The project I’m working on with David Brown and Bryden Cais is the first thing I’ve ever done involving computational experiments as a serious part of the development of the ideas. We’re trying to formulate a reasonable conjecture about the limiting distribution of some arithmetic objects, and I thought we’d arrived at a pretty good heuristic — call it conjecture A — which fit nicely in a context with other conjectures and theorems about similar objects.
But the data we’d collected didn’t fit conjecture A very well. In fact, when we looked at the data carefully, it appeared to be pointing strongly at conjecture B, which has a similarly clean formulation but for which we didn’t have any theoretical justification.
So I spent much of the weekend thinking about this, and by the end of it, I felt pretty confident that conjecture B was pretty reasonable — even, in a way, more reasonable than conjecture A — though we still didn’t have a strong justification for it.
Today it turned out that there was a mistake in the data collection, and conjecture A looks good after all. But it’s a sobering reminder that my intuitions about “what ought to be true,” which I think of as rather rigorous and principled, are in fact quite malleable.