do atoms actually exist as they are depicted at times in the scientific models?
I ask because I read CS Lewis saying that when Jeans or Edington want to explain the atom, they use a picture, but they don't actually believe the picture, they believe a mathematical equation.
Atoms exist as sums of field interactions.
The atom is, in fact, a combination of principle mathematical equation describing the nature. For example, even in elementary quantum chemistry, the probability that an electron exist in a certain space is based on Legendre functions, Bessel functions, and spherical harmonics. This probability space, or electron cloud makes up the perceived "hard" positioning of the electrons so that we can draw a Lewis structure, Bohr atomic model, or even a quantum model of an atom. But, in reality even the electron itself isn't a hard "dot" of charge, but rather a sum of interactions in fields creating the effect of charge - specifically charge with mass, and energy of the electron.
In field theory, a sum of a specific type of principle field can describe the equations of motion for the system. This is usually done by another mathematical equation called the Lagrangian. For example, in field theory the most basic description for a fermionic field permeating space in
d dimensions is free particle model (like free electron). It is a more mathematically rigorous equation that describes more details of the particle interaction than the Sommerfeld model for free electron.
For something like mass, field theory uses the mass potential as the field descriptor - so that one can write the Lagrangian for the mass field.
For something like electro-magnetic potential, there are special experimental terms that describe the perturbations of the potential on surrounding fields. One such Lagrangian perturbation is the Yukawa potential (like Coulomb potentials.) We use mathematical rigor (even more equations) to determine how this potential effects the entire Lagrangian system (equation) in
d dimensions.
So, to describe an electron's motion as a free particle (similar to metallic bonding, or an ionized free radical,) we could write down one single equation - the total Lagrangian. For electrons we would include the terms for potential, mass and fermionic fields - as well as a base free electron term.
So, then since protons are also fermions, slightly more massive, and have similar charge, a mathematical (Lagrangian) equation for protons since they exist as fermionic fields.
We can do tell similar with neutrons.
And, for atoms with more than one of each subatomic particle, we can sum up the interactions for each respective field term.
There are also other boson fields interactions, and they describe scalar/longitudinal fields - which are associated with quantum fluctuations in the vacuum energies, for example. This also has a term, and can be added to the total mathematical equation describing fundamental partial motion (total Lagrangian.)
Now how does this relate to the OP? Field theory describes the existence of these "particles" as interactions between fields producing what ultimately seems like a "hard" particle. This is actually Newton's law in action on a quantum scale - that the sum of the forces applied
can be, or exists up to an equal force in the opposite direction. Since the Lagrangian field equation describes the sum of the equations of motion of fundamental particles, we can use it to find the forces involved. We see that what we perceive as "hard" is a system of field interactions that have an energy capable of resisting the forces of other field interactions.
So, you are a sum of mathematical equations describing the motion of the complex field interaction system you call your body. You are hard, or soft, in some places due to a myriad of differences described by mathematical equations (e.g. the Lennord-Jones potentials, fermionic fields, mass, etc.)
We have the idea of a "hard, discrete solid" for convenience.