There are such things as mathematically necessary truths: 1=1, say; and logically neccessary truths: the law of modus ponens, say.

But can there be one in physics?

In Lewis's plural worlds where worlds can have possibly different physical laws; one might suggest that a neccessary law holds when it holds for all worlds.

A suggestion though presents itself from Aristotles Physics, in his investigation of change; that is a motion requires a mover; which can be an other (that which moves is moved by something else; for example, a push of a pendulum) or it can be reflexive (it keeps moving).

Can there be a possible world where this does not hold?

That this has occurred to someone suggests itself by the paradoxical question: what happens if an irresistable force meets an immovable object.

Note: It is probably worth pointing out that Mach considered Newton's First Law was a 'tautology' (Robert Desilles essay on Newton's philosophy space and time).

This cannot be the same sense of tautology in logic, philosophical or mathematical; or otherwise. I'd suggest it is stripping a notion of as much contingency as possible.

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    The law of the excluded middle is an axiom of two-valued logic. But it does not hold in many-valued logics. Hence it is not necessary for a consistent logic calculus. – Jo Wehler Aug 22 '15 at 16:42
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    Motion is a relative concept. A body at rest in one coordinate system may have non-zero velocity with respect to a second system of coordinates. In post-Newtonian physics there is no absolute space which could serve as a point of reference for motion. – Jo Wehler Aug 22 '15 at 16:47
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    That's false. Motion means "being absolutely accelerated", case closed. – sure Aug 22 '15 at 17:52
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    @wehler: agreed; however I'm distinguishing motion intrinsically; that is by rest (unaccelerated motion) and motion (ie accelerated); these are frame (coordinate) independent concepts (so long as one uses inertial frames, for general ones it won't hold). – Mozibur Ullah Aug 22 '15 at 19:36
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    "Motion means "being absolutely accelerated"", while that is what Mozibur Ullah meant, that is not how Aristotle defined motion, making the confusion understandable. – Cicero Aug 22 '15 at 20:43

Strictly speaking there are no absolute necessities in physics. But strictly speaking there are no absolute necessities in mathematics and logic either. Mathematical theories have axioms, necessity of conclusions is relative to them, and to logic used. The law of excluded middle is rejected by intuitionists, the law of non-contradiction by dialetheists (see inconsistent mathematics), and although it is rare even the identity law a=a is sometimes rejected as well ("we do not step into the same river twice" - Heraclitus).

But notice the dependence, mathematics is necessary relative to the logic adopted, similarly physical theories are necessary relative to mathematics and logic adopted. For example, if Newton's laws are adopted as axioms then conservation of mechanical energy and momentum become necessary truths. There are more subtle necessities. Newton's laws are only valid in "inertial frames", so the entire theory is empirically meaningless without the presupposition that inertial frames exist. This presupposition is then a necessary law relative to Newtonian mechanics. When Einstein sought to discard it, he had to discard all of Newtonian mechanics along with it.

Reichenbach named such presuppositions relativized a priori, and Friedman developed a whole theory of them to analyze logical structure of scientific theories. There is a long tradition behind it involving Kant, Marburg neo-Kantians, Reichenbach, Carnap and Kuhn. Thus, a theory is roughly stratified into empirical claims, that are directly testable, coordinating principles, that relate them to theoretical predictions (like existence of inertial frames), theoretical principles needed to derive predictions (like laws of motion and Euclidean geometry), mathematics (like calculus) and logic. Coordinating and theoretical principles are the relativized a priori, they are not directly testable because they need to be assumed within a theory to produce claims and relate them to empirical tests.

But they are tested by a theory's success overall, and therefore historically revisable, dynamic. Interestingly, what may happen under revision is that a mere empirical fact of an older theory is elevated to a relativized a priori in a new one. For example, Friedman characterizes the equivalence principle as a relativized a priori that makes general relativity empirically meaningful, replacing the assumption of inertial frames, whereas in the Newtonian mechanics is was only an experimental brute fact. Also, revised theories satisfy a downward correspondence principle (inspired by Bohr's, but more elaborate): the older theories can be emulated within them as limiting cases but without the stratification, their relativized a priori may be discarded (absolute space/time) or downgraded to approximate claims (inertial frames). In this way Friedman avoids Kuhn's incommensurability of paradigms, and the resulting relativism.

See his Einstein, Kant, and the Relativized a Priori for a short version and the book Dynamics of Reason for a long one.

