Here is my annual roundup of some of the most interesting things I came across in 2021. As usual, here I’m focusing on topics that caught my eye which I haven’t already written about earlier.

Ben Eater’s Channel on Electronics

This absolute gem of a channel has been around for a few years, but I had discovered it just this year. Ben is an engineer, and makes quality, highly informative yet approachable educational videos on various electronics. His videos cover both the hardware and software sides, offering a true “end-to-end” look at how things like computers work. I find the depth and amount of details he covers in the videos to be very well balanced, showing enough to not conceal important information while avoiding tedium.

To start, I suggest his two-part series on building a video card from scratch (part 1, part 2). Using just some very basic chips, he constructs (step by step) a rudimentary video card on breadboards, capable of outputting an image on a VGA bus. If you mostly work with software in the higher levels of abstraction, these videos can help de-mystify a lot of the “magical” bottom layers of computing.

The Revolutionary Galois Theory

The discovery of intimate connections between seemingly disparate subjects of study is one of the key drivers of progress. These connections often reveal deeper levels of truth that are more fundamental than what we’ve known before. Maxwell’s equations, for example, unified magnets, electricity, and light, ushering in a new era with inventions that were previously unimaginable.

Galois theory is one such significant discovery in mathematics, where a correspondence between fields and groups were discovered. This article is fairly introductory and gives a very good overview of what the theory is along with its significance. One thing I found to be almost as intriguing as the theory itself is that Évariste Galois himself died in a duel at the age of 20, over a lover. Today no one dies from dueling anymore, let alone genius mathematicians. And as barbaric as duels were, I wonder if it does indicate a decline in people’s level of passion about things.

A Meta-Scientific Perspective on “Thinking, Fast and Slow”

Daniel Kahneman’s Thinking, Fast and Slow has been one of the most popular books since its publication in 2011. Its formulation of the two modes of thinking - dubbed system 1 and system 2 - has become so ubiquitous that people nowadays assume that it’s common knowledge. R-Index, a blog run by a psychology professor in Toronto that investigates the replicability of published studies in psychology, took a look at this influential book. And it seems that, like many other works in the field of psychology, some of the research referenced in the book turned out to be quite shaky. The primary thesis of the book does seem to be based on more solid studies, but some chapters are especially bad.

There is a related site that tracks a broader list of reversals in psychology which is quite an interesting read as well. Note that, as I’ve written before On Debunking, a failure to replicate does not prove the negation of the original result, only that we are back in square one, ignorance.

More generally, it seems that we really lack the intellectual tools to study things that don’t have exact answers, or where the exact answers lie in very high dimensional spaces (i.e. influenced by a large number of factors). The former is perhaps a fundamental limit of mathematics and rationality, while the latter is undoubtedly a limitation of our current brain power.

Bertrand’s Paradox (Part 2)

Not to be confused with Bertrand Russell’s famous paradox in set theory nor the identically named one in economics. This paradox in probability theory is quite an interesting one, and it showed an aspect of mathematics that I had not considered before. The main video goes through the original formulation of the paradox from geometry, with part 2 expanding and generalizing it further (which is a must-watch).

Broadly speaking, Bertrand’s paradox shows that statements like “choose something randomly from a collection” or “perform this action with a random outcome a number of times” are not as innocuous as they might seem, even when we are dealing with infinite samples (and assuming the axiom of choice). We are inclined to take these statements for granted, and move on to the more interesting parts of the problem. But sometimes the exact way we make the choice (or sample from the distribution) critically affects the answer to the problem, and without specifying it explicitly, the question is actually ill-formulated.