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We had students the other day from Maths for Information Technology and their task was to form the contrapositive of a several statements. Given a particular statement of the form "If A, then B", the contrapositive is "If not B, then not A", so mathematically the problem is not actually very difficult. However grammatically the problem is much harder than it looks.

Consider this statement: "If it is raining, then there are clouds." If we compare this to my generic example above, we see that A is "it is raining" and B is "there are clouds", so by my own rule, the contrapositive ought to be this: "If not there are clouds, then not it is raining." This is obviously not a grammatical English sentence! A correct version is, "If there are not clouds, then it is not raining." By giving people a generic rule, we are getting them into trouble with their grammar. This may seem like a small thing, but there are plenty of students for whom English is not their first language, and even students whose first language is English often don't have a very good command of the rules of grammar!

There's no easy way around this in the way we present these generic rules, except to make them aware that they need to think about the grammar of the sentence that they write when they do this, in particular, where the word "not" has to go.

But it gets worse! Consider the statement "If f: R → R is differentiable, then f is continuous". According to my above rule, A is "f:R → R is differentiable" and B is "f is continuous", so taking into account the grammatical placement of the word not, we get the contrapositive is "If f is not continuous, then f:R → R is not differentiable." The students we worked with did this exact thing, and they could tell there was something odd about it, but they couldn't quite figure out what it was.

The problem is that the part saying ":R → R" is not technically part of the if-then construction. It could have been stated in a completely different sentence like this: "Let f: R → R. If f is differentiable, then f is continous." Then that lead-in sentence isn't an if-then construction, so it isn't part of the contrapositive.

And here's where the grammar gets particularly tricky. The fact that this little bit of the sentence can be pulled out into a sentence of its own means that grammatically it is called a "relative clause". A relative clause gives more information about a noun in a sentence, without interfering with the verb. You see it in sentences like "My brother, who is in Canada at the moment, says hi." I could have said: "My brother is in Canada at the moment. He says hi." Of course it wouldn't have had quite the same impact as the first sentence, which is why we say it the way we do. Another example is "If Catherine, your wife, is a kindy teacher, then she is clearly awesome." The relative clause here is "your wife", which is telling more information about who Catherine is before you go on to say stuff about her. This sentence is closer to the maths sentence above, but it has one very important difference. In the maths sentence I mentioned f twice; in this English sentence I didn't mention Catherine twice. Instead, I used the pronoun "she". We could have done the same in the maths sentence too: "If f:R → R is differentiable, then it is continuous" would become "If it is not continuous, then f:R → R is not differentiable." It would have been much more obvious what was wrong with this sentence – we haven't told the reader what f is, or indeed even mentioned f at all until the end. This makes it obvious that we ought to move the relative clause to the first part of the sentence when we form the contrapositive.

Other than the strange tendency of mathematicians to not use pronouns, there is something else that prevents us from seeing ":R → R" as a relative clause: the flexibility of the notation itself. Maths notations that include a verb in their meaning can be read aloud in multiple ways depending on context. Many students do not actually realise this, mainly because they never read their maths aloud. For example, consider this bit of maths: "Let x = 12. Then x = 4×3 = 2×2×3." This is read aloud as "Let x  be equal to 12. Then x is equal to 4 times 3, which is equal to 2 times 2 times 3". That "=" was read loud as "be equal to", "is equal to" and "which is equal to". In our contrapositive example,  consider these three sentences: "Let f: R → R.", "Suppose f: R → R." and "If f:R → R is differentiable...". The first is read aloud as "Let f be a function sending R to R", the second is read aloud as "Suppose f is a function sending R to R", and the third is read aloud as "If f, which is a function sending R to R, is differentiable...". But they all look the same!

The flexibility of our maths notation makes for easy writing, but sometimes it makes for difficult grammar, especially when it masks those pesky relative clauses!

This comment was left on the original blog post:

John Baez 8 September 2014:
Very interesting analysis. A digression: you write “Other than the strange tendency of mathematicians to not use pronouns….” This reminds me of something the computer scientist Tom Payne told me: “mathematicians are people with an extraordinary ability to keep track of many pronouns”. His point was that variables in mathematics serve as pronouns. Instead of saying “he”, “she”, and “it”, which breaks down when you have more than one he, she, or it, we introduce new pronouns (variables) as needed. Ordinary people lose track of all these pronouns.