What should philosophers know about math? I know, I know... this is not a technical question. Nevertheless, I believe this is the right place to ask such question.  
I am sure many of you read about philosophy, including philosophy of science, ethics and so on. If you read works from the ancient times, especially those written by non mathematicians, you may find mistakes simply because they used to think on the science the knew at the time. Actually, until a couple of hundred years ago, most natural philosophers (if not all of them) were scientists as well.  
My question is:

Does the lack of knowledge about math or science in general, have an impact on a philosopher's ability to reason about the world and figure the right path to follow?

I know many philosophers or at least, people who have a degree in philosophy (either master or phd), and I find that they have no basic understanding of very simple facts, from physics all the way down to mathematics. For example, many of them would guess that, even in absence of friction, an object heavier than another would reach the ground faster. But, again this is only one of the many examples. I'm not even going to mention absence of both knowledge and intuition about markov-based processes, probability in general and so forth.  
What is your opinion about this. Do you think philosophers should have a deeper basis in natural sciences in order to better understand the world and develop better ways of thinking?
 A: A short and dirty answer: they should read Imre Lakatos' Proofs and refutations (Google books), to get an idea of how mathematics develops/is built/is discovered. It is little use delving into the thornier issues of (in)completeness, for example, without knowing how mathematicians think about and develop their subject.
The book uses the simple example of the Euler characteristic of a (convex) polyhedron, usually expressed by the formula $V - E + F = 2$, and proceeds by dialogues. I don't believe any huge leaps of mathematical understanding are needed.
A: I think that those who claim to be philosophers of science or math should have had some real experience with doing research in those fields. It seems that a first-rate philososphy of math can only come from a practioner of the discipline. (I would suggest the same for philosophy of science.) Given this, a basic training would involve graduate work in those disciplines, not philosophy by itself.
A: This commits a complex question fallacy. It presumes that philosophers are ignorant of mathematical and scientific issues pertinent to their work. We are not. The recent literature on structuralism in the philosophy of mathematics and the philosophy of science, for example, contains deep engagement with category theory, quantum field theory, and other deep and difficult mathematics and scientific topics. A case might be made that some traditional epistemologists should learn more probability theory, but recent years have seen an ascendant formal epistemology that is deeply informed by probability theory and an increased interaction between traditional and formal methods. Just about every semi-important epistemologist these days knows at least a bit of probability and is familiar with Bayes theorem. To be sure, there may be some philosophers who don't find gravity or Markov processes really pertinent to, let's say, GE Moore's open question argument in meta-ethics, but then there are physicists, no doubt, who believe silly things about ethics. I object to the presupposition of this question that philosophers are not, to the extent that it is pertinent, learning the things we need to from mathematics. A better question (which I have begun to answer) is not what should but rather what are philosophers drawing from engagement with mathematics.
A: I do not see math or physics as a requisite for a philosopher. For example, one can look to the ancient eastern philosophers and see that they possessed a profound perception of the world without any knowledge of modern physics or mathematics. And for those not as philosphically inclined (as well as those suitably apt), knowledge of modern mathematics and physics can only be a good thing, as they are languages of nature. However, I do not see one's philosophy being hampered by lack of knowledge of the fact that rocks of a different size dropped from the top of the tower of Pisa should hit the ground roughly at the same time.
A: I wish some day a philosopher would come who would make my naive ways and habits of thinking more efficient but the more I look at their works, the more I am aware of the abyss that separates "natural sciences" and, say, "epistemology". I want to know and they want to know what it means to know. I ponder whether some particular way to approach the problem is right or wrong and they ponder over what "right" and "wrong" might mean. Sometimes it seems to me that, despite we (occasionally) use the same words, our languages describe two non-overlapping parts of "reality". 
This observation makes me somewhat disagree with David: the philosophy of mathematics has about as much to do with the mathematics itself as history has to do with ruling a kingdom and as car reviews have to do with the engine manufacturing or winning a car race. A good historian may be a dismal warrior or organizer. A top journalist for "Car and driver" may have a very faint idea of how the internal combustion engine works and may skid into the ditch on the first patch of ice when driving any of the cars he recommends for Winter driving. So a graduate degree in mathematics is neither necessary, nor sufficient for a philosopher of mathematics, IMHO. 
What is crucial for the outside look the philosophy takes (or, at least, pretends to take) is not the knowledge of the nitty-gritty of the thing under consideration but the knowledge of the place of that thing in some larger entity. A mechanic doesn't care about the atomic structure of the metal the pistons are made of but he makes sure they are in the right position when assembling the engine. The driver may have only a remote idea of how to assemble the engine but he rings an alarm if the response to controls is unusual. And so on. The philosophers are (or, at least, seem to be) near the top of that "chain of blissful ignorance". I have no more ability to see what they are looking at than a metallurgist has to see how the aluminum sheet he produced is connected with other sheets on the plane wing or the engineer has to see how the air traffic controllers direct the plane to a safe landing at JFK. Like any craftsman, all I can do is to say "I've made this thingy. It needs this to operate and will give you that if you use it in such a way". 
This brings me to an uneasy question of what exactly is that larger entity that is seen by philosophers and where our craft fits in it. I guess the best thing to do would be to let the philosophers answer that themselves. Once that answer is given, we can, probably see much better which sides of mathematics have importance to them and which don't. Without that, we just risk to have an attitude of a miner who insists on a jewelry expert's being at ease with a sledgehammer.
Note that I am not talking about (ab)using mathematical methods in philosophy at all here. It is a separate story having nothing to do with the subject I tried to address in my unsophisticated English. All I'm really saying is that the word "mathematics" has a different meaning for them altogether and I'd better figure out what exactly it is before giving any opinion on the issue.  
