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.

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

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    $\begingroup$ Two examples I can think of that might be slightly outside your scope (because even strong mathematicians can do this from time to time) are not considering all possible cases when dividing the proof into separate cases, and in overlooking sometimes problematic exceptional situations (what if the variable is zero, what if the set is the empty set, etc.). $\endgroup$ – Dave L Renfro Feb 17 '20 at 19:33
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    $\begingroup$ @amWhy The fallacies you link to are rhetorical fallacies, which is explicitly not what I'm asking about. I suppose it's possible that an undergraduate student could include an instance of the ad hominem fallacy in a proof they write, but I have yet to see it. $\endgroup$ – Patrick Lutz Feb 17 '20 at 22:01
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    $\begingroup$ There is far more overlap than you acknowledge (circular reasoning, argument by authority, etc.) $\endgroup$ – amWhy Feb 17 '20 at 22:03
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    $\begingroup$ Personally I would focus attention on good proofs, and if a taxonomy, only one of the more important forms of good proof (in/direct, non/constructive existence, induction, etc.). $\endgroup$ – user615 Feb 18 '20 at 1:31
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    $\begingroup$ I love this. I'm a TA for vector calc and I most often see backwards proofs and "the tangle". Beautiful example for explaining backwards proofs is one I saw last year: The question asks you to prove that if $r(t)$ is a parametric curve and $r(t)$ is always orthogonal to $r'(t)$, then $r(t)$ traces out a circle. Many students started with $r(t) = \langle R \cos(t), R \sin(t) \rangle$ and then proved $r'(t) \cdot r(t) = 0$ to conclude that $r(t)$ and $r'(t)$ are orthogonal. $\endgroup$ – Dark Malthorp Feb 25 at 16:39

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.

  • $\begingroup$ Those are both great! I think "Wishful Thinking" is different from "The Tangle" since in the tangle, the errors are usually not caused by trying to get to any particular goal; it's more like lots of unnecessary complexity is introduced which often leads to errors, which often in turn leads to a false proof. And I also think "Using What You Are Proving" is different enough from "Backwards Proof" to constitute its own entry. $\endgroup$ – Patrick Lutz Feb 17 '20 at 18:50
  • $\begingroup$ Also, it occurs to me that "The 'Obvious' Worst Case," "The Mysterious Stranger" and "The Incantation" are all at least sometimes caused by cases of "Wishful Thinking." $\endgroup$ – Patrick Lutz Feb 17 '20 at 18:52
  • $\begingroup$ I suppose "using what you are proving" can be better described as "circular reasoning"? $\endgroup$ – GoodDeeds Feb 17 '20 at 18:57
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    $\begingroup$ @GoodDeeds It's true that "Using What You Are Proving" is an instance of "Circular Reasoning," but what is interesting to me about it is the circumstances that cause students to make that mistake. Most students can recognize bald instances of circular reasoning and know it is wrong. But they still end up engaging in circular reasoning anyway and I am interested in how that occurs. $\endgroup$ – Patrick Lutz Feb 17 '20 at 19:32

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).
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    $\begingroup$ I'm accustomed to tangles being much longer than this. $\endgroup$ – Andreas Blass 6 hours ago
  • $\begingroup$ @AndreasBlass Yeah, I agree. That's why I just said "there is an element of the tangle." But it would be nice to have better examples. $\endgroup$ – Patrick Lutz 6 hours ago

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