Maybe this is both mathematical and philosophical (and physical?)
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A thought experiment is a scenario you can imagine to help focus on a specific problem or issue. The point of a thought experiment isn’t to actually find a practical solution to a question, but to try to imagine an ideal situation that lets you isolate a process or interaction.
For instance, “Maxwell’s demon” is a thought experiment in which a demon controls a massless, frictionless gate between two containers and lets slow-moving molecules pass only in one direction and fast-moving molecules pass in the other: the result is that one side of the chamber gets hot and the other side gets cold, even though no work is actually being done. Where would you get a massless, frictionless door? Or a demon, for that matter? The point of the thought experiment isn’t actually about the demon, it’s about trying to isolate what it is specifically about entropy in a system that makes it increase with time.
A proof is a series of logically consistent statements that reach a conclusion based on given assumptions. You can use a thought experiment to zero-in on what it is you’re testing mathematically, but the thought experiment all by itself doesn’t really prove anything.
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