There was no controversy about Wiles' proof of FLT
It's the Appel-Haken 4CT proof I was actually thinking about, my bad. There was controversy about that not being a "proper" proof, as I recall, and it's been (unfavorably) compared to Wiles' proof in that respect (which helped me mix up the two - I'm no mathematician!).
My underlying question is "what counts as a controversy", and more directly "how would I go about checking the facts of your claim about the correlations between a field's distance to objective truth and proneness to controversy".
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, ...
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.