Poincaré conflated mathematical consistency with physical applicability; competing geometries may be logically compatible but yield different experimental predictions about real phenomena.

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Conflated(as used in logic and argumentation): Mixing up two different things or treating them as if they're the same when they're actually distinct.
Logically compatible(describing the relationship between decomposability and different ethical systems): Able to exist together or work together without creating a contradiction or logical problem.
Poincaré(as the scientist whose work is being referenced): Henri Poincaré was a French mathematician and physicist (1854-1912) who discovered that tiny, unmeasurable changes in a system's starting conditions can lead to completely different outcomes—a key insight that helped create chaos theory.
competing geometries(as the example of systems that can both be logically sound but give different real-world predictions): Different mathematical systems for understanding space and shapes that contradict each other—like Euclidean geometry versus non-Euclidean geometry.

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