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    +1 for the insightful answer, especially for citing the Dynamics of Reason by Friedman. – Cicero Aug 22 '15 at 21:01
  • This "relativity" you're speaking about is actually really subtle: even though people are always free to come up with different axiomatic systems and different way of reasoning about what it means to deduce something (i.e., another logic), it seems that the results obtained from one or another systems are quite similar. In particular, notice that there's some kind of stratification of deepness of mathematical theories, in the same way general relativity is a deeper theory than classical mechanics. Deepness meaning that it explains things at a much more foundational level, and in a clearer way. – sure Aug 23 '15 at 8:11
  • 'Strictly speaking there are no absolute neccessities in physics': this is why I said 'more or less speaking'; the relativised a priori looks a useful suggestion in this respect. – Mozibur Ullah Aug 24 '15 at 3:05
  • @sure The depth you describe is reflected by Friedman's downward correspondence principle I think, see edit. He also stratifies into levels within a theory by presuppositional dependence (this replaces and enhances the old analytic/synthetic distinction). – Conifold Aug 24 '15 at 20:21

There are a few conservation laws that can be proved through Noether's theorem, if you only postulate some symmetries. Thus, if a systems behaves at the same way at all times, you have conservation of energy. If it behaves the same way no matter where in space we place it, you have conservation of momentum. This does not mean that conservation of energy is necessary in all possible universes, but it is as close that you will get to a proof in physics.

Ultimately, physics is a science that makes statements about the natural world; if there would be some axiomatic part of it, it would no longer be physics, but mathematics.

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    Yes, Noether's theorem holds true because - being a mathematical statement - it is proved. But to apply Noether's theorem in physics, one has to verify the existence of a differentiable group of symmetries. There is no a priori reason why such symmetries should exist. On the opposite, the violation of parity symmetry in weak interactions shows that symmetries do not necessary exist, even at a fundamental level - for the sake of argumentation one can neglect that here the group is discrete. - I agree with your last passage, +1 upvote :-) – Jo Wehler Aug 22 '15 at 22:37
  • 'Ultimately...physics makes statements about the natural world'; true but see additional note. – Mozibur Ullah Aug 23 '15 at 6:56

In physics as we know it one did never prove any general theorem. Instead, successfull scientific theories are hypotheses confirmed by correct predictions in many cases. But no scientifc theory is protected from later falsification.

Of course one can imagine different possible worlds with different physical laws. The most simple case are physical laws with different values of our fundamental physical constants. But I do not see any reason why a certain physical law should hold in all possible worlds.

As remarked in my comment I do not consider Aristotle's example correct from the viewpoint of today's physics.

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    @sure Aether theory is not as predictive as special relativity, as shown by the Michelson–Morley experiment. Einstein himself stated that "No amount of experimentation can ever prove me right; a single experiment can prove me wrong." While a physical theory does not merely make predictions, a critical part of a physical theory is to make predictions later confirmed (or rejected) by experiment. – Cicero Aug 22 '15 at 20:14
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    This is why we adopted QM over Newtonian mechanics; because the former theory made more accurate predictions. And for large scale phenomenon, we adopted General Relativity because it made predictions more accurate than Newtonian theory (for instance in regard to the Mercury perihelion). – Cicero Aug 22 '15 at 20:14
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    There are numerous sources of that quote, which really is a paraphrase. However, here is a better quote straigh from Einstein's paper Induction and Deduction: – Cicero Aug 22 '15 at 20:30
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    @sure I know of the Quine-Duhem argument, that is why bundles of hypothesis have to be disproven. Now, as for complexifying models, I completely agree with you, which is why scientists prefer testable theories that are easily falsifiable (read the wiki article on Occam's razor and 1st and 10th source which cite a SEP article and an arvix article). What is ironic is that your sentence mirrors many statements made by Popper, i.e. you seem to be going down a similar line of thought that Popper did.You should read the comments with quotes from Einstein's paper, on falsifiability and predictions. – Cicero Aug 22 '15 at 20:55
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    @sure How are you supposed to make sense of the world if you don't check your theories match reality (ie. experiment)? Perhaps the sticking point here is whether physical theories describe reality or whether they just happen to produce the same results as reality? Either way, this is way beyond comments for this answer and should be taken to its own answer, question or chat. – Schwern Aug 23 '15 at 4:09

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