# Clarification on the idea of Mathematical Statements

The two statements below were left as an exercise to the viewers on this Youtube video about Mathematical statements.

1. Sum of 2 integers is positive.
2. Sum of 2 integers can never be negative.

Most of the answers in the comments say that both statements are mathematical statements. After giving enough thought on these statements, I believe that the first one is NOT a mathematical statement because it is ambiguous while the second one is indeed a mathematical statement.

I believe that it is valid to interpret the second statement as equivalent to The sum of any pair of integers can never be negative because of the presence of the word 'never'. However, I think the same is not true for the first one. I don't know how to explain it but I think Sum of 2 integers is positive cannot be interpreted as The sum of any pair of integers is positive.

• So what is the question here exactly? One could also argue there's an implicit "every" in the first point of interest. Persistent warning: careful when mixing language with mathematical logic. There's not always a smooth carryover. Commented Feb 3, 2020 at 5:00
• First of all, we have the ambiguity about Integer: if we stay with integer=not a fraction, the two are simply false. Having said that, without context the issue about the reading of "The sum of 2 integers is positive" and that of "The sum of 2 integers can never be negative" is not with "can" and "never". We may rewrite the second one without them and it becomes: "The sum of 2 integers cannot be negative", that has the same grammatical form of the fist one. Commented Feb 3, 2020 at 14:19
• In common mathematical language "is" and "must be" are the same : "two plus two is four" and "two plus two must be equal to four" does nor mean different facts. Commented Feb 3, 2020 at 14:20

There are unfortunately several ambiguities in the two sentences. The first one is the precise definition of the word integer. In mathematics, an integer is an element of $$\Bbb Z$$. But in the context of this video, I wonder whether it should not be understood as a natural number, that is, an element of $$\Bbb N$$. On the other hand, the second question refers to negative numbers... The second ambiguity is indeed the implicit use (or not) of quantifiers.

First question. Neither the sentence "The sum of two integers is positive" nor the sentence "The sum of two natural integers is positive" is a mathematical statement because it is false for the pair $$(0, 0)$$ and true otherwise.

Second question. The sentence "The sum of two natural integers can never be negative" is true (and hence is a mathematical statement), and the sentence "The sum of two integers can never be negative" is false (and hence is also a mathematical statement), if you consider there is an implicit universal quantifier: it is not true that the sum of two any integers can never be negative, because, for instance, $$(-1) + 0$$ is negative.

However, the sentence "The sum of two natural integers is not negative" is true (and hence is a mathematical statement) but the sentence "The sum of two integers is not negative" can be either true or false (and hence is not a mathematical statement).

In conclusion, I fully subscribe to Daniel W. Farlow's persistent warning: be careful when mixing language with mathematical logic!

• wow, thanks for that technically insightful comment. Commented Feb 3, 2020 at 6:46

Sum of 2 integers is positive.

This statement is ambiguous. It could be interpreted as

$$\exists x, y \in Z: x+y\gt 0\space\space$$ (a true statement)

or as

$$\forall x, y \in Z: x+y\gt 0\space\space$$ (a false statement)

Sum of 2 integers can never be negative.

This statement can only be interpreted as

$$\forall x, y \in Z: x+y\ge 0\space\space$$ (a false statement)