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Non-commutative geometry deals with spaces where the uncertainty principle of quantum mechanics thwarts even one's ability to simultaneously measure two position co-ordinates. It finds applications in models where geometry is emergent including the matrix model approach to string theory.

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Is the quantization of gravity necessary for a quantum theory of gravity?

Jonathan, the extension to your question makes your original question far more clear. Your ideas are in line with what many people have thought and are thinking about quantum gravity. My take on th …
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