I don't think there's any place other than math where you can learn how unbelievably creative you get to be in a purely deductive setting. Here are two of my favorite examples:
100 ants crawl left or right on a meter stick at 1 cm/s. When two ants meet, they bounce and both change directions. Ants fall off when they reach the ends. Show that regardless of the starting distribution of ants and directions, they all fall off the stick by the end of 100 seconds. Solution: vaqvivqhny vqragvgl vf na vyyhfvba.
A regular hexagon with side length is tiled with diamonds of side length (a diamond is two equilateral triangles glued along a side.) Show that there will always be the same number of diamonds in each of the three possible orientations. Solution: gur Neg bs Ceboyrz Fbyivat ybtb.
My first thought was your second option, that they both require high IQ and thus being good at one lets you be good at the other.
However, I didn't agree with the prediction that you'd find this in many other pairs of fields, as many fields require getting a lot of background knowledge before you can contribute. What I'd predict here is that (a) this is fortunately/unfortunately less true in philosophy so mathematicians can easily get to the forefront, and (b) top philosophers who're interested in mathematics do not do the sort of mathematics that requires many years of training (e.g. they would talk more about what mathematics means, and what it means to prove something).
If you'd found a trend where the top philosophers had been contributing to cutting edge math research that had typically required multiple years of training, that would change my mind (and surprise me).
Training in different disciplines will teach you different skills. Maths is great for leaning to think precisely as if you mess up, someone will be able to show you a formal proof of why you are wrong. In contrast, it is much harder to learn to think precisely purely by studying philosophy as the claims and arguments are less well defined. It's very hard to get the logic flow clear enough that someone could formally prove that your argument is either logically sound or logically unsound. Without the same feedback loop, your ability to think precisely simply won't advance as fast.
However, someone who has trained extensively in maths might face the opposite problem of being stuck in that paradigm. Perhaps they want ever claim to be put into a formal model before they are willing to consider it, without understanding that this is too much of a burden for some areas of philosophy and that they would be missing out on valuable lessons.
So I expect this would lead to a bipolar distribution where some mathematical philosophers crash and burn, whilst others soar.
4) The skill to produce great math and skill to produce great philosophy are secretly the same thing. Many people in either field do not have this skill and are not interested in the other field, but the people who shape the fields do.
FWIW I have reasonably strong but not-easily-transferable evidence for this, based on observation of how people manipulate abstract concepts in various disciplines. Using this lens, math, philosophy, theoretical computer science, theoretical physics, all meta disciplines, epistemic rationality, etc. form a cluster in which math is a central node, and philosophy is unusually close to math even considered in the context of the cluster.
What do you think of the idea that mathematics is exactly philosophy with consistency and precision carried to logical extremes?
I guess I'll just add that I made the transition from mathematician to philosopher and, for what it's worth, the things I learned to be a mathematician do prove useful in doing philosophy in that without mathematics I would not be able to think and write as precisely about ideas as I do. I'm not sure how much correlation there is between the two fields, though, beyond it being typical that people smart enough to do one are smart enough to do the other.
I have nothing that looks like an answer, but I wanted to put the question out to start a discussion. I especially hope that this leads to someone here researching the question well. I also have almost nothing to back up the claims below.
I became curious about this from thinking about how to find people who can make progress in MIRI's Agent Foundations questions. It seems to require very high levels of both math and philosophy, and I want to know how to select for both of these skills simultaneously.
It seems like many of the great mathematicians at least had enough of an interest in philosophy to share their philosophical opinions. Also it appears that the people who substantial progress in philosophy often use math. (There are of course disagreements about what makes good philosophy, and I am biased in favor of the mathematical flavored philosophy.)
I am especially surprised by the fact that the tails seem to come together rather than apart. It seems like mediocre mathematicians and philosophers are much less likely to be interested in the other field.
Some possible theories:
1) I am noticing a pattern that is not there.
2) This is a consequence of the fact that all intelligence is correlated, and you will notice a similar pattern between many pairs of fields.
3) Mathematics and Philosophy are very old fields. Historically, fields did not really exist, and people were more interdisciplinary. If you look at modern mathematicians and philosophers, the trend goes away.
4) The skill to produce great math and skill to produce great philosophy are secretly the same thing. Many people in either field do not have this skill and are not interested in the other field, but the people who shape the fields do.
Thoughts?