Are the fundamental constants of nature independent?

Background: I am a layman and I just watched the video "Infinite Worlds: A Journey through Parallel Universes" (http://www.youtube.com/watch?v=OO4uzgiRHkE) featuring a discussion with Andrei Linde, Brian Greene and others. This got me somewhat confused.

At one point they are talking about the "fine-tuning" problem, and it was said that the "huge" number of solutions of string theory (about $10^{500}$) which all supposedly give rise to different laws of nature does explain fine-tuning (applying the anthropic principle of course).

Now, while $10^{500}$ may seem like a huge number, it is finite: There is only a discrete finite set of possible laws of nature, including fundamental constants. So my first question is: Why is it finite and so small (from a combinatorial perspective)(can be compressed into 250 Bytes of data)? Why isn't it infinite? Why are there no real valued free parameters creeping in somewhere?

I always thought one could "choose" the fundamental constants of physics quite freely and independently from one another. So for example take the fine-structure constant $\alpha^{-1}$. Its value is approx. 137 in this universe. Is it possible that in another universe the value is 274 while all the other constants (and all laws of nature) are exactly the same as in our universe?

If so, it should be also possible to have other values of $\alpha^{-1}$ say in the interval $[137, 274]$, but there can only be a finite set of allowed values for $\alpha^{-1}$ because the number of different string vacua is finite. To realize different values for $\alpha^{-1}$ we need to choose different ST solutions out of a finite set of about $10^{500}$.

But then there are many fundamental constants of physics, some Susy-Theories have even hundreds. To vary any of these numbers, a distinct solution of ST is needed out of just $10^{500}$. Andrei Linde confirms this in the video at 1:22:00 with the cosmological constant: To fix just this one constant we need to use up $10^{120}$ or more solutions.

Is this correct so far, or am I missing something?

Now my conclusion: If a ST solution encodes what forces of nature there are in a universe and at the same time fixes the values of all the constants governing these forces, those constants cannot possibly all be "chosen" independently, by a counting argument: The number of possible combinations grows exponentially with the number of constants and would obviously far exceed $10^{500}$.

So it seems that ST implies that there must exist subtle dependencies between the constants of classical physics that (at least in principle) could be computed from the theory. Is this correct? But can this really be the case? It seems strange: If there are such strong constraints on what is possible and there is "only" a total of $10^{500}$ possibilities, it still looks like a small wonder that at least one of them admits life...

Can you point me to texts about these implications of the string theory landscape?

• The fine tuning problem is simply a signature of a theory breaking down at the edges. All it means is that Brian Greene and colleagues have to work harder AND smarter. – CuriousOne May 4 '15 at 23:02
• See NIST: "Thus α depends upon the energy at which it is measured, increasing with increasing energy". The fine-structure constant is a running constant. So it isn't constant. Then see this about the speed of light, and note that the cosmological constant is "the energy density of the vacuum of space", which varies in a gravitational field. Remember this next time you hear somebody waxing lyrical about the Goldilocks multiverse. – John Duffield May 5 '15 at 20:28