Great article. Personally I have been learning more about the mathematics of beyond-CLT scenarios (fat tails, infinite variance etc)<p>The great philosophical question is why CLT applies so universally. The article explains it well as a consequence of the averaging process.<p>Alternatively, I’ve read that natural processes tend to exhibit Gaussian behaviour because there is a tendency towards equilibrium: forces, homeostasis, central potentials and so on and this equilibrium drives the measurable into the central region.<p>For processes such as prices in financial markets, with complicated feedback loops and reflexivity (in the Soros sense) the probability mass tends to ends up in the non central region, where the CLT does not apply.
You (and others) may enjoy going down the rabbit hole of universality. Terence Tao has a nice survey article on this which might be a good place to start: <a href="https://direct.mit.edu/daed/article/141/3/23/27037/E-pluribus-unum-From-Complexity-Universality" rel="nofollow">https://direct.mit.edu/daed/article/141/3/23/27037/E-pluribu...</a>
As to ye philosophy of “why” the CLT gives you normals, my hunch is that it’s because there’s some connection between:<p>a) the CLT requires samples drawn from a distribution with finite mean and variance<p>and b) the Gaussian is the maximum entropy distribution for a particular mean and variance<p>I’d be curious about what happens if you starting making assumptions about higher order moments in the distro
Hot take: bell curves are everywhere exactly <i>because</i> the math is simple.<p>The causal chain is: the math is simple -> teachers teach simple things -> students learn what they're taught -> we see the world in terms of concepts we've learned.<p>The central limit theorem generalizes beyond simple math to hard math:
Levy alpha stable distributions when variance is not finite, the Fisher-Tippett-Gnedenko theorem and Gumbel/Fréchet/Weibull distributions regarding extreme values. Those curves are also everwhere, but we don't see them because we weren't taught them because the math is tough.
I hate Quanta a lot<p>a vast amount of fluff for less than a college statistics professor would (hopefully) be able to impart with a chalkboard in 10 minutes, when Quanta has the ability to prepare animated diagrams like 3Blue1Brown but chooses not to use it<p>they could go down myriad paths, like how it provides that random walks on square lattices are asymptotically isotropic, or give any other simple easy-to-understand applications (like getting an asymptotic on the expected # of rolls of an n-sided die before the first reoccurring face) or explain what a normal distribution is, but they only want to tell a story to convey a feeling<p>they are a blight upon this world for not using their opportunity to further public engagement in a meaningful way
Quanta is true trash. Remarkably a reader can walk away knowing <i>less</i> about a topic than they did before reading their article!
A lot of times on HN when a math topic comes up that isn't <i>about</i> 3b1b, someone will jump in to say "this isn't as good as 3b1b". Last time I saw that, I was moved to comment:<p><a href="https://news.ycombinator.com/item?id=45800657">https://news.ycombinator.com/item?id=45800657</a><p>3b1b doesn't have the same goal as Quanta, or as introductory guides. It's actually not that great a teaching tool (it's truly great at what it is for, which is (a) appreciation and motivation, and (b) allowing people to signal how smart they are on message board threads by talking about how much people would get out of watching 3b1b).<p>This is prose writing about math. It's something you're meant to read for enjoyment. If you don't enjoy it, fine; I don't enjoy cowboy fiction. So I don't read it. I don't so much look for opportunities to yell at how much I hate "The Ballad of Easy Breezy".
I don’t fault Quanta (or 3b1b) for being the way they are. Each is serving their goal audience pretty well.<p>My compliant is only that there should be a dozen more just like them, each competing with each other for the best, most engaging math and science content. This would allow for more a broader audience skillevel to be reached.<p>As it stands, we’re lucky even to have Quanta and 3b1b.<p>I think there is hope though, quite a few new-ish creators on YouTube are following in Grant’s footsteps and producing very technically detailed and informative content at similar quality levels.