This word is acceptable for play in the US & UK dictionaries that are being used in the following games:

• Lexulous
• Wordscraper
• Wordosaur

n. The state or quality of being incompatible.

n. Mutually exclusive or antagonistic qualities or things.

n. The quality or state of being incompatible; inconsistency; irreconcilableness.

n. The quality or state of being incompatible; inconsistency; irreconcilableness.

n. The quality or condition of being incompatible; incongruity; irreconcilableness.

n. (immunology) the degree to which the body's immune system will try to reject foreign material (as transfused blood or transplanted tissue)

n. the relation between propositions that cannot both be true at the same time

n. the quality of being unable to exist or work in congenial combination

"In all of the written pieces I found Coyne specifically points out that the incompatibility is of a philosophical nature."

contradictoriness   

inconsistency    irreconcilableness    antipathy    inconsistence   

accessibility    advisability    availability    believability    collectibility    comparability    convertibility    credibility    culpability    deniability   

dissimilarity    antagonism    incongruity    dissimilitude    repugnancy    misunderstanding    disagreeableness    unfitness    impossibility    divergence