Writing up a proof that assumes what is to be proven?

Question

I was working on this question on math, where (among other things), the OP was asked to prove that $$x \oplus y=\sqrt[3]{x^3+y^3}$$ is associative.

After some prompting, the offered proof was $$\begin{align} x \oplus (y \oplus z) &= (x \oplus y) \oplus z\\ \\ & \implies x \oplus\sqrt[3]{y^3 + z^3}\\ \\ & \implies \sqrt[3]{x^3 + (\sqrt[3]{y^3+z^3})^3}\\ \\ & \implies \sqrt[3 ]{x^3 + y^3 + z^3} \\ \\ &\implies \sqrt[3]{x^3 + y^3} \oplus z \\ \\ &\implies(x \oplus y) \oplus z\end{align}$$
How big of a sin is this? The algebra is right, but the implications start with what is to be proved. If we just deleted the conclusion at the start and changed the implications to equalities, I would be happy. As it is we start with what is to be proved and have implications between expressions, not between sentences. I have seen this kind of write-up a number of times and would like to know how to advise the poster.

Answer 1

I always provide the following example whenever a student assumes what they want to prove:

1. Suppose 0=1.
2. Then 1=0 must be true.
3. Then we can add both equations to deduce that 1=1.
4. This is a true statement (1=1) and therefore our original assumption was valid, so 0=1.

In our "Intro to Proofs" course for math majors, I show this on the first day of class. Any time later in the semester, either during class or on a homework submission, when someone attempts to assume the desired conclusion, I start to write this proof on the board. Eventually, it gets to the point where I write "0=1" and a student says, "Okay, okay, yeah yeah, I get it."

I think it's beneficial to have a "standard incorrect example" like this. I get the feeling that students learn to catch this mistake themselves by thinking about this very example.

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I don't usually get into it further, but I'm looking at @Jasper's answer where they mention "wrong usage of implications" and thinking that this example could spark further discussion. You can point out that this is a "valid proof" in the sense that if it really were true that 0=1 then we could deduce that 1=1. However, this is separate from whether or not 1=1 is true on its own. Moreover, you can discuss whether any of these $\implies$ implications are reversible. Specifically, in my example as outlined above, (1) and (2) imply (4) (while (3) is just stating what is being done in that step). However, one cannot take (4) and deduce that both (1) and (2) are true. That is the implication whose converse is false, and this example can be used to motivate and explain all of these concepts and terminology.

Answer 2

This sort of writing is quite common, in my experience, among young mathematicians (in their early years of college) who have had little feedback or training in formal mathematical writing.

The first line is simply a statement of the problem, boiled down to its algebraic bones, so to speak.

By the arrow, which in mathematical writing is read as an implication (or "approaches" in the case of a single arrow), students usually mean something like "the next step is...." The arrow is a nice, natural indication of the order in which they thought up the steps and you should read them.

There is no real sentence structure here, and probably the student is not accustomed to writing out their thinking. Instead they've written down only the expressions they've arrived at by means of their thinking. This is how they interpret the instruction "show all steps" from their high school and perhaps early college days. I gather it is accepted in many high schools by how often my first-year students write in this way. I don't have a good feel for how often it is accepted in college. I endeavor to get my students to write correctly, but some of these habits are so ingrained that they are difficult for them to kick.

Reading the proof as a relatively easy puzzle given as a sequence of hints, one might find it a innocuous pastime. The main problem is when there is an error in their thinking and one cannot easily fill in the gaps, but that is not an issue here.

Given that the proof came from a comment, I might have written, if I wanted to address their writing, "I think this reads better: $x \oplus (y \oplus z) = x \oplus\sqrt[3]{y^3 + z^3} =$ $\sqrt[3]{x^3 + (\sqrt[3]{y^3+z^3})^3} =$ $\sqrt[3 ]{x^3 + y^3 + z^3} =$ $\sqrt[3]{x^3 + y^3} \oplus z =$ $(x \oplus y) \oplus z$," and leave it to them the appreciate the difference. Perhaps add, "The proper use of the equal sign makes clearer the connection between the steps." I guess at some point I'd wonder how much effort would be effective trying to help the user write mathematics better via StackExchange comments. The user might have been writing the comment less formally than on an exam, say. In class, in which I see the students for many weeks, I take a different approach.

Answer 3

I think there are two problems that have to be addressed individually:

1. Wrong usage of implications. Implications just don't describe a relation between terms of the kind that appear here. They can be used between things that have a truth value such as equations or boolean expressions. I'd say there is no way other than fixing this to use only the correct symbols. This Question about symbols and category errors seems closely related.
2. Starting with what is to prove. This might be OK depending on the target audience and if the relations linking the individual steps are symmetric (i.e. equivalence or equality), so the order could be arranged so that the goal of the proof appears at the end. The Question at Mathexchange sounds as if OP there is in some sort of beginner algebra course. I hope that proper proof etiquette is taught there, so for this specific case, I'd also say that "starting with the goal" is also wrong.

Edit: I now realize that once the first point is taken care of, the second one becomes a non-issue because the proof then will be a chain of equations. I'd still mention the second point as a general advice.

Answer 4

How big a sin is this?

It's flat-out wrong and nonsensical for multiple reasons (as you point out here). The person writing those things is in need of correction.

Note that similarly, the person in question also has serious problems writing standard-grammar English sentences. E.g.: Inconsistent use of a capital for the personal pronoun "I", inconsistent capitals to start sentences, inconsistent use of punctuation to end a sentence, etc.

They seem to be aware that they're a beginner and have written comments such as, "It was a typo, i'm pretty tired and determined to understand this".