Are integers unphysical/unnatural?
When I first looked into number theory, I was surprised about why mathematicians are so obsessed about integers, primes and so on. As far as I understand, (almost) all integers in the real world arise as an approximate description of composite dynamics.
I think anyone here understands why individual macroscopic objects, like rocks, buildings, etc. are approximations that we can count, so I'll dive directly into quantum world.
The obvious argument about where integers could naturally arise is a quantum harmonic oscillator, which dynamics results in the energy states that we can count with integers: $$E=(n+1/2)\hbar\omega$$ Or the energy states of atoms in the same way.
However, there are no free quantum harmonic oscillators in reals life! In real life quantum fields are interacting, and discrete energy states are simply an approximation. To see this clearly consider quantum particle. We can count individual particles, right?
No! If we take full interacting theory, the particle states polarise the vacuum and, if we start looking closer (at higher energies), we will start seeing that there is no individual particles and what we see from far away as a single particle actually is some kind of a superposition of multiparticles states. (We actually don't really know what "particles" are. Abstractly this is captured in the <PERSON>'s theorem, stating that the state space of an interacting QFT is not a Fock space.) Moreover such state will never be localized.
Looking at the world from the perspective of interacting QFT allows us to find similar arguments for any system where integers arise. Hence, integers are approximations and not natural.
The ONLY place, where I see integers as fundamental building blocks are the dimensions of symmetry groups of our world. The integer dimensions of the <PERSON> group result in the 4-dimensional spacetime, the integer dimensions of the SM groups $SU(3)\times SU(2)\times U(1)$ result in the integer number of different particle types.
Update 1: A very good point were made about the charge. If we measure the number of elementary charges in a given area, it will always be an integer number, even though the number of particles is not conserved (in QFT). 298
It means, that any physical state is an eigenstate of a charge operator with an integer eigenvalue. Consider the simplest case, where the minimal set of mutually commuting operators consist of the Hamiltonian, H, and the charge operator, Q. Then we can label our states in terms of their eigenvalues $|E,q\rangle$. However, are there any requirement, that $q$ is only an integer?
Update 2: I think, I still need to elaborate on macroscopic objects to stop discussing the distribution of windows =) Consider, say, rocks, fingers or even particles. You can count them, for example, by counting the number of distinct maxima in their density distribution. This counting is an approximation. First problem is that we need to agree what exactly will count for a single object. Second problem is that these objects are not stable in time: stones will break down and the probability distribution of a particle will become highly delocalized. The next problem is that if you zoom in, you can loose your definition of a single particle: due to vacuum polarizations any electron is surrounded by a cloud of other particles. I'm ok with integers arising from approximate description, they are just not fundamental to the theory.
Update 3: The question was temporary closed as being opinion-based. So here are two questions that can, in principle, be answered with facts and citations:
1) Following "Update 2", are there any mechanisms that relate the fact that the symmetry space, say $U(1)$, has integer dimensions and the fact that all the physical state are eigenvalues of the generator of $U(1)$ with integer eigenvalues? Simply put, why the charged is quantized and quantized evenly?
2) Is there a way, to construct a symmetry group, such that it doesn't have an integer number of dimensions? Such that the number of generators is not an integer number (whatever that means =)) 976