Let $f(x)$, $f \in \mathrm{C}^{\infty}$, be the following infinitely continuously differentiable function over the space of real numbers:
\[
f(x) \doteq
\sum_{n=0}^{\infty} \frac{e^{-2} \, 2^n}{n!}
+ \frac{1}{\sqrt{2 \, \pi}}\int_{-\infty}^{+ \infty} x \, \exp(-y^2/2) \, dy;
\]
then, applying Taylor's theorem and the Newton–Leibniz axiom,
f(x) \doteq
\sum_{n=0}^{\infty} \frac{e^{-2} \, 2^n}{n!}
+ \frac{1}{\sqrt{2 \, \pi}}\int_{-\infty}^{+ \infty} x \, \exp(-y^2/2) \, dy;
\]
then, applying Taylor's theorem and the Newton–Leibniz axiom,
\[f(1) = 2.\]
Time out! What the Heck?!?!
Okay. Breathe.
Let's restate the above in non-pompous terms.
Let $f(x)$ be the following function
\[f(x) = 1 + x\]
then $f(1) = 2$.
All the words between "following" and "function" in the first paragraph mean "smooth," which this function certainly is; $f \in \mathrm{C}^{\infty}$ is the formal way to say all the words in that sentence, so it's redundant.
As for the complicated formula, it uses a series and an integral that each compute to one. Eagle-eyed readers will notice that the first is the Taylor series expansion of $e^2$ times the constant $e^{-2}$ and the second is $x$ times the integral of the p.d.f. for the Normal distribution for $y$, which by definition of a probability has to integrate to 1. Taylor's theorem and Newton–Leibniz axiom are used to get the values for the series and the integral from first principles, as is done in first-year mathematical analysis classes, and which no one would ever use in a practical calculation.
I took a trivially simple function and turned it into a complicated, nay, scary formula. With infinite sums, integrals, and theorems. Taylor is relatively unknown, but Newton and Leibniz? Didn't they invent calculus? (Yes.) So my nonsensical formula acquires immense gravitas. Newton! And Leibniz!!
And that's the problem with an increasing number of public intellectuals and technical material.
There are some genuinely complex things out there, and to even understand the problems in some of these complex things one needs serious grounding in the tools of the field. There's no question about that. But there's a lot of deliberate obfuscation of the clear and unnecessary complexification of the simple.
Why? And what can we do about it?
Why does this happen? Because, sadly, it works: many audiences incorrectly judge the competence of a speaker or writer by how hard it is to follow their logic. And many speakers and writers thus create a simulacrum of expertise by using jargon, dropping obscure references and provisos into the text, and avoiding simple, clear examples in favor of complex and hard-to-follow, "rich," examples.
What can we do about it? This is a systemic problem, so individual action will not solve it. But there's one thing we each can do: starve the pompous of the attention and recognition they so crave. In other words, and in a less pompous phrasing, when we realize someone is purposefully obfuscating the clear and complexifying the simple, we can stop paying attention to them.
Simplicity actually requires more competence than haphazard complexity; it requires the ability to separate what is essential from what's ancillary. To make things, as Einstein said, as simple as possible, but no simpler.
It's also a good thinking tool for general use. Feynman describes how he used to follow complicated topological proofs by thinking of balls, with hair growing on them, and changing colors:
As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball)—disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!”
If it’s true, they get all excited, and I let them go on for a while. Then I point out my counterexample.
“Oh. We forgot to tell you that it’s Class 2 Hausdorff homomorphic.”
“Well, then,” I say, “It’s trivial! It’s trivial!” By that time I know which way it goes, even though I don’t know what Hausdorff homomorphic means.
Excerpt From: Richard Feynman, “Surely You’re Joking, Mr. Feynman: Adventures of a Curious Character.”
Let's strive to be like Einstein and Feynman.
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This post was inspired by an old paper that starts with $1+1=2$ and ends with a multi-line formula, but I've lost the reference; it might have been in the igNobel prizes collection.
This post was inspired by an old paper that starts with $1+1=2$ and ends with a multi-line formula, but I've lost the reference; it might have been in the igNobel prizes collection.
