Is there a Connection Between Greatness in Math and Philosophy?

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.