My underlying question is "what counts as a controversy"
"A state of prolonged public dispute or debate." How prolonged? How much disputed? Look at the various disciplines I listed and see how they compare. Agreed, for mathematics, Appel-Haken was a controversy. Compared with politics, it was animated conversation over afternoon tea at the vicarage. Also, judging from the Wikipedia account, the controversy progressed steadily to a resolution.
how would I go about checking the facts of your claim
If you want numbers and experiments, obviously I haven't done any of that, but just recounted what it seems to me that I have seen. You, or someone, would have to work out an objective measure of the existence and intensity of a controversy, and survey publications in various disciplines. I don't know if you could devise a method of detecting controversies just from citation patterns, but the more you could automate this the easier it would be to collect data.
Richard, do you think Pearlian causality is mathematics or something else? Because I think Pearlian causality is extremely controversial by your definition (to be fair not as a piece of math, but how applicable it is to practical scientific problems).
Half-closing my eyes and looking at the recent topic of morality from a distance, I am struck by the following trend.
In mathematics, there are no substantial controversies. (I am speaking of the present era in mathematics, since around the early 20th century. There were some before then, before it had been clearly worked out what was a proof and what was not.) There are few in physics, chemistry, molecular biology, astronomy. There are some but they are not the bulk of any of these subjects. Look at biology more generally, history, psychology, sociology, and controversy is a larger and larger part of the practice, in proportion to the distance of the subject from the possibility of reasonably conclusive experiments. Finally, politics and morality consist of nothing but controversy and always have done.
Curiously, participants in discussions of all of these subjects seem equally confident, regardless of the field's distance from experimental acquisition of reliable knowledge. What correlates with distance from objective knowledge is not uncertainty, but controversy. Across these fields (not necessarily within them), opinions are firmly held, independently of how well they can be supported. They are firmly defended and attacked in inverse proportion to that support. The less information there is about actual facts, the more scope there is for continuing the fight instead of changing one's mind. (So much for the Aumann agreement of Bayesian rationalists.)
Perhaps mathematicians and hard scientists are not more rational than others, but work in fields where it is easier to be rational. When they turn into crackpots outside their discipline, they were actually that irrational already, but have wandered into an area without safety rails.