Would a Theory of Everything allow us to calculate the constants of physics? I have heard several times in the past that a Theory of Everything would allow us to calculate (as opposed to measure) the fundamental dimensionless physical constants.
What I mean is, e.g., the fine structure constant would just be an algebraic expression containing no unknowns. In other words, we would be able to calculate the constant's value as precisely as we wish, just like we calculate the value of the mathematical constant $\pi$, for example.
Is my understanding correct?
 A: Would a theory of everything allow us to calculate any constant?
Yes. A theory of everything has to be completely self-contained, so anything it predicts must be derivable from a set of "axioms".
In practice, this could mean any variety of things. You could have an infinite series to calculate $\hbar$. Or, you could just start off by defining $\hbar$. When you get down to the axiomatic level, the only requirement is that the constants be deduced from the axioms. Thus, $\hbar$ could be defined, while $\alpha$ could be derivable from knowing the "true" value of $\hbar$.
I think the real question that you're trying to ask, is whether or not all the constants we need can be related to each other with a finite number of axioms, or constant definitions. I'd say theoretically yes, but practically no. Although we have an idea of what fundamental constants would be candidates to be axiomized, we still don't know if those constants are the best picks.
Here's a thought experiment. Assume we have a TOE. Assume now that we can ask any physical question and have it answered precisely. I'd ask how many fundamental/axiomatic constants are there? If there are a finite number, then I should have numerical constant that counts how many fundamental constants there are. However, there should be an infinite number of regular constants. This is because after asking my question, a new constant was created to "count" the other constants. Thus, there should also be a constant that counts the counting constant!
