What Is a Mathematical Statement?

Before Asking for a Proof

In What Is Mathematics?, we distinguished a pattern from a proof and proved that the sum of two even integers is even. Both ideas depend on knowing exactly what is being claimed.

Before asking whether an argument is correct, we should therefore ask a more basic question: Does the sentence express a definite mathematical claim?

“Add these numbers” gives an instruction. “Are these numbers equal?” asks a question. Neither asserts something true or false. By contrast, “\(4+6=10\)” makes an assertion that we can evaluate.

Statements and Truth Values

A declarative sentence asserts that something is the case. Its truth value is “true” or “false,” according to whether the assertion holds.

We use the usual classical framework of mathematics throughout this course. Once a statement's meaning and context are fixed, its truth value does not depend on who reads it or whether that person knows how to justify it.

The second row is particularly important. A statement does not have to be true. It has to be the kind of assertion that has a definite truth value.

Expressions Are Not Assertions

A mathematical expression represents an object, such as a number. A mathematical statement asserts something about objects.

For example, the expression

has the numerical value \(25\). But the expression by itself is not true or false. It is simply a way of writing a number.

Adding a comparison can produce an assertion:

This is a true statement because \(3^2=9\), \(4^2=16\), and \(9+16=25\). The sentence \(3^2+4^2=24\) is also a statement, but it is false.

Symbols are not required. “The sum of nine and sixteen is twenty-five” expresses the same claim in words. What matters is the meaning of the sentence, not whether it contains an equation.

Context Must Fix the Meaning

Consider the sentence:

“This number is greater than five.”

If “this number” has not been identified, the sentence does not yet express a definite claim. If it refers to \(8\), the claim is true. If it refers to \(2\), the claim is false.

The same issue occurs with letters. Written on its own, with no information about \(x\), the formula

does not have a single truth value. Substituting \(x=3\) gives the true statement \(3+2=5\). Substituting \(x=1\) gives the false statement \(1+2=5\).

This does not mean that any sentence containing a letter is incomplete. The surrounding words may explain exactly what is intended.

Earlier, we described \(a+b=b+a\) as a general statement after asking about arbitrary real numbers \(a\) and \(b\). That surrounding context supplies the intended meaning: the equality holds for every pair of real numbers.

Not every detail has to be repeated in every line of a proof. However, any context needed to interpret a claim must be available and unambiguous.

Precision Is Different from Complexity

A sentence can be short and precise, or long and vague. Compare “\(100>20\)” with “One hundred is a very large number.”

The first has a definite truth value. The second does not specify what counts as “very large.” As written, it does not provide a precise mathematical condition to evaluate.

We can replace vague language with a definite comparison:

These replacements make particular choices of meaning. They do not reveal a unique meaning already contained in the vague wording. When clarifying someone else's claim, we must check that the precise version is the claim they intended.

Having a Truth Value Is Not the Same as Knowing It

Suppose you encounter the assertion

Before calculating, you might not know whether it is true. Nevertheless, the assertion already has a definite meaning and truth value.

We can settle the question by calculation:

The calculation establishes that the statement is true. It does not turn a nonstatement into a statement.

The same distinction applies to conjectures discussed in the previous tutorial. A precise conjecture remains a mathematical statement even when no proof or counterexample is known. “We do not know” describes our knowledge; it is not a third truth value in the classical framework used here.

From a Statement to a Theorem

Recognizing a statement and proving it are different tasks. The first concerns what the sentence asserts. The second concerns whether the assertion follows from accepted facts.

Here is a simple example that builds directly on the result proved in the previous tutorial.

Proof. Let \(a\), \(b\), and \(c\) be arbitrary even integers. By the result established earlier that the sum of two even integers is even, \(a+b\) is an even integer.

We can apply that same result to the two even integers \(a+b\) and \(c\). Their sum \((a+b)+c\) is therefore even. By the associative law of addition,

Thus \(a+b+c\) is even. Since \(a\), \(b\), and \(c\) were arbitrary even integers, the claim holds for every such triple.

The proof includes zero and negative even integers. Nothing in the argument required positivity or required the three integers to be different.

For a claim about every member of a finite collection, checking every member can establish the claim. For a claim about all integers, checking only finitely many individual integers leaves other cases unexamined. The theorem above handles all allowed triples through one general argument.

A Method for Reading Mathematical Sentences

When a new sentence appears, separate its meaning from its justification. The following sequence helps prevent several common mistakes.

The essential distinction is between what is asserted, whether it is true, and how we justify our answer. Keeping these questions separate is a first step toward reading and writing rigorous proofs.