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.

Solution 1:

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.

Solution 2:

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.