Is it ever harmful to submit extra recommendation letters from somebody outside of my department?
I read about 200 graduate applications a year. In my department we take a fairly dim view of people who do not conform to the specified application process. For one, it displays a wilful ignorance of the departments' requests, which are typically not arbitrary. Secondly, it can be construed as arrogant—as if you think you somehow deserve or require something that everyone else does not get. Lastly, each letter of reference takes about two or three minutes to read. If every candidate was permitted to flout the rules then our admissions panel members would each have to spend an extra ten hours reading redundant recommendation letters.

Others may take a more relaxed view of course, but I would urge you to submit exactly the material requested by the departments to which you are applying. That includes complying with the requested length of any research proposal, etc.

As far as letters are concerned, select whichever three you believe will make the strongest case for admission to your target programmes. It may well be that the optimal strategy is to send two mathematics letters along with the philosophy one. After all, getting a PhD is about more than subject knowledge: it requires originality/creativity, self-discipline and motivation, ability to work with the literature, etc. All of these are strengths that the philosophy professor could possibly comment on. A good strategy might be to ask a neutral party—such as the head of admissions in your current department—for advice as norms vary from discipline to discipline.