Taxonomy of bad proofs

Question

I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in examples of poor writing that introduce logical errors into the proof as well as those that just make the proof hard to read.

I am aware that there is already a question on this site asking for examples of bad proofs. I find the answers there kind of unsatisfactory (one answer simply says to look at student assignments and the other just consists of a broken link), but perhaps it's because there aren't many great examples of poorly written proofs on the internet. Also, that question is only focused on proofs that are wrong or "could be simplified" whereas I am also interested in more general problems with the way proofs are written.

In case there are no good examples online of poorly written proofs, I would like to try to generate some myself. Ideally, for each general type of mistake that undergraduates are prone to make when writing proofs, I want to write a bad proof that isolates and exemplifies that mistake. So I am interested in compiling a "taxonomy" of ways that proof writing by students can go wrong.

Here's my current taxonomy:

1. Backwards proofs: Proofs that start from the desired conclusion and work backwards until they arrive at the premises. Not only can this be confusing to read, it is easy to introduce errors when some step of the proof is not reversible.
2. The "obvious" worst case: It is popular to warn students against "proof by example" but I have found that students are usually already aware that an example is not a proof... unless of course they think that the example they chose is obviously the worst case scenario without bothering to justify it.
3. The tangle: The proof that keeps going and going, introducing a complicated series of manipulations, definitions, hand-waving, etc. until somehow arriving at the right conclusion. Often some step in the middle is wrong (or possibly nonsensical) but it's hard to find because it's buried under a pile of more-or-less-correct but irrelevant details. A special favorite of students taking exams.
4. The mysterious stranger: Some object is used without ever being introduced or defined. Often there is a reasonable way to fill in the definition of the object but the proof becomes a guessing game of what the student meant the object to be. Sometimes this is caused by a student trying to use a variable outside of its proper scope. In that case it can also lead to the student giving the same name to two distinct objects.
5. Inappropriate types: The student claims that some ideal is an element of the ring. Or tries to divide by a set of real numbers. Etc. The point is that they are trying to do something with an object which is of the wrong type to do it with. Or sometimes they are trying to perform a typecast that cannot automatically be done in an unambiguous way like if they try to apply a function of type $\mathbb{R} \to \mathbb{R}$ to a complex number. Often in these cases the student does have in mind a way to do the typecast (although sometimes they have just misunderstood a definition or don't even really grok that mathematical objects have types) but problems occur when there are multiple mutually incompatible ways to do it.
6. The incantation: You've just learned a big, important sounding theorem in class so you should probably use it on the homework, right? You don't quite understand it or know when it applies, but it seems like you should be using it so you just go ahead and invoke it anyways. It's pretty common for students to invoke big theorems without checking that the hypotheses of the theorem hold and sometimes in situations where it doesn't even make sense to invoke the theorem.
7. Implication confusion: The student is supposed to prove $X \implies Y$ but instead proves $\lnot X \implies \lnot Y$ or $Y \implies X$ or tries (and typically fails) to prove $X\land Y$. Or the student is told to assume that $X \implies Y$ and then concludes that $Y \implies X$. Most common when the statements of $X$ and $Y$ are a bit complicated or involve implications as well.

Which leads me to my question.

Question: What other common pitfalls of student proof writing should be added to this taxonomy? What are some poorly written proofs that exemplify one of these pitfalls (either one that I have already listed or a new one)?

Feel free to argue with my current list. Also, if anyone wants to add an answer to the old question on "bad proofs" I'd be very happy.

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EDIT: After discussing this with one of my friends, I've added two more entries to the taxonomy.

Answer 1

Perhaps related to "The Tangle" is what I call "Wishful Thinking". This most often happens when the student has a correct algebraic expression/equality and knows the correct final expression/equality, but makes great (or incorrect) leaps to get from one to the other, e.g. in proofs by induction.

There is also the "Using What You Are Proving" (maybe similar to your "Backwards Proof," but more subtle). Example: in proving two elements $a$ and $b$ commute, they use $(ab)^2 = a^2b^2$. Or in an onto proof where they assume a general element of the range has the form of an image.

Answer 2

The Q&A so far has gone into a lot of relatively sophisticated stuff. The students I teach are mostly engineering majors at a community college, so their mistakes are a lot simpler. They have generally been exposed to proof writing in high school geometry. However, my experience is that this does not translate well into a more general skill. Many of them learned "two-column proofs," which are pretty artificial and constrained.

Often they will "prove" something by giving an example, e.g., they'll prove that the dot product is commutative by making up a couple of vectors and doing the computations. This also comes up in physics labs, where they claim that their lab results prove Newton's second law. They don't want to believe that a counterexample can disprove a general statement, but one piece of evidence can't prove it.

