Working at Cross Purposes

I’ve been reading the book How Not to be Wrong: The Power of Mathematical Thinking, by Jordon Ellenberg. Toward the end of the book, he briefly picked up one of my favorite themes, working to disprove hypotheses.  He started of with a quote from F. Scott Fitzgerald-

The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function.

I can’t vouch for that actually being able to sort out a “first-rate intelligence,” I don’t think it needs to be taken quite that literally to draw something useful from Fitzgerald’s statement. What we can take from the quote is a hint at the idea that we should be trying to both prove and disprove our hypotheses. One can only know in retrospect that work to prove, or disprove, a hypothesis was done in vain. “Retrospect” literally means “back looking.” The question is how much “back looking” one wants to risk needing to do. Working for years to prove something, only to learn that it was not true, is a lot of “back looking.” At the same time working to disprove something, only to find out that it really was true after all, means that time might have been better spent.

According to Ellenberg, the folk wisdom passed down from advisor to mathematics student is that one should work to prove an idea by day and to disprove it by night. Working like that might seem odd (and the day night split is not the mandatory way of handling it) but there are good reasons to work on hard problems this way. One I’ve already hinted at. It is a way of hedging one’s bets and reducing the time spent on the wrong thing. Another is that assuming that the attempts at disproof fail, those failures are informative. In genealogy they can start to build up negative evidence for your idea. If something could reasonably be disproven by land sale X, birth record Y or census entry Z and none of those things seem to exist, it might be telling you something. That thing could be that you’ve forgotten to check for probate records that would disprove your hypothesis, but it could also be telling you that there is very little wiggle room for your hypothesis to be wrong, and that is valuable knowledge. You have negative evidence for your hypothesis in that lack of evidence for other possibilities.

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