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Do physicists ever expect to be able to derive the fundamental constants of nature from theory? For example, if string theory or some other theory unites the four forces, would the theory be considered complete if it relies on these measured constants, or would a true theory of everything (TOE) require that these constants come out of the theory itself?

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Only dimensionless values are interesting (you may always choose a unit system, with $G=\hbar=c=1$). For instance, standard model coupling constants are supposed to converge, at some value, at some energy scale, (in a supersymmetric theory). So, this (common) value should be calculated from a fundamental theory. – Trimok Nov 6 '13 at 10:17

A physical theory needs many experimental inputs - form of equations and constants in them. A theory claiming to be able to calculate all constants in its equations is mathematics, not physics. A recent take on it is presented by S. Weinberg here.

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Is there any motivation to reduce the number of constants needed in a theory. For instance, why would there need to be more than 1 fundamental constant? – Chris L. Nov 5 '13 at 21:58
We use equations and boundary conditions. The latter are simplified solutions of equations in some limiting cases. These boundary conditions vary from one system to another, so do equations. Factually physics deal with compound unique systems so each system has its own constants, roughly speaking. TOE is not reachable for a human mind (too many equations and constants). – Vladimir Kalitvianski Nov 5 '13 at 22:03
But if a theory of anything can't derive, say, the Gravitational constant, and this constant turns out not to actually be constant in time, wouldn't there be a drive in the physics community to understand why it changes (or why it doesn't)? – Chris L. Nov 5 '13 at 22:07
@user1708: Some constants are approximate solutions (closure relationships) obtained form a more complicated theory. – Vladimir Kalitvianski Nov 5 '13 at 22:17
@user1708: heat a spring up and watch its spring constant change :) – lionelbrits Nov 13 '13 at 14:56

In this case, opinions diverge.

Some people claim that only two constants are needed: the speed of light $c$ and the string length $\lambda_s$.

Other think that you also need $h$.

You might want to have a look at:

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The following constants might vanish (e.g., place-dependent behaviors along a dimension, built-in identifiers that assign access along a string [ push-forward ], carrier-entropy) or transform--as in a qubit (a decision) flinching characteristically from dimensional memory, where the amount of the flinch is just the carriernormal length--when the initial interior entropy & the final exterior curvature are equivalent: (1) the inertia (here to be the expected mass of the up quark; (2) the fundamental curvature=1/the Planck length; (3) dV sub E,G (electrical, gravitational work rate beyond the exterior mass on the Total Universe (the canonical sum of all current universe's [ tensor and/or Higgs fields? ]); (4) -1/qubit which approaches the inertia at our current Universe observable edge; (5) a Total Count times the carrierhole coupling constant such that all events approach all decisions (the current Universe is an event; and (6) the 8 lower and 8 upper dimensions but still normalized to 11?

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