In praise of failure

I had a recent back and forth on Math.SE with a user that asked whether or not some exercise he found in some textbook is correct. The OP asked not to provide a proof, but rather to confirm if this statement is at all provable. When I asked why not just try and prove the damn thing, the reply was that if there is a typo or a mistake and the statement is in fact not provable, then they would have wasted their time trying an impossible task.

Well. Actually no. When I was a dewy eyed freshman, I had taken all my classes with 300 students from computer science and software engineering (Ben-Gurion University has changed that since then). Our discrete mathematics professor was someone who was renowned as somewhat careless when it comes to details in questions and stuff like this (my older brother took calculus with the same professor about ten years before, one day he didn’t show up to class, when my brother and two others went to see if he is at his office, he was surprised to find out that today is Tuesday).

So needless to say, our homework came rife with mistakes. Some obvious, others less obvious. And people were rightfully furious about it, since nobody would actually do their homework until the day before the submission, nobody would find the mistakes until the very last moment, which would cause great panic and annoyance.

I can’t say that I took anything from that course in discrete mathematics. What little graph theory, generating functions, and basic combinatorics that we learned there passed through my brain without leaving any footprints. But I learned a valuable lesson from that professor one time when he was confronted about the homework mistakes in class.

If you try to prove something, and you couldn’t prove it, and later you find out that it was in fact a false statement, then you did an excellent job not lying to yourself.

As I learned, both in the hard way of failure, and later as a teacher, students will often convince themselves that their proof is correct just because it seems to reach the wanted conclusion. In that sense, homework, as we often present them fail to teach students how to identify a dead-end, a mistake, a problematic statement, and how to correct for these problems. This is what research is for, and this is why a good advisor when you first start researching is an indispensable treasure.

Trying to prove something, and failing to do that, is a wonderful lesson for the future. If you later learn that the statement you’re after is false, then you’ve done a good job not lying to yourself (or to others) about it and about your proof. This develops self-confidence, which is something that is often lacking with beginners (and I felt it lacking in myself when I was a young masters student). If you do manage to prove the statement, and your proof is correct, that is a very good achievement, you’ve discovered something [almost] on your own, again this is a step towards building confidence. And if you did end up lying to yourself, or somehow made a mistake and proved a false statement (or your proof isn’t good), well, then you learn how to be more careful.

In either case, it is a good thing to encourage students to try and tackle “naturally occurring” statements, even if they are false, even if the students are not sure. There is no substitution for hard work, and a lot of failure and “wasted time” when it comes to learning mathematics.