Kochen-Specker theorem: Nature is contextual

The most important theorems in the foundations of quantum mechanics, which is arguable the most central topic in the philosophy of physics, are Bell's theorem and Kochen-Specker theorem. These are the first mathematical theorems in all of metaphysics with Bell's theorem being the most important of the two since it applies to all interpretations of quantum mechanics suffering from small loopholes. Quantum metaphysics based on these theorems should then be properly thought of as experimental metaphysics.

Bell's theorem, which is called by  Stapp the most important discovery in science, is based on the two assumptions of locality and realism and it applies to all interpretations of quantum mechanics. A theory (hidden variables theory or otherwise) which satisfy these two assumptions will contain certain inequalities which are  seen to be violated in quantum mechanics and by Nature (experimentally). Thus, the two conditions of locality and realism are contradictory and typically people abandon the condition of locality., i.e. Nature is non-local


On the other hand, the Kochen-Specker theorem applies only and specifically to hidden variables theories of quantum mechanics and it relies on three assumptions:

1) Projection postulate which states that physical properties can be mapped one-to-one to projection operators on the Hilbert space. This simply states that a yes-no measurement can always be devised on a physical system to reveal whether or not it has a certain property.
2)Value definiteness which states that all observables have definite values at all times. This relies explicitly on the existence of a hidden variables theory of quantum mechanics and it partially reflects the reality condition. 
3)Non-contextuality which states that the outcome obtained in the measurement of an observable does not depend on the order, i.e. context in which that observable is measured. For example, if we measure the squared spin component in the x-direction as a part of the triad (x,y,z) or if we measure it as a part of the triad (x,y1,z1), where (y1,z1) is obtained from (y,z) via a rotation about the x-axis, we should get the same answer.

Kochen and Specker showed that the above three conditions are contradictory in any hidden variables theory of quantum mechanics. Typically, people tend to relax the conditions for the Kochen-Specker theory by dropping the requirement of non-contextuality. In other words, quantum mechanics and by consequence Nature seem to be contextual, i.e. the context of measurement is crucial.