What makes a constant of nature fundamental? I came across a video on youtube in which Alexander Unzicker argues that in order to have a scientific revolution, one of the constants of nature should be eliminated(at 23:00). He gives the example of Electromagnetism where by the equation
$\epsilon_0\mu_0 = \dfrac{1}{c^2}$
one can eliminate $\mu_0$. According to him, since we can relate 3 constant to each other we can eliminate one of them.
He also says in Newton's theory we eliminate the constant $g$ with equation
$g = \dfrac{GM}{r^2} $
Finally he says boltzmann constant was eliminated by relating kinetic energy to temperature by
$\frac{1}{2}mv^2 = \frac{3}{2}kT $
So, this video made me think: What makes a physical constant truly fundamental? Does relating one constant to another means that we can eliminate one of the constants? For example is fine structure constant $\alpha = \frac{k e^2}{\hbar c}$ not  fundamental? On the other hand one can argue that dimensionless constants are more fundamental.
And I don't understand anything from boltzmann example, I think there is a huge problem in it, or am I missing something?
 A: 
What makes a physical constant truly fundamental?  Does relating one
constant to another means that we can eliminate one of the constants?

First of all keep in mind that everything we, humans, describe is all based on our interpretation of the universe and we try to formulate it is such a way that we can best understand and describe how the universe works. In other words allow us to predict as exactly as possible what should happen given a set of initial conditions.
A fundamental constant is described as a value that is constant over time and space. A better term is universal constant. Basically if you set up your experiment correctly you should always measure the same value, now, in the past, in the future in any location of the universe. (as long as you take space-time warping into account in your experiment/calculations)
However not everyone finds this a satisfying or good description of what a fundamental constant should be. Some say that a fundamental constant should not be able to be expressed by other fundamental constants. In other words, every fundamental constant should only be possible to obtain by measuring it, even if you know all other constants. All other things should then be possible to calculate from fundamental constants and a complete theory of how the universe works.
The problem with that is, how do you decide which constant is fundamental or not. let's look at the case of $\epsilon_0\cdot\mu_0=1/c^2$, how do you determine which two would be fundamental and which one would not be?
Objectively there isn't much of a difference between the three, so the choice would be rather arbitraty. Therefor that description is unlikely to be adopted any time soon.
So what is a fundamental physical constant or in better words an universal constant?: 
A constant that is the same in the entire universe that doesn't change over time.
If these constants would be different then the universe would look completely different and it's highly likely life would be impossible, our universe/constants are therefor sometimes also described as a goldi-lock universe, which is an often used argument for people who believe in multi-universe theories.
Also keep in mind that just because an "expert" in a field says something, doesn't mean it's true. Experts also often disagree, especially the more complex the subject is. He's also talking a lot of "empty air" and mostly talking extremely philosophical and proposing HIS view/theory.
Just take a look at all the comments below already and how many relative downvotes he has 108/630 (20-7-2020), which is 17% of the people downvoted the video. However only a very specific target group will even watch this video, indicating that many of the people that watch it (strongly) disagree with him.
It feels to me more that he is proposing if you know equation X, then you won't need to know constant Y. However that's a bit ambiguous because that also works the other way around then. Basically he's bashing many physicists and saying that they are focussing on the wrong things. So take what he says and how he says it with an open mindset but questioning.
A: There is a long history of experimental measurements of the speed of light, starting back in 1726 with John Bradley which measured it to be 300,000 m/s! It took many many iterations in measurement before the value settled to precisely 299,792,458 m/s with laser instrument measurement in 1979. In the relation that associates c with Ɛ0 and µ0 , those two putative constants have undergone rather arbitrary adjustments in their values over time in order to keep a match with the measured value of c thru the established relation. While c is a directly measured experimental constant, Ɛ0 and µ0 are arbitrary. They are established to fit c. Their existence, as permittivity and permeability of the vacuum, is motivated by Maxwell Theory of Electromagnetism which stipulates that light is an electromagnetic radiation similar to radiation spread around a rod with alternating current flow, with the ability to even travel thru the vacuum. Many people in physics wrongly believe that Ɛ0 and µ0 characterize the vacuum in its ontology. Some even believe that they pop out of Maxwell equations. They absolutely do not. So if it is the desire of any physicist to eliminate Ɛ0 and µ0, I guess they can since those are conventional quantities. But to suggest to eliminate c because one wrongly understood that the existence of this constant is dependent on Ɛ0 and µ0, and to keep one or the both of these two, would be oblivious.
There is no one in fundamental physics advocating eliminating experimental constants, to my knowledge, without clear understanding of their correlations between one another ALTOGETHER. But it is long consecrated practice in dimensional analysis to give them the value of 1 in the equations  in order to have deeper visibility in those formulae and the meaning in their association of variables.
A: It is important for everyone, including Unzicker, to keep in mind that Ɛ0 and µ0 are not true empirical constants but conventional constants. Their values have been arbitrarily chosen in order to match the current measured value of the speed of light constant, a true experimental constant, according to the mentioned relation. Ɛ0 and µ0 do not say anything at all about the structure of spatial vacuum. They only exist to satisfy Maxwell Theory of Electromagnetism.
Now the idea of eliminating an experimental constant because it is apparently a function of another constant is not sound at all, I am afraid. It is only because of the insufficiencies of our physics that we define or  more precisely express a constant by relating it to other constants in certain cases. An experimental constant, whether dimensional or dimensionless, is a legitimate, true and inalienable quantity that exists on its own in nature in its very essence. It tells us how physical elements in the natural world intimately correlates with one another. It is not up to us to eliminate them in our system of interpretation, aka physics. We know that there is something serious, even crucial about them, in terms of what they are telling us about the structure and organization of the natural world. That is why modern physics is in a quest to infer those quantities and their correlations from a single towering system of interpretation, which when achieved will unmistakably be the Unified Theory of Physical Law long sought.
So no,  relating one constant to another does not means that we can or should eliminate one of the constants. What we need is a Theory that can explain and derive them all from stringent mathematical physics logic.
