This week, I learned the value of public ignorance (or, perhaps, relearned it). One of my math students wanted to complete a proof by saying, half-way through, “Since all these steps are reversible, just reverse them for the second half of the proof.” He asked me if I would let him do so, and I replied, “Certainly not!” There was a little more give-and-take, but unless you’re into math, you know all about the situation you need to know for this story.
I posted his comment/request and my response on Facebook, and soon former students took me to task. Some said they’d heard of the approach before. My response was that it might do for a class discussion (something along the lines of the infamous “the rest of the proof has been left as an exercise for the reader” line), but it would not be accepted in a formal proof.
One guy, now a teacher himself, then started covering my wall with examples of the approach’s use in published journals. None of the journals he mentioned was at the level of PNAS, but still…
As a result of my post, I was shaken from my complacency (and either complimented or insulted, depending on your perspective, when I was compared to Weierstrass), but I have now learned something I wouldn’t have otherwise known. Not just digging in my heels and upping the ante by saying “those journals aren’t rigorous enough to budge me from my position” when the evidence started accumulating against my point of view led me to new knowledge and a slightly more open mind.