They write something that they claim is a proof, but they never make use of one of the assumptions, and this assumption is necessary.

In my experience students in elementary classes like first-semester calculus need to get some experience writing really simple, easy proofs so that they can get past these initial stages. Unfortunately, I don't think most instructors assign any such work in these courses. I would imagine that this is a big reason for the meltdowns I hear about when math majors start their upper-division coursework.

Here's an example of what I would like them to be able to do. Suppose they know enough about the dot product to know how to compute it from cartesian components. Then if I ask them to prove that the dot product is commutative, ideally they would be able to write something like this: "The dot product can be expressed in terms of the products of the two vectors' components. Since the components are real numbers, and multiplication of real numbers is commutative, it follows that the dot product is commutative."

But even this extremely simple-looking argument contains some pretty sophisticated mental activity. Students are trying to relate everything to their previous education, and their previous education contains a couple of possible models of this. One model is concrete computation involving numbers, so they produce an example involving numbers and call that a proof. Another model is two-column proofs in high school geometry, but that model doesn't carry over here, because it doesn't help them to focus on the essentials, ignore inessential elements, and write a sentence meant for a human to read.

Answer 3

My friend, who is currently teaching an abstract algebra course, recently provided me with an example of an incorrect proof that I think showcases a few types of faulty reasoning that are common among students. I'm sharing it here with his permission.

Proposition: Suppose $R$ is a ring and $I$ is a prime ideal in $R$. For all ideals $J$ and $K$ in $R$, if $JK \subseteq I$ then either $J \subseteq I$ or $K \subseteq I$.

Proof. Suppose $JK \subseteq I$. Then $\sum_{n, m} a_nb_m \in I$. Since $I$ is an ideal, it is closed under addition so each $a_nb_m$ is in $I$. Since $I$ is prime, either $a_n$ or $b_m$ is in $I$. Therefore either $J \subseteq I$ or $K \subseteq I$.

There are a few obvious problems with this proof.

• First, it is not clear what $a_n$ and $b_m$ are. Presumably the $a_n$'s are some finite set of elements in $J$ and the $b_m$'s are some finite set of elements in $K$, but this should be stated explicitly. This is an instance of The Mysterious Stranger.
• Next, saying that "since $I$ is closed under addition, each $a_nb_m$ is in $I$" is not correct. It is mixing up "closed under addition" with its converse, so this is an instance of Implication Confusion.
• The conclusion of the proof is more or less a non-sequitor. All that the proof has really established is that either $J$ or $K$ contains some element which is in $I$, which is not enough to show that either $J$ or $K$ is entirely contained in $I$. I would say this is an example of Wishful Thinking.
• Finally, there is an element of The Tangle to the whole proof. It is not really clear what the strategy of the entire proof is, why the sum $\sum_{n, m} a_nb_m$ is being introduced or what the goal of each step is. It's likely that such a proof is mostly the result of pattern-matching the appearance of proofs given in class in the hopes that this will lead to an approximation of a correct proof (and thus partial credit).

Answer 4

Three categories of error I often encounter are "Incorrect negation", "trouble with quantifiers" and "Confusing general parameters with specific objects". Everybody knows that students have trouble dealing with negating quantifiers in the context of, for example, an $\epsilon-\delta$ proof in an Analysis course. We also see these issues arise in Linear Algebra, where student work like the following is rampant:

Let $\vec{v}$ be any nonzero vector; prove that $\{\vec{v}, \vec{0}\}$ is linearly dependent.

Solution. Let $a\vec{v} + b\vec{0} = \vec{0}$. If $a = b = 0$ then this is a trivial relation; therefore the vectors are linearly dependent.

What's happening in this muddle is something like this: Students know that "linearly independent" means "there is no nontrivial linear relation on the vectors"; they try to negate it, but get tangled up in the triple negative ("it is not true that there is no nontrivial..."), and end up thinking that "linearly dependent" means "there is a trivial linear relation on the vectors". The use of the letters $a$ and $b$, which seem like general parameters in the first sentence, but are subsequently revealed in the very next sentence to have been very specific constants all along, helps to obscure the obvious error.

In fact on our most recent final exam we included, as a true/false question,the statement

"If $S = \{v_1 \, \dots, v_m \}$ is a set of vectors, and $a_1v_1 + \cdots a_nv_n = 0$ is a trivial relation, then $S$ is linearly independent."

A distressingly large fraction of our students marked this as true. They seem to have trouble understanding that "There are no nontrivial relations..." means something different from "There is a trivial relation..."

Answer 5

I once was a teaching assistant for a class which included a lot of proof by induction. Many of the students would give more than one base case. They seemed to think that the base case was some kind of "example" and more examples would be better.