I have been attempting to clarify the difference between an axiom and a postulate, but I keep running into almost contradictory answers. Some will say they're equivalent, some will say a postulate may not known to be unprovable (cannot be derived?) while axioms are? Some will say it depends on the field of science and one is used over the other due to historical context. But I have noticed that most often, it is said to be the 'Postulates of Quantum Mechanics' and not the 'Axioms of Quantum Mechanics'. Can someone clarify for me why that is?
As this question was made a duplicate of a new one, I will add an answer.
Mathematics is axiomatic, it is a self contained theory, that has axioms and then theorems, dependent on the axioms, in a closed system with very many valid solutions. Axioms can be turned into theorems by making a theorem an axiom and that does not affect the mathematical solutions.
Physics uses mathematics in order to fit measurements and observations. This means it needs statements that will pick up from the mathematical solutions , those solutions that fit data and observations and, very important will be predictive of new data and observations, plus define the relation of physical units to the values calculated by the mathematical formulas.
These axiomatic statements have various names, because they developed historically: principles , laws, postulates,.... For example, the postulates of quantum mechanics ( a formulation here) pick the subset of the huge number of solutions of the mathematically rigorous wave equations, that can describe measurements of quantum effects and predict new ones.
I think both answers are slightly unsatisfactory as they stand. Here is an extended comment/answer.
You may wish to begin by consulting the words axiom and postulate in this excellent website https://mathshistory.st-andrews.ac.uk/Miller/mathword/a/. Note how there has historically been a blurring between the two words.
Heath translates Euclid's elements has having both postulates (also translated as common notions) as well as axioms. The latter appear to be logical identities, whilst the former describe what is assumed to be possible.
The OED describes
- Axiom as a proposition which commends itself to general acceptance. Crucially however, it also gives a 2nd meaning to axiom in the context of logic: a proposition (true or false), or requiring no demonstration to prove its truth.
- Postulate as a thing demanded or claimed. Again, it gives a second meaning in logic: something to be assumed or granted, or b) an unproved assumption or c) a self-evident fact (it says 'hardly distinct from axiom'). In geometry, as something which is possible (as opposed to axiom, which is a self-evident theorem).
[I am paraphrasing and cutting drastically, so do consult the OED yourself for complete definitions].
Clearly there is a lot of murkiness here. But there is a case to be made that one should use the word postulate for an observation which may prove to be incorrect, whilst axiom should be used in the context where its truth value is irrelevant. (This is in line with the OED).
For example, in constructing axiomatic QFT, you are in a context where you want to try to make mathematical sense of some framework arising in physics. So you want to use axioms for the same reasons as a mathematician uses axioms - you are constructing a framework whose truth value is irrelevant, one just wants something consistent. In other words, one should probably use the word axiom in contexts where we are not interested in declaring the principle to be actually true in nature.
In contrast, I would reserve the word `postulate' for physical laws which are asserted as being true in nature; thus postulates in principle ought to be able to be disproved. For example, it makes more sense to declare the principle that nothing can travel faster than light to be a postulate of special relativity rather than an axiom - in the context of experimental physics it can in principle be disproved. But if you are constructing a mathematical framework of relativity you might instead want to introduce an axiom that spacetime is Minkowski.
It is all a bit subtle and context is the key. I think it would be particularly interesting to study the history of the two words.
To my understanding, an axiom is a statement that something is the case. A postulate is more like an assumption that something is the case.
When you are talking about a physical theory meant to be predictive of reality built on experimental and theoretical work (i.e. Quantum Mechanics), I don't think it makes much sense to use axioms since then you'd be claiming that reality is such and such way.