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Is there any difference between an axiom and a postulate?

I would say that while an axiom is any proposition fixed as true without premises in a theory, a postulate is (also according with its etymology postulatum = something that is requested) something that is necessary to assume to make a theory fit (also in a consequential way) with some kind of experimental measurement. Is there any convention about it?

Qmechanic
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anna
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    that does not seem to be question about Physics – aaaaa says reinstate Monica May 05 '23 at 19:53
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    Related: Can I say Axioms of Quantum Mechanics instead of Postulates? – Ghoster May 05 '23 at 19:54
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    @aaaaasaysreinstateMonica : I interpret this as a question about how physicists (as opposed to, say mathematicians or philosophers) generally use certain words. That's not strictly a physics question, but it seems as on-topic as, say, a question about what some standard notation means. (On the other hand, the answer in this case is probably that there is no standard convention.) – WillO May 05 '23 at 20:43
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    @WillO But to be honest, for that there would need to be a reason to assume the meanings are different across these fields. Honestly, whether a mathematician talks about that on a math platform or an engineer on a physics platform in an argument with a chemist… makes no difference. So, we might as well address it here. – Marcus Müller May 05 '23 at 21:14
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    @WillO, no, this is a convention that is argued between mathematicians and physicists and is generally accepted as the correct words to use. Postulates are the assumptions of physics, and so need to be as simple and as small as we can pick them to be. They can be invalidated by new experimental findings, and we would have to rebuild new theories when that happen. Axioms can often be provably correct, e.g. group axioms, or "must be correct just so you can talk about stuff" e.g. equality and transitivity axioms. – naturallyInconsistent May 06 '23 at 02:56
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    My answer to a similar question here might help https://physics.stackexchange.com/questions/266077/how-do-we-know-that-energy-and-momentum-are-conserved/266085#266085 – anna v May 06 '23 at 05:48
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    please see the duplicate, I added an answer – anna v May 06 '23 at 06:12

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Axiom is rather well-defined. It's a unprovable cornerstone statement of a logical system. You cannot disprove it – if you can show an axiom is wrong with the tools from the logical system it founds, you didn't disprove the axiom, but the logical system (or made a mistake).

A postulate is something you mentally work with as a "given" to form a tractable hypothesis or derive a falsifiable result, prior to being able to prove it.

So, a postulate can turn out wrong (in which case it's simply a fallacy) or right (in which case it becomes a theorem).

In a practical context, we often do not inspect the truth of a postulate any further – it's a didactic tool that we often use as shortcut as in

The apple falls from the tree in a straight downwards line; you can prove that with Newtonian mechanics based on gravity, but this is 8th grade of school so maybe stick with us while we learn how a ball flies when thrown horizontally.

"The apple falls from the tree in a straight downwards line" is definitely not an axiom. No system I'm aware of needs fruit dynamics as fundamental statement to be somewhat complete. We postulate it does, because as soon as we look closer, things just become intractably complicated within the context we're working in.

Marcus Müller
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