The withdrawal of philosophy into a “professional” shell of its own has had disastrous consequences. The younger generation of physicists, the Feynmans, the Schwingers, etc., may be very bright; they may be more intelligent than their predecessors, than Bohr, Einstein, Schrodinger, Boltzmann, Mach and so on. But they are uncivilized savages, they lack in philosophical depth — and this is the fault of the very same idea of professionalism which you are now defending.With some hesitation Carroll concludes that "I tend to think that knowing something about philosophy — or for that matter literature or music or history — will make someone a more interesting person, but not necessarily a better physicist." (See comment 56 by Lee Smolin and comment 64 by Craig Callender for some useful replies.)
Beyond that debate, it's worth wondering how knowing some science and mathematics helps the philosopher of science and mathematics. Pretty much everyone in these areas of philosophy would agree that it does help, but exactly how is probably a controversial issue.