Is it ever harmful to submit extra recommendation letters from somebody outside of my department? I'm a senior applying for admissions to Ph.D. programs in mathematics. The departments I'm applying to all ask for three recommendations and encourage (but don't usually require) them to be from math professors. 

I have three math professors whom I plan to ask for letters, and I believe all of them will be strong, but perhaps not superlative.  In addition, I know a professor who is primarily from the philosophy department (whose interests are related to math) who I believe would write a stronger letter than the other three. (He supervised an unusual independent study course for me and it was fairly successful).

The subjects I studied with him were entirely mathematical (logic), but his own work is more distinctly on the philosophy side (philosophy of science/math and related topics, not mathematical logic) .

Could asking this professor to write a fourth letter harm my application in any way? I think the letter itself would represent me in a good light and highlight an unusual aspect of my resume, but I don't want it to replace a letter from a mathematician for fear of admissions committees looking negatively at a letter from someone in the wrong department. 

EDIT In response to the comment: I've worked closely will all three math professors. I've done long-term research projects with two of them (my university requires a junior and senior thesis; they were my advisors), and a year-long independent study with the third. I believe they were all reasonably happy with my performance, but I'm not sure I was able to blow them away. The independent study project I did with the professor from the philosophy department was more unusual, took more initiative on my part, and I believe I made a stronger positive impression.
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.