# Theories, Corollaries, and Models

I apologize if this question seems overly basic. I was wondering how to recognize what a theory is really saying, as opposed to the explanation/corollaries that are drawn from it.

As an example, take Einstein's theory of special relativity. Would the theory be that inertial reference frames are indistinguishable and the speed of light is constant? Would the theory be that the coordinates of an event can be transformed from one system to another using the Lorentz transformation? Would the theory be a statement of time dilation and length contraction? I apologize if this question is vague. I am just wondering if there is any way to say that "this is the theory" and from "this", I can derive everything else.

Another, rather basic question, I have is on the relationship between theories and models. To my understanding, models are tools used to understand/visualize theories. I was wondering how physicists go about creating such accurate models. For instance, the atomic model of the universe is extremely powerful and is consistent with all of the physical theories I can think of. In fact, it seems that this model of the universe can be used to predict phenomena that are not immediately obvious from the theoretical equations. I was wondering how it is possible for us to create such accurate models. Have there been instances where we have come up with invalid models for valid theories? Have there been instances where we have come up with valid models whose underlying theories were found to be invalid?

• You appear to be trying to understand physics from a mathematics perspective. Nov 20, 2018 at 18:46
• Related: physics.stackexchange.com/q/87239/2451 and links therein. Nov 20, 2018 at 19:04
• FYI, Einstein didn't predict that the speed of light would be a universal constant. That was predicted by Maxwell, and Einstein took it as a postulate. Nov 20, 2018 at 19:05

Like in math, you can have the same theory starting with different axioms, so the choice of axioms is a matter of taste. Once you define the axioms you can demonstrate the corolaries, but axioms and corolaries can be exchanged in many circumstances.

A model is an application of the theory to a specific circumstance. Say you want to model a star, then you use the theory in addition to specific circumstances, such as assuming an equilibrium situation, specific mass and materials, making some approximations, etc, at the level of detail that you want (or can). And yes you can have a good model even if the theory is later shown not to be exact, and viceverza, a correct (that is, currently accepted) theory and your model does snot work because you included the wrong assumptions and simplifications.

• Thank you for the response! In regards to your first paragraph, do you mean that, as long as the different axioms are consistent, they can each be used?
– dts
Nov 20, 2018 at 18:58
• Yes, in fact it is not unusual to find different axiomatizations of a theory
– user65081
Nov 20, 2018 at 18:59
• Thank you! Sorry, just one more question. Let's say I was to solve a special relativity problem (as an example) using two different sets of axioms. If I were to arrive at two different solutions, how would I know which one is "correct"?
– dts
Nov 20, 2018 at 19:00
• No, if the theories are equivalent you will get the same solution. Otherwise they are different theories, and experiment will determine which one is correct, if any
– user65081
Nov 20, 2018 at 19:01
• @dts There are two criteria for a given set of axioms to be used. First, the set must be logically consistent. Second, the axioms themselves must be consistent with our observations of nature. We are constantly checking both our theories/models and our axioms to make sure that they are not disproven by increasingly precise measurements. If an axiom fails to be consistent with observations, then the theories that depend on that axiom need to be modified (or their applicability needs to be limited). If a theory fails, then either there was a mistake in the reasoning or an axiom has failed. Nov 20, 2018 at 19:09