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If I were asked to give an example of an axiomatic mathematical theory, I'd be able to answer: set theory, probability theory, maybe group theory (assuming the elements of the definition of a group can be considered as axioms).

But it seems more difficult to give an example of an axiomatic physical theory.

I'm not necessarily talking about some " grand theory" aiming at unifying the whole field of physics. To the contrary, I'm looking for some basic and modest physical theory that starts from rather easy to understand axioms.

The 2 only examples I know do not really satisfy this requirement : Newton's Principia and Einstein Special Relativity Theory. ( Is Thermodynamics axiomatic? I couldn't say, though its fundamental laws are often referred to).

Is there a " little" physical theory dealing with only a small part of physics (say optics, or electricity, etc.) that one could give as a sample of natural science that develops deductively?

In case there is, what are its axioms? and what does the derivation of a theorem look like in this theory?

Note : I'm not asking for a totally formalized theory.

I'm thinking about examples like this :

https://www.researchgate.net/publication/259693325_Axiomatic_Reformulation_of_Maxwell%27s_Theory_of_Electromagnetism_Axiomatic_Reformulation_of_Maxwell%27s_Theory_of_Electromagnetism>

though this one is mathematically too sophisticated for me.

Qmechanic
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Floridus Floridi
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  • Would you say Newton's Laws count as "axioms"? – Nihar Karve Nov 22 '20 at 16:49
  • With very important but rather few exceptions, most physical theories can be axiomatized to the extent that you can write down and prove theorems. Ex: quantum mechanics with its postulates as axioms fed into operator theory. The example you provided is NOT how this is done. – BB681 Nov 22 '20 at 16:53
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    I believe that optics is a very good example. Originally it developed rather inductively, eventually explaining many different observations by Maxwell equations. Except from quantum optics and technical details, all optics can be axiomatically built from few postulates. – dominecf Nov 22 '20 at 17:05
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    Related: https://physics.stackexchange.com/q/87239/2451 and links therein. – Qmechanic Nov 22 '20 at 17:10
  • @NiharKarve. - I'm not sure at all that modern physics considers Newton's laws as axioms. In his Principia, Newton states them under this heading. – Floridus Floridi Nov 22 '20 at 17:45
  • This is going to strongly depend on what you count as qualifying for 'axiom' status. Your post at the moment sounds like the qualifying feature is just being called an axiom rather than some other word (ie you think Principia is axiomatic but are hesitant about group theory). – jacob1729 Nov 22 '20 at 22:03

2 Answers2

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It is my understanding that ultimately the realm of theories of physics is ill suited to axiomatization.

There is an answer by stackexchange contributor knzhou about derivability in physics that I think happens to touch well on the axiomatization issue.


In the wider realm of abstract thought the demarcation between mathematics and not-mathematics is that we qualify something as mathematics if any form of self-contradiction is disallowed. Example in geometry: spherical geometry, euclidean geometry, and hyperbolic geometry are incompatible with each other, but each is subject to the constraint that it must be free from self-contradiction.

It would appear that the constraint of being free from self-contradiction is what gives rise to axiomatizability.

(Then again: even when you do have axiomatizability, there is still freedom to exchange axiom and theorem without changing the content of the system. In general: which statements to classify as the axioms and which to classify as theorems is a judgement call.)


Physicists assume that fundamentally Nature is self-consistent, from that assumption follows an expectation that properties of Nature can be mathematically modeled. As we know, in the history of physics this expectation has been overwhelmingly corroborated.

In physics, in order to make progress, one must allow for tentative theories to be incomplete, or even apparently self-contradicting. A particularly striking example of that is the Bohr atom. The Bohr atom accounts for many of the known properties of Hydrogen, including accounting for the Balmer series.

As a theory of physics the Bohr atom is painfully incomplete, but given how powerful it was the Bohr atom was clearly doing something right. For the physicists of the time the challenge was on to try and understand what it was that the Bohr atom was doing right.


I believe that while the Bohr atom is a particularly striking example of a theory that is powerful even though incomplete, in general you never know in advance at what level of completeness the current reigning theory stands. I believe it isn't possible either to know in advance whether the current reigning theory is fully self-consistent.

Physics theories do not need to get everything right, it is sufficient to do something right, and do that right in a way that is empowering.


In physics, the concept of 'axioms' has largely a narrative function. When teaching a particular theory what is referred to as the 'axioms' are what are regarded as the core ideas. It is a way of giving focus to a narrative.

I do believe that what in mathematical physics is referred as 'axioms' of a theory are very important, but they don't serve the same function as in mathematics.

Cleonis
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One of the fruits of an axiomatic theory is that some results that were not obvious at first, comes up from the axioms and definitions.

The classical electromagnetism is a good example. The Maxwell equations + Lorentz force law can be regarded as axioms. The electromagnetic radiation was derived from them, and soon later confirmed by experience.

Moreover, the notion of the speed of light in the vacuum as a constant was also embedded in the same axioms, and leads to SR.

Claudio Saspinski
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