You can't understand the why of the normal distribution without Fourier analysis though - which is pretty heavy going for anyone who's not hardcore science or engineering.
Can't you introduce the normal distribution as the limit of the binomial distribution? I think you can prove the central limit theorem without using terribly advanced math.
That's only the most limited version of the theorem which has since been renamed the de Moivre-Laplace theorem [1]. The rabbit hole goes much deeper when you talk about the most general form which works for any set of independent and identically distributed random variables, not just binomial random variables.
That’s moving the goalposts. The original claim was about getting a complete picture. A full understanding of the “why” of statistics going all the way to the bottom.
The only proof of the central limit theorem I know of relies on it. Iterated convolutions tend toward a Gaussian. Essentially you take the Fourier transform of any nice probability distribution, do a Taylor expansion, and Bob's your uncle.
Another child poster pointed out that in special cases (like the limiting case of the binomial distribution) you can get away with other arguments. And you can certainly 'prove' it via simulation. So maybe you don't need the full generality of Fourier analysis in the end.
Probably a semantic difference that I don't equate proof with understanding.
I would call that a colloquial definition of understanding. Very different from mathematical understanding.
If you ask a mathematician what you’d need to know to understand Fermat’s last theorem, he won’t say “high school pre-algebra.” That’s only enough for you to understand the basic statement of the theorem. It doesn’t get you to the why. To understand the problem involves a deep dive into both algebraic number theory and analytic number theory.
On the contrary, I use "understand" quite precisely. It's interesting to study what practitioners of various disciplines mean by "I understand."
For example, most mathematicians would say they understand large numbers of theorems that they cannot prove off the top of their heads. On the other hand they probably have what Borovik in "Mathematics under the microscope" called a "recovery procedure." That is, a set of constraints or path that reproduces the result. They know they can reconstruct the proof if they need to from the various gambits and skills they keep polished.
Also, the proof via the normal distribution being an attractive fixpoint of convolution is fine, but it only works on a particular subset of functions. We know the theorem applies beyond that subset, and there's a cottage industry of extending it in bits and pieces and calculating better convergence bounds. There is no proof available today that says central limit theorem applies iff conditions x, y, and z. So in this case the proof really can't be said to be understanding.
Now, that's well and good for probability alone, which is a field of mathematics. Statistics isn't a subfield of math, or is a subfield of math the way physics is. For a statistician, understanding the central limit theorem is much more about knowing what kind of observations it is reasonable to expect it to approximately apply to, what kind of tests rely on it and which don't, how to check if it applies in a rigorous way, what kind of visualizations and exploratory data analysis is enabled if it does, how the normal distribution and convergence to it fits into a whole family of distributions and features thereof...