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I was wondering if there was a way to prove the invariance of the space-time interval just from postulating a constant speed of light and an isotropic space-time. Most arguments go "from the Lorentz transformations it can be seen that the interval is invariant." Can we show the invariance without first appealing to the Lorentz transformations? Thanks.

I ask because many texts define L.T. as linear transformations that leave the interval invariant. But if we use L.T. to show the invariance, then the observation of invariance is trivial since that's how we definied our L.T. So I guess I'm looking for an independent proof/argument.