What mathematical questions or areas have philosophical implications outside of mathematics? Please list both the problem/area and justify why it is important philosophically. This question doesn't cover questions that are only important within the philosophy of mathematics itself.
 A: Goedel's incompleteness theorem is used by some (e.g. Roger Penrose) as part of a justification for why computers will never achieve consciousness.
The fact that all infinite dimensional separable Hilbert spaces are isomorphic has philosophical implications for the metaphysics of quantum mechanics.
Various results in dynamical systems theory related to chaotic systems limit what can be said about predictability and about what it means for a system to be deterministic. For example this paper by Ornstein and Weiss (warning: it's huge and will take a long time to download on slow connections) has been used to suggest that the distinction between deterministic and stochastic systems is flawed.
A: Existence and uniqueness theorems for things like differential equations (the big canonical one being for ODE's) can be thought of as a philosophical foundation for a weak type of determinism.  
A: Game theory and fair division have implications of sorts for moral (and political) theory. The foundations of mathematics cut to the heart of logical possibility. Noam Chomsky sparked research into formal grammar, and while I don't have much knowledge of this area I believe it holds promising theory for work in the philosophy of language. The models of biological neural networks could hold implications for the philosophy of mind and consciousness. Computability theory is suggestive towards metaphysics (see e.g. Church-Turing thesis) and the philosophy of mind.
A: Wikipedia has a more detailed description for each one of them, therefore I will just list them and the main ideas.
Aumann's agreement theorem
Two people under certain prior conditions can not honestly disagree forever. In fact, Scott Aaronson have proved they don't have to exchange too much information to lead to an agreement. If the prior conditions are met and the disagreement lasts too long, then one side has to be dishonest.
Arrow's impossibility theorem
In short, there is no perfect voting system.
Free will theorem
Under certain assumptions, if we have free will, so does elementary particles.
Gödel's incompleteness theorems
There are statements in a sufficiently strong formal system that can't be proven true or false within the system. Some people use this to justify humans must be different from machines, since humans can prove theorems by using another formal system.
Tarski's undefinability theorem
Similar to the theorem above, it states truth in a sufficiently strong formal system can't be defined by that formal system. For people who believe people are machines, this implies people can't define truth.
The following theorems might be a stretch, but it looks like someone can use them in philosophical arguments.
CAP theorem
It shows there is no distributed system such that each machine store the same information, can operate while some machines are broken, and can operate even when some messages are lost.
Rice–Shapiro theorem
There is no algorithm to check if an infinite set have some non-trivial property. 
Shannon's source coding theorem 
The theorem states there is a hard bound on data compression.
A: Some examples were given in: Non-Scientific questions solved by mathematics
A: Mathematical Realism is philosophically relevant and (imo) important.
If mathematical realism is plausible (and true) then many things about the natural reality can be infered by mathematical reasoning.
Conversely if mathematical realism holds many open mathematical problems might be resolved in un-expected ways and provide technological progress in important areas of life (or death, of course it is up to the use)
