Are constants from natural science of any importance at all in mathematics as real numbers?

Question

In mathematics, people have discovered constants that are proven to be of great importance mathematically. For example,

1. Archimedes' constant, which is approximately $3.14159265$.
2. Euler's number, which is approximately $2.718281828$.
3. Pythagoras' constant, which is precisely $\sqrt{2}$.
4. The golden ratio, which is precisely $\frac{1+\sqrt{5}}{2}$.
5. The Euler–Mascheroni constant, which is approximately $0.57721566$.

In science, people have also discovered and defined contants, such as

1. Avogadro constant, which will be precisely $6.02214076\times 10^{-23}$ starting 20 May 2019.
2. Gravitational constant, which is approximately $6.674\times10^{−11}$. (Can it be proven to be an irrational number?)
3. Planck constant, which is approximately $6.62607015\times10^{−34}$. (Can it be proven to be an irrational number?)

and many others. From a pure mathematical point of view, are those scientific constants be of any importance in mathematics at all? After all, scientific constants appears naturally in nature and "mathematical reasoning can provide insight or predictions about nature".

Answer 1

Most physical constants such as your examples depend on your unit system. Therefore there is no reason to expect that their values in the SI system have any special meaning or are of any mathematical interest. In particular, for each of these constants, I can define a unit system such that they have a rational or integer value.

However, the relations of some of these constants to each other can yield dimensionless physical constants that are independent of your unit system and where it is at least conceivable that you can can compute their value by purely mathematical means. However, so far, nobody has succeeded with this.

As an example, let’s look at the most prominent example for such a constant, namely the fine-structure constant. It stands out because:

• It is very fundamental as opposed to other constants such as the mass ratios of elementary particles, which boil down to complex interactions and more fundamental constants.
• We can measure it with a sufficiently high precision (11 digits) such that it is an actual challenge to find a numeric match.

If you can find a simple mathematical recipe to arrive at this constant, this would be noteworthy – even more so, if this correctly predicts future, more accurate measurements of the constant’s value. However, finding this is only the first step to coming up with a physical understanding of its value.

Mind the simple though. Nobody will be impressed by something that is obviously the result of a brute-force approach.

Sidenote: Near Misses

There are some types of dimensionless constants with a physical meaning, which are probably not what you want:

• The Madelung constants describes the electric forces experienced by an atom in an infinite crystal lattice (where all atoms are idealised as point charges). Since their definition employs a mathematical idealisation, they are purely mathematical objects and can be investigated as such. (Of course, it is conceivable that with sufficient physical knowledge, the fine-structure constant and others are of this type.)

• The row sizes of the periodic table as well as the magic numbers of nuclear physics are integer numbers that can be explained by models from atomic or nuclear physics. If you wish, you can ignore the physics and treat these as purely mathematical objects