Switching to philosophy after a PhD in mathematics
Maybe you do not know what an undergraduate degree at an American university means.  You do not study, say, philosophy, you "major" in it.  Depending on the university and your own choices, you may take anywhere between 1/3 and 2/3 of your courses in the area.  You will also be taking basic courses on writing, social sciences, very broad courses in the sciences, and so forth, that to many well educated Europeans would feel like high school all over again.  The idea of getting a PhD and then coming back for an undergraduate degree would be regarded as crazy by most American universities.  I agree.

In general, yes, a PhD in subject X should get you out of having to do an undergraduate degree in Y and you should be thinking of enrolling directly in a PhD program in Y.  But forgive me while I pick apart your question a bit.


  I've always been torn between philosophy and mathematics, and I'd like to give philosophy a chance (now that I have the "safety net" of a PhD in mathematics).


The idea that someone regards a PhD in mathematics as a safety net is very amusing (or bemusing?) to me.  If you have always been torn between mathematics and philosophy and got a PhD in mathematics, surely you must have devoted some serious study to philosophy, right?  If the answer is "no" then I don't really understand the situation.  If the answer is "yes" then I hope you have been exploring the connections between mathematics and philosophy, which is something appropriate for someone considering doing multiple PhDs.  

If you want to study an area of philosophy with connections to mathematics (most obviously, philosophy of mathematics, but there are many other areas of modern philosophy which draw upon mathematics), then having a PhD in mathematics at a "world top 10 university" should be a tremendous advantage.  The level of mathematical expertise you bring will probably be superior to that of some of the faculty you would be working with.  Thus at the very least you should be looking for some kind of specially worked out, accelerated graduate program.  I would encourage you to also look for postdoctoral and other temporary faculty positions: these can be places to transition from one field to another.

The desire to start up an undergraduate program after having received a PhD at a top place is frankly a bit worrisome to me.  You've probably spent your entire adult life as a student, roughly 10 years of it.  You now want to turn around and spend another roughly more 10 years as a student?  That's about a third of your adult life.  Don't you have other things that you want to do?

Added: Although I do not think you should consider enrolling in an undergraduate degree program, if you really haven't taken any courses in philosophy at the undergrad level [by the way, in the US it would be really screwy for someone to have a lifelong interest in X and Y, do a PhD in X and never take an undergraduate course in Y; but maybe less so in Europe] then I think you should enroll in -- or audit, or whatever -- at least one or two as a non-degree student.  You just need some amount of assurance that academic philosophy is anything like what you think it is.