When two equivalent algebraic statements have two "different" meanings

Question

Suppose I want to prove $\sqrt{7}$ is not a rational number. I suppose it is and it brings me to a contradiction. Here how it goes line be line:

First line. $\sqrt{7}=\frac{m}{n}$

Second line (after squaring both sides of the equality above). $7n^2=m^2$

Third line. A solution for this equality contradicts the uniqueness part of the fundamental theorem of arithmetics, thus it has no solution.

I used the argument above in my number theory class today. It was hard for one of the students to accept it for an interesting reason I had never seen before.

No solution for $7n^2=m^2$ means $7n^2$ is not a square number, but no solution for $\sqrt{7}=\frac{m}{n}$ means $\sqrt{7}$ is not a rational number. These two statements are not the same, are they? Asked the student!

The lecture that happened to be me had not an answer, or at least, at that moment, and even now, could not find a similar chain of algebraically equivalent statements that each one brings a different meaning to mind, while it is "easy" to see those meanings refer to the same thing.

I guess my question is do you know an example like the one that I tried to summarize in the title and explain in the body of the post. Or, should I ask, if you were the lecturer what was your answer to the student?

Answer 1

The two statements aren't literally "the same", because as the student observed, they say different things. However, they are logically equivalent: each implies the other. (Similarly, "4/2" isn't literally the same expression as "5 - 3", but they provably represent the same number.)

If we could only prove things by repeating the same statement, we couldn't get anywhere. The whole point is that different statements can imply each other, so knowing one statement to be true lets us logically deduce the truth of other statements.

From another perspective, we could say that the meaning of a statement is determined by its implications, in which case logically equivalent statements do mean the same thing — when we look more closely at what "$\sqrt{7}$" and "rational number" mean, we see that "$\sqrt{7}$ is a rational number" really does mean the same thing as "there exist integers $m$ and $n$ with $n \neq 0$ such that $7n^2 = m^2$".

If the student has trouble accepting this, it might be because they aren't in the habit of thinking about what words or expressions mean in such a precise way. They might be taking "square root" and "rational number" as fuzzy notions whose general sense or usage is commonly understood — the way words are used in everyday, nontechnical language — rather than as notions specified by mathematically precise definitions. This misconception about the nature of mathematical definitions is documented in the literature, even among math majors: see, for example, [Edwards–Ward 2004].

• B. Edwards and M. Ward, "Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions". American Mathematical Monthly, vol. 111, no. 5 (May 2004), pp. 411–424. DOI: 10.2307/4145268 [PDF] [JSTOR]

Answer 2

There may be an issue with quantifiers here. "No solution for $7n^2=m^2$" does not mean "$7n^2$ is not a square number". It means, perhaps, "there is no $n$ so that $7n^2$ is a square number". But it's important to remember that "no solution" means "no simultaneous choice of n and m". The phrasing "$7n^2$ is not a square number" suggests that, by separating the variables, the student has gotten confused about the variables, and is now treating n as a constant and is looking for solutions in m.

Answer 3

The two statements are equivalent, assuming care with quantifiers. In the original form $\frac{n}{m}=\sqrt{7}$ we have an equation over the real numbers. We trade this for the second form $7m^2=n^2$ which is an equation over the integers. In this second form, we can make use of integer factorization to attack the problem.