Luke_A_Somers comments on Can You Prove Two Particles Are Identical? - Less Wrong
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Consider: carbon. It has six electrons. If they are identical, none of them can be in the same state by the Pauli exclusion principle, and organic chemistry is a fairly direct consequence. If they are distinct, they all fall into an S1 orbital, and Carbon chemistry is just like Hydrogen chemistry but more so.
Would you say that's an observable difference?