A chap called Jim Wiseman at some university in Georgia (the American one) has collected together some quotations from linear algebra class and some of them are rather lulsome.
I particularly like
See if you can use this proof to show the square root of three is irrational. Then try the square root of four. If it works, you did something wrong.
There is great wisdom hidden in that statement and it works at a research level as much as with homework problems. It’s very often you think you have proven something (give or take some trifling details to fill in) but then you realise that the same argument would prove something known to be false.
Hopefully you can then pick out why it fails, note what is different between the two cases and mend the argument. Normally this doesn’t happen; normally you just end up banging your head on the desk and crying. All the same, spotting that you have apparently proven a contradiction is a fairly good way of moving forward with a problem and not staying stuck on an idea that can’t work.
This was put nicely by Tony O’Farrell, in the following form:
I consider the day wasted in which I have not proven that zero equals one.*
To see how this is the same you need to know that (in “ordinary” logic) if you were to prove one false statement true (i.e. prove a contradiction) you can infer the truth of ANY statement. Hence we can pick “0=1” as a canonical false statement and refer to all apparent proofs of false statements as “proofs that zero equals one”. I guess mathematicians’ humour can be kind of odd (see Dolphin’s law but note that all the really funny Gauss facts are mine).
Of course this is a rule that doesn’t only work in mathematics. If the an apparently sound argument in any setting can prove nonsense by substituting different terms into it (assuming the specific choice weren’t important to the logical form of the argument) then the argument is flawed. So if your proof that God exists also show the existence of a perfect island, it isn’t really a proof of anything, even if it’s hard to pick out where it fails. There are plenty of day-to-day examples as well; if you refuse to go to one to retailer because you know of bad ethical practices but go to their competition who behave in the same fashion, you’re probably not acting sensibly. (I confess, I’ve found myself doing this).
In short. I think it’s a good rule of thumb to keep with you: “does my thinking here also show that zero equals one”. It can help you to be wrong slightly less often
* Thanks to my PhD supervisor Joel Feinstein for recounting this to me; it’s helped defeat despair many times.