed so much about Portuguese in a Mathematics class.
A) Start each sentence with a word not a mathematical symbol.
The reason is that sentences start with capital letters, but math symbols are case sensitive. Since a and A can have totally different meanings, placing these symbols at the beginning of a sentence can lead to ambiguity.
Here are some examples of misuse (marked with X) and good use (marked with ok).
A is a subset of B(X)
The set A is a subset of B (okay)
x is an integer, so 2x + 5 is an integer (X)
Since x is an integer, 2x + 5 is an integer (OK)
x²-x + 2 = 0 has two solutions (X)
X²-x + 2 = 0 has two solutions (XXXXX)
The equation x²-x + 2 = 0 has two solutions (OK)
B) End each sentence with a period, even when the sentence ends with a symbol or mathematical expression.
Euler proved that V – A + F = 2 (X)
Euler proved that V – A + F = 2. (okay)
Mathematical statements (equations, etc.) are like English sentences that contain special symbols, so use normal punctuation.
C) Separate mathematical symbols and expressions with words.
Failure to do so can cause confusion, causing distinct expressions to appear to merge into one. Compare the clarity of the following examples.
Because x²-1=0, x=1, or x=-1. (X)
Since x²-1=0, it follows that x = 1 or x = -1. (OK)
Unlike A∪B, A ∩ B = ∅. (X)
Unlike A∪B, the set A ∩ B = ∅. (OK)
D) Avoid misuse of symbols.
Symbols like =, ≥, ⊆, ∈, etc., are not words. While it is appropriate to use them in mathematical expressions, they are out of place in other contexts.
Since both sets are =, one is a subset of the other. (X)
Since the two sets are the same, one is a subset of the other. (OK)
The ∅ is ⊆ of each set. (X)
The ∅ is a subset of each set. (OK)
Since a is odd and x is odd → x² is odd, a² is odd. (X)
Since a is odd and any odd number squared is odd, so a² is odd. (OK)
E) Avoid using unnecessary symbols.
Math is messy enough without them. Let’s try not to muddy the water any further.
No set X has negative cardinality. (X)
No set has negative cardinality. (OK)
F) Use the first person plural.
In mathematical writing, it is common to use the words “we” and “our” instead of “I”, “you”, or “mine”. It is as if the reader and writer are talking, with the writer guiding the reader through the details of the proof.
G) Use the active voice.
This is just a suggestion, but active voice makes your writing more vivid.
The value x = 3 is obtained by dividing both sides by 5. (X)
Dividing both sides by 5, we get the value x = 3. (OK)
H) Explain each new symbol.
When writing a proof, you must explain the meaning of each new symbol introduced. Failure to do so can lead to ambiguity, misunderstandings, and errors. For example, consider the following two possibilities for a sentence in a proof, where a and b were introduced on a previous line.
Since a|b, it follows that b = ac. (X)
Since a|b, it follows that b = ac for some integer c. (OK)
If you use the first form, then a reader who is carefully following your proof can momentarily scan backwards, looking for where the c entered the text. Not realizing at first that it came from the definition of divisions.
I) Beware of “it”.
The pronoun “this” can cause confusion when it is not clear what it refers to. If there is any possibility of confusion, you should avoid the word “it”. Here’s an example:
As X entirely contained in Y, and 0 <| X |, we see that It is not empty. (X) Is “it” X or Y? Either option would make sense, but what do we really mean?
As X entirely contained in Y, and 0 <| X |, we see that Y is not empty. (OK)
J) Since, because, as, for, so.
In proofs, it is common to use these words as conjunctions that unite two statements, meaning that one statement is true and, consequently, the other is true.
All of the following statements mean that P is true (or assumed to be true) and, as a consequence, Q is also true.
Q since P
P leads to Q
Q because P
From P we have Q
Q, as P
Because P, then Q
Q, for P
P implies Q
P, then Q
By P, then Q
Note that the meaning of these constructions is different from that of “If P, then Q”, as they are claiming not only that P implies Q, but also that P is true. Be careful when using them. It must be the case that P and Q are statements and that Q really follows from P.
If x belongs to Naturals, then x belongs to Integers. (X)
As x belongs to Naturals, so x belongs to Integers. (OK)
K) So, therefore, consequently.
These adverbs precede a statement that logically follows from previous sentences or clauses. ENSURE that a statement follows them.
So 2k + 1. (X)
So a = 2k + 1. (OK)
L) Clarity is the gold standard of mathematical writing.
If you believe that breaking a rule makes your writing clearer, then break the rule.
Generally speaking, mathematical writing will evolve with use and practice. One of the best ways to develop a good math writing style is to read other people’s proofs and solve those famous “exercises left to the reader”. Keep these tips in mind, as it will help other people understand more clearly what you wrote (and assess what is right or wrong).