Proofs in math and physics

Suppose we have the case of a proof in math or physics and we want to compare the status of the derived information. I know that in math mostly all derived information or deduced details are a priori. Now the point where I wonder is that in physics all or a lot of the rules are called laws and they seem to have some kind of connotation that is very similar to the status of deduced details in math.

Now my question is can there be such a thing as an "a priori"-law in physics? That is to say it is not derived from experience or experiment.

• Possible duplicate of Can there be a priori truths for science? Jul 1, 2018 at 5:43
• It seems unusual to talk about "Proofs" in Physics at all. Have you a particular reference in mind? Jul 1, 2018 at 8:21
• We can agree that physical axioms are assumed a priori; consider e.g. the Newtonian law of inertia: we have hardly experienced such a phenomenon. And then we derive by way of logic and math consequences from axioms. BUT, in the end, we use the derived consequences to explain known empirical facts and predicts unknown empirical facts. So, in the end, we contrast the consequences of axioms with reality: if they do not "fit", we have to discard the assumed a priori axioms as wrong. Jul 1, 2018 at 11:23
• I made an edit to help clarify the question. You are welcome to roll this back or further edit. Jul 1, 2018 at 13:55
• @Mr.Kennedy I don't think this is a duplicate. The one you reference seems to be asking if there can be any a priori laws in science. This one seems to be asking aren't they all a priori deductions? Jul 1, 2018 at 14:00

It is an interesting question. Some people who are pursuing the so called "theory of everything" have suggested that it may be unique. That is, it may be that there is only one way that such a theory could be formulated (or one collection of functionally equivalent ways, perhaps) and so there might be a way to derive everything from first principles.

The poetic way this is sometimes expressed is to wonder if God had any choice.

As yet, this has certainly not been demonstrated. Only speculated.

But there is a strong line of thought that a very small collection of observed things should be able to nail down just about everything about such a theory. You start with some such thing as the beginnings of Leibniz's ideas about monads. And you add in some interactions. And it seems that these interactions must have certain kinds of symmetry to look even vaguely like what we see. And having done that we seem to be able to get a large portion of the standard physics model. That is, certain symmetries seem required, and that's a large part of the story.

Notice the "what we see" bit there. So far, we still seem to need to look at things, somehow, a little at least, to get anywhere in physics.

• Specific references for any claims you make would strengthen your answer. Who are the "some people" you mentioned above? What works by them can you reference so others can get further information? Jul 1, 2018 at 13:42
• May you recommend a text for that subject? Jul 4, 2018 at 13:48
• Well... No specific text. Any intro text on quantum field theory should tell you that conservation of particle number requires a unitary Hamiltonian. Being unitary is a very restrictive symmetry. Similarly, any good text on Noether's theorem will tell you that a variety of symmetries are related to conservation laws. Energy to time, momentum to translation, rotation to angular momentum, gauge to charge, etc. So if you build a theory that respects all of these, and let's you see, you pretty much are forced into quantum electro dynamics.
– user34017
Jul 4, 2018 at 14:46
• As to the "did God have a choice" thing, it is supposed to be something Einstein said. But I don't have a source.
– user34017
Jul 4, 2018 at 14:52

According to the Anthropic principle, we can know some things by virtue of being able to experience them. A simple example is in the section on dimensions, showing that a Universe we can be in has three spatial and one temporal dimension. By being here, then, we know the fundamentals of spacetime. Whether this qualifies as a priori is up to the reader.

Not by modern standards. Popper's characterization of the demarcation boundary around science is now widely accepted by scientists and most philosophers of science. By that standard, something with no empirical content just isn't physics, it is math. If it is a-priori and would not be dismissed due to experience, it cannot be falsified, and it is not science.

Every theory includes the assumption that a certain part of math applies to the situation at hand, and how. If the computations do not predict the outcome, that part of the theory is wrong. For instance, the theory of relativity determined that Euclidean geometry does not apply to things moving at high speeds. That did not make the geometry wrong, because the geometry is not part of the physics. It just called for physicists to pull in higher-powered geometry to fit those cases.

Examples that basic seem to make the point that things we think are a-priori parts of physics can still always be displaced by experimental results, no matter how deeply embedded or hard to question they may to be.