What exactly is a mathematical statement?

Question

A mathematical statement, so I have heard, is a statement that is either true or false but not both. My initial source for this is not a book on the philosophy of mathematics; instead, it is a first-year, undergraduate level exercise in Algebra from my university; this being their first introduction to university level Algebra after A Levels, assuming things like division of polynomials has been seen before, they know what an integer is, they can prove basic theorem, etc. I am a TA for the course. My students are very smart and are likely to ask questions. I have taught the course before, and often resort to saying that this is supposed to be an exercise in using definitions and to encourage discussion; I don't know that for sure.

Other sources, at a cursory glance, give the same definition; so says Google when I search, "what is a mathematical statement?".

This seems an inadequate definition. Too many things are included. Unless I am mistaken, the statement, "if Elvis Presley was the Prime Minister of the United Kingdom, then I am a microwave" is, vacuously, a true one; what has it to do with mathematics?

And what of the mathematics of non-classical logics? This is not an idle question: topos theory handles logics that have statements that are neither true nor false, that, fair enough, yes, we can make statements about that are either true xor false - but the statements themselves don't fit the definition!

So my question is a fundamental one:

What is a mathematical statement?
More importantly: why is the/your definition the way it is?

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The exercise:

A mathematical statement is a sentence that is either true or false, but not both. For each of the following, determine whether they are such a statement. For those that are, state whether they are true or false:

(Then a list of sentences. For example: "if $n$ is even, then $n^2$ is even" and "if $1=2$, then the Earth crashed into the moon last night".)

Answer 1

I wish to bring forward an idea first noted by Giuseppe Peano that could be a part of an answer, or at least, helpful for it. Because an adequate answer should also be capable of distinguishing a mathematical statement from a non-mathematical one (for example, a theoretical statement in physics).

Before that, I should remark that such a description as "mathematical statement is a statement that is either true or false but not both," by itself, is too naïve, and not distinctive of mathematics (consider other formal scholarly disciplines). The view that a mathematical statement must ultimately (that is, even if it remains conjectural for a while) have a strictly determinate truth value is traditionally cited as one of its key characteristics (however, a topic of philosophical debate in modern times; there are a lot of fine distinctions around this, let us leave it off here), but there are a lot more to say about it. Probably, the mentioned words have been expressed wrapped in a context that focuses on the nature of mathematical discourse stressing that it has to be exact and precise (avoiding vagueness or ambiguity), express clear and distinct ideas, and the like.

No doubt a mathematical statement has to satisfy the grammatical conventions of mathematical discourse, otherwise, it could not be even meaningful, let alone exact and precise in expression.

Notice that we begin to talk about mathematical discourse in connection with the notion of mathematical statement. So, a question suggests itself: What are the constituents of mathematical discourse? Very crudely, it involves rigorous arguments built on mathematical principles, established facts, concepts and terms. Thus, we have come to Peano's idea. It is more explicitly articulated by F. P. Ramsey, so I shall refer to Ramsey. Ramsey distinguishes semantic paradoxes from the purely logical by the notions involved in their construction (see "The Foundations of Mathematics" in The Foundations of Mathematics and Other Logical Essays, pp. 20-21):

It is not sufficiently remarked, and the fact is entirely neglected in Principia Mathematica, that these contradictions fall into two fundamentally distinct groups, which we will call A and B. The best known ones are divided as follows :—

A. (1) The class of all classes which are not members of themselves.

(2) The relation between two relations when one does not have itself to the other. (3) Burali Forti’s contradiction of the greatest ordinal.

B.

(4) ‘I am lying.’

(5) The least integer not nameable in fewer than nineteen syllables.

(6) The least indefinable ordinal.

(7) Richard’s Contradiction.

(8) Weyl’s contradiction about ‘heterologisch’.

The principle according to which I have divided them is of fundamental importance. Group A consists of contradictions which, were no provision made against them, would occur in a logical or mathematical system itself. They involve only logical or mathematical terms such as class and number, and show that there must be something wrong with our logic or mathematics. But the contradictions of Group B are not purely logical, and cannot be stated in logical terms alone; for they all contain some reference to thought, language, or symbolism, which are not formal but empirical terms. So they may be due not to faulty logic or mathematics, but to faulty ideas concerning thought and language. If so, they would not be relevant to mathematics or to logic, if by ‘logic’ we mean a symbolic system, though of course they would be relevant to logic in the sense of the analysis of thought.

Likewise, it appears that an indispensable feature of a mathematical statement is that it be set down in the terms and norms of mathematics (as practised by the community of mathematicians). Indeed, we witness in the history of mathematics such problems that have been originated and defined outside of mathematics, but redefined in mathematical terms and solved by mathematicians, thus amassed into the body of mathematical knowledge along with those emerged from within mathematics.

For those who want to reflect on such issues further, I recommend David Bostock's reader-friendly book Philosophy of Mathematics: An Introduction.

Answer 2

A mathematical statement is a sentence that is either true or false, but not both. For each of the following, determine whether they are such a statement. For those that are, state whether they are true or false:

1. If $n$ is even, then $n^2$ is even.
2. If $1=2,$ then the Earth crashed into the moon last night.

It is apparent that the author is not using the term "mathematical statement" in a strict, pedantic sense—for example, as a well-formed formula, containing no free variable, comprehensible assuming just mathematical axioms—because, ironically, as such, Sentence 1 wouldn't qualify (it has a free variable) while Sentence 2 arguably would ("the Earth crashed into the moon last night" could be a conceptual abstraction in some mathematical framework).

The given exercise, regardless of its intention in introducing the upcoming material, appears to require a "mathematical statement" to pass precisely two checks:

1. Is it an assertion (admitting tacit universal implication if applicable)? (An assertion always potentially has a truth value, even if the value is ambiguous in the absence of further context.)

   ✗ If $n$ is even, then or $n^2$ is even.

   ✗ Tell me what the square root of 3 is.

   ✓ 45 # 5 = 53.

   ✓ Every number has a square that's smaller than $5.$ (True in the domain of purely imaginary numbers)

2. If yes, then is the assertion mathematically conventional?

   ✗ 45 # 5 = 53.

   ✓ Every number has a square that's smaller than $5.$

   ✓ If $n$ is even, then $n^2$ is even.

   ✗ If $1=2,$ then the Earth crashed into the moon last